Slope Filtrations Revisited

Kiran S. Kedlaya^{1}

Received: July 24, 2005

Communicated by Peter Schneider

Abstract. We give a “second generation” exposition of the slope filtration theorem for modules with Frobenius action over the Robba ring, providing a number of simplifications in the arguments. Some of these are inspired by parallel work of Hartl and Pink, which points out some analogies with the formalism of stable vector bundles.

2000 Mathematics Subject Classification: 14F30.

Contents

1 Introduction 448 1.1 The slope filtration theorem . . . . . . . . . . . . . . . . . . . . 449 1.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449 1.3 Purpose of the paper . . . . . . . . . . . . . . . . . . . . . . . . 450 1.4 Structure of the paper . . . . . . . . . . . . . . . . . . . . . . . 450

2 The basic rings 451 2.1 Witt rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451 2.2 Cohen rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 2.3 Relation to the Robba ring . . . . . . . . . . . . . . . . . . . . 457 2.4 Analytic rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460 2.5 Reality checks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462 2.6 Principality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466 2.7 Matrix approximations and factorizations . . . . . . . . . . . . 469 2.8 Vector bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . 472 2.9 The Bézout property . . . . . . . . . . . . . . . . . . . . . . . . 474 3 σ-modules 477

1Thanks to Olivier Brinon for providing detailed feedback on [19, Chapter 3], to Francesco Baldassarri and Frans Oort for some suggestions on the exposition of this paper, to Mark Kisin for suggesting the statement of Theorem 6.3.3, and to Bruno Chiarellotto for pointing out an error in a prior version of Proposition 3.5.4. Thanks also to Baldassarri and to Pierre Berthelot for organizing two workshops on F -isocrystals and rigid cohomology in December 2004 and June 2005, which helped spark the aforementioned discussions. The author was supported by NSF grant number DMS-0400727.

Documenta Mathematica 10 (2005) 447–525

3.1 σ-modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478 3.2 Restriction of Frobenius . . . . . . . . . . . . . . . . . . . . . . 479 3.3 σ-modules of rank 1 . . . . . . . . . . . . . . . . . . . . . . . . 480 3.4 Stability and semistability . . . . . . . . . . . . . . . . . . . . . 483 3.5 Harder-Narasimhan filtrations . . . . . . . . . . . . . . . . . . . 485 3.6 Descending subobjects . . . . . . . . . . . . . . . . . . . . . . . 486

4 Slope filtrations of σ-modules 487 4.1 Standard σ-modules . . . . . . . . . . . . . . . . . . . . . . . . 487 4.2 Existence of eigenvectors . . . . . . . . . . . . . . . . . . . . . . 488 4.3 More eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . 492 4.4 Existence of standard submodules . . . . . . . . . . . . . . . . 495 4.5 Dieudonné-Manin decompositions . . . . . . . . . . . . . . . . . 497 4.6 The calculus of slopes . . . . . . . . . . . . . . . . . . . . . . . 500 4.7 Splitting exact sequences . . . . . . . . . . . . . . . . . . . . . . 501

5 Generic and special slope filtrations 502 5.1 Interlude: lattices . . . . . . . . . . . . . . . . . . . . . . . . . . 503 5.2 The generic HN-filtration . . . . . . . . . . . . . . . . . . . . . 504 5.3 Descending the generic HN-filtration . . . . . . . . . . . . . . . 505 5.4 de Jong’s reverse filtration . . . . . . . . . . . . . . . . . . . . . 506 5.5 Comparison of polygons . . . . . . . . . . . . . . . . . . . . . . 508

6 Descents 510 6.1 A matrix lemma . . . . . . . . . . . . . . . . . . . . . . . . . . 510 6.2 Good models of σ-modules . . . . . . . . . . . . . . . . . . . . 512 6.3 Isoclinic σ-modules . . . . . . . . . . . . . . . . . . . . . . . . . 513 6.4 Descent of the HN-filtration . . . . . . . . . . . . . . . . . . . . 514

7 Complements 515 7.1 Differentials and the slope filtration . . . . . . . . . . . . . . . . 515 7.2 The p-adic local monodromy theorem . . . . . . . . . . . . . . 517 7.3 Generic versus special revisited . . . . . . . . . . . . . . . . . . 519 7.4 Splitting exact sequences (again) . . . . . . . . . . . . . . . . . 519 7.5 Full faithfulness . . . . . . . . . . . . . . . . . . . . . . . . . . . 522

1 Introduction

This paper revisits the slope filtration theorem given by the author in [19]. Its main purpose is expository: it provides a simplified and clarified presentation of the theory of slope filtrations over rings of Robba type. In the process, we generalize the theorem in a fashion useful for certain applications, such as the semistable reduction problem for overconvergent F -isocrystals [24].

In the remainder of this introduction, we briefly describe the theorem and some applications, then say a bit more about the nature and structure of this particular paper.

1.1 The slope filtration theorem

The Dieudonné-Manin classification [18], [29] describes the category of finite free modules equipped with a Frobenius action, over a complete discrete valuation ring with algebraically closed residue field, loosely analogous to the eigenspace decomposition of a vector space over an algebraically closed field equipped with a linear transformation. When the residue field is unrestricted, the classification no longer applies, but one does retrieve a canonical filtration whose successive quotients are all isotypical of different types if one applies the Dieudonné-Manin classification after enlarging the residue field. The results of [19] give an analogous pair of assertions for finite free modules equipped with a Frobenius action over the Robba ring over a complete discretely valued field of mixed characteristic. (The Robba ring consists of those formal Laurent series over the given coefficient field converging on some open annulus of outer radius 1.) Namely, over a suitable “algebraic closure” of the Robba ring, every such module admits a decomposition into the same sort of standard pieces as in Dieudonné-Manin ([19, Theorem 4.16] and Theorem 4.5.7 herein), and the analogous canonical slope filtration descends back down to the original module ([19, Theorem 6.10] and Theorem 6.4.1 herein).

1.2 Applications

The original application of the slope filtration theorem was to the p-adic local monodromy theorem on quasi-unipotence of p-adic differential equations with Frobenius structure over the Robba ring. (The possibility of, and need for, such a theorem first arose in the work of Crew [11], [12] on the rigid cohomology of curves with coefficients, and so the theorem is commonly referred to as “Crew’s conjecture”.) Specifically, the slope filtration theorem reduces the pLMT to its unit-root case, established previously by Tsuzuki [35]. We note, for now in passing, that Crew’s conjecture has also been proved by André [1] and by Mebkhout [30], using the Christol-Mebkhout index theory for p-adic differential equations; for more on the relative merits of these proofs, see Remark 7.2.8. In turn, the p-adic local monodromy theorem is already known to have several applications. Many of these are in the study of rigid p-adic cohomology of varieties over fields of characteristic p: these include a full faithfulness theorem for restriction between the categories of overconvergent and convergent F -isocrystals [20], a finiteness theorem with coefficients [22], and an analogue of Deligne’s “Weil II” theorem [23]. The pLMT also gives rise to a proof of Fontaine’s conjecture that every de Rham representation (of the absolute Galois group of a mixed characteristic local field) is potentially semistable, via a construction of Berger [3] linking the theory of (ϕ, Γ)-modules to the theory of p-adic differential equations. Subsequently, other applications of the slope filtration theorem have come to light. Berger [4] has used it to give a new proof of the theorem of Colmez-Fontaine that weakly admissible (ϕ, Γ)-modules are admissible. A variant of Berger’s proof has been given by Kisin [26], who goes on to give a classification of crystalline representations with nonpositive Hodge-Tate weights in terms of certain Frobenius modules; as corollaries, he obtains classification results for pdivisible groups conjectured by Breuil and Fontaine. Colmez [10] has used the slope filtration theorem to construct a category of “trianguline representations” involved in a proposed p-adic Langlands correspondence. André and di Vizio [2] have used the slope filtration theorem to prove an analogue of Crew’s conjecture for q-difference equations, by establishing an analogue of Tsuzuki’s theorem for such equations. (The replacement of differential equations by q-difference equations does not affect the Frobenius structure, so the slope filtration theorem applies unchanged.) We expect to see additional applications in the future.

1.3 Purpose of the paper

The purpose of this paper is to give a “second generation” exposition of the proof of the slope filtration theorem, using ideas we have learned about since [19] was written. These ideas include a close analogy between the theory of slopes of Frobenius modules and the formalism of semistable vector bundles; this analogy is visible in the work of Hartl and Pink [17], which strongly resembles our Dieudonné-Manin classification but takes place in equal characteristic p > 0. It is also visible in the theory of filtered (ϕ, N)-modules, used to study p-adic Galois representations; indeed, this theory is directly related to slope filtrations via the work of Berger [4] and Kisin [26]. In addition to clarifying the exposition, we have phrased the results at a level of generality that may be useful for additional applications. In particular, the results apply to Frobenius modules over what might be called “fake annuli”, which occur in the context of semistable reduction for overconvergent F -isocrystals (a higher-dimensional analogue of Crew’s conjecture). See [25] for an analogue of the p-adic local monodromy theorem in this setting.

1.4 Structure of the paper

We conclude this introduction with a summary of the various chapters of the paper. In Chapter 2, we construct a number of rings similar to (but more general than) those occurring in [19, Chapters 2 and 3], and prove that a certain class of these are Bézout rings (in which every finitely generated ideal is principal). In Chapter 3, we introduce σ-modules and some basic terminology for dealing with them. Our presentation is informed by some strongly analogous work (in equal characteristic p) of Hartl and Pink. In Chapter 4, we give a uniform presentation of the standard Dieudonné-Manin decomposition theorem and of the variant form proved in [19, Chapter 4], again using the Hartl-Pink framework. In Chapter 5, we recall some results mostly from [19, Chapter 5] on σ-modules over the bounded subrings of so-called analytic rings. In particular, we compare the “generic” and “special” polygons and slope filtrations. In Chapter 6, we give a streamlined form of the arguments of [19, Chapter 6], which deduce from the Dieudonné-Manin-style classification the slope filtration theorem for σ-modules over arbitrary analytic rings. In Chapter 7, we make some related observations. In particular, we explain how the slope filtration theorem, together with Tsuzuki’s theorem on unit-root σ-modules with connection, implies Crew’s conjecture. We also explain the relevance of the terms “generic” and “special” to the discussion of Chapter 5.

2 The basic rings

In this chapter, we recall and generalize the ring-theoretic setup of [19, Chapter 3].

Convention 2.0.1. Throughout this chapter, fix a prime number p and a power q = p^{a} of p. Let K be a field of characteristic p, equipped with a valuation v_{K}; we will allow v_{K} to be trivial unless otherwise specified. Let K_{0} denote a subfield of K on which v_{K} is trivial. We will frequently do matrix calculations; in so doing, we apply a valuation to a matrix by taking its minimum over entries, and write I_{n} for the n × n identity matrix over any ring. See Conventions 2.2.2 and 2.2.6 for some further notations.

2.1 Witt rings

Convention 2.1.1. Throughout this section only, assume that K and K_{0} are perfect.

Definition 2.1.2. Let W (K) denote the ring of p-typical Witt vectors over K. Then W gives a covariant functor from perfect fields of characteristic p to complete discrete valuation rings of characteristic 0, with maximal ideal p and perfect residue field; this functor is in fact an equivalence of categories, being a quasi-inverse of the residue field functor. In particular, the absolute (p-power) Frobenius lifts uniquely to an automorphism σ_{0} of W (K); write σ for the log_{p}(q)-th power of σ_{0}. Use a horizontal overbar to denote the reduction map from W (K) to K. In this notation, we have \overline{u^{σ}} = \overline{u}^{q} for all u ∈ W (K).

We will also want to allow some ramified extensions of Witt rings.

Definition 2.1.3. Let O be a finite totally ramified extension of W (K_{0}), equipped with an extension of σ; let π denote a uniformizer of O. Write W (K, O) for W (K) ⊗_{W (K_{0})} O, and extend the notations σ, \overline{x} to W (K, O) in the natural fashion.

Definition 2.1.4. For \overline{z} ∈ K, let [\overline{z}] ∈ W (K) denote the Teichmüller lift of K; it can be constructed as lim_{n→∞} y_{n}^{p^{n}} for any sequence \{y_{n}\}_{n=0}^{∞} with \overline{y}_{n} = \overline{z}^{1/p^{n}}. (The point is that this limit is well-defined: if \{y′_{n}\}_{n=0}^{∞} is another such sequence, we have y_{n}^{p^{n}} ≡ (y′_{n})^{p^{n}} (mod p^{n}).) Then [\overline{z}]^{σ} = [\overline{z}]^{q}, and if \overline{z}′ ∈ K, then [\overline{z}\overline{z}′] = [\overline{z}][\overline{z}′]. Note that each x ∈ W (K, O) can be written uniquely as ∑_{i=0}^{∞}[\overline{z_{i}}]π^{i} for some \overline{z_{0}}, \overline{z_{1}}, ⋯ ∈ K; similarly, each x ∈ W (K, O)[π^{−1}] can be written uniquely as ∑_{i∈ℤ}[\overline{z_{i}}]π^{i} for some z_{i} ∈ K with \overline{z_{i}} = 0 for i sufficiently small.

Definition 2.1.5. Recall that K was assumed to be equipped with a valuation v_{K}. Given n ∈ ℤ, we define the “partial valuation” v_{n} on W (K, O)[π^{−1}] by

v_{n} (∑_{i}[\overline{z_{i}}]π^{i}) = min_{i≤n}\{v_{K}(\overline{z_{i}})\};  (2.1.6)

it satisfies the properties

v_{n}(x + y) ≥ min{v_{n}(x), v_{n}(y)}  (x, y ∈ W (K, O)[π^{−1}], n ∈ ℤ)
v_{n}(xy) ≥ min_{m∈ℤ}{v_{m}(x) + v_{n−m}(y)}  (x, y ∈ W (K, O)[π^{−1}], n ∈ ℤ)
v_{n}(x^{σ}) = qv_{n}(x)  (x ∈ W (K, O)[π^{−1}], n ∈ ℤ)
v_{n}([\overline{z}]) = v_{K}(\overline{z})  (\overline{z} ∈ K, n ≥ 0).

In each of the first two inequalities, one has equality if the minimum is achieved exactly once. For r > 0, n ∈ ℤ, and x ∈ W (K, O)[π^{−1}], put

v_{n,r}(x) = rv_{n}(x) + n;

for r = 0, put v_{n,r}(x) = n if v_{n}(x) < ∞ and v_{n,r}(x) = ∞ if v_{n}(x) = ∞. For r ≥ 0, let W_{r}(K, O) be the subring of W (K, O) consisting of those x for which v_{n,r}(x) → ∞ as n → ∞; then σ sends W_{qr}(K, O) onto W_{r}(K, O). (Note that there is no restriction when r = 0.)

Lemma 2.1.7. Given x, y ∈ W_{r}(K, O)[π^{−1}] nonzero, let i and j be the smallest and largest integers n achieving min_{n}{v_{n,r}(x)}, and let k and l be the smallest and largest integers n achieving min_{n}{v_{n,r}(y)}. Then i + k and j + l are the smallest and largest integers n achieving min_{n}{v_{n,r}(xy)}, and this minimum equals min_{n}{v_{n,r}(x)} + min_{n}{v_{n,r}(y)}.

Proof. We have

v_{m,r}(xy) ≥ min_{n}{v_{n,r}(x) + v_{m−n,r}(y)},

with equality if the minimum on the right is achieved only once. This means that:

• for all m, the minimum is at least v_{i,r}(x) + v_{k,r}(y); • for m = i+k and m = j +l, the value v_{i,r}(x)+v_{k,r}(y) is achieved exactly

once (respectively by n = i and n = j);

• for m < i + k or m > j + l, the value v_{i,r}(x) + v_{k,r}(y) is never achieved.

This implies the desired results.

Definition 2.1.8. Define the map w_{r} : W_{r}(K, O)[π^{−1}] → ℝ ∪ {∞} by

w_{r}(x) = min_{n}{v_{n,r}(x)};  (2.1.9)

also write w for w_{0}. By Lemma 2.1.7, w_{r} is a valuation on W_{r}(K, O)[π^{−1}]; moreover, w_{r}(x) = w_{r/q}(x^{σ}). Put

W_{con}(K, O) = ∪_{r>0}W_{r}(K, O);

note that W_{con}(K, O) is a discrete valuation ring with residue field K and maximal ideal generated by π, but is not complete if v_{K} is nontrivial.

Remark 2.1.10. Note that u is a unit in W_{r}(K, O) if and only if v_{n,r}(u) > v_{0,r}(u) for n > 0. We will generalize this observation later in Lemma 2.4.7.

Remark 2.1.11. Note that w is a p-adic valuation on W (K, O) normalized so that w(π) = 1. This indicates two discrepancies from choices made in [19]. First, we have normalized w(π) = 1 instead of w(p) = 1 for internal convenience; the normalization will not affect any of the final results. Second, we use w for the p-adic valuation instead of v_{p} (or simply v) because we are using v’s for valuations in the “horizontal” direction, such as the valuation on K, and the partial valuations of Definition 2.1.5. By contrast, decorated w’s denote “nonhorizontal” valuations, as in Definition 2.1.8.

Lemma 2.1.12. The (noncomplete) discrete valuation ring W_{con}(K, O) is henselian.

Proof. It suffices to verify that if P (x) is a polynomial over W_{con}(K, O) and y ∈ W_{con}(K, O) satisfies P (y) ≡ 0 (mod π) and P′(y) ≢ 0 (mod π), then there exists z ∈ W_{con}(K, O) with z ≡ y (mod π) and P (z) = 0. To see this, pick r > 0 such that w_{r}(P (y)/P′(y)^{2}) > 0; then the usual Newton iteration gives a series converging under w to a root z of P in W (K, O) with z ≡ y (mod π). However, the iteration also converges under w_{r}, so we must have z ∈ W_{r}(K, O). (Compare [19, Lemma 3.9].)

2.2 Cohen rings

Remember that Convention 2.1.1 is no longer in force, i.e., K_{0} and K no longer need be perfect.

Definition 2.2.1. Let C_{K} denote a Cohen ring of K, i.e., a complete discrete valuation ring with maximal ideal pC_{K} and residue field K. Such a ring necessarily exists and is unique up to noncanonical isomorphism [15, Proposition 0.19.8.5]. Moreover, any map K → K′ can be lifted, again noncanonically, to a map C_{K} → C_{K′}.

Convention 2.2.2. For the remainder of the chapter, assume chosen and fixed a map (necessarily injective) C_{K_{0}} → C_{K}. Let O be a finite totally ramified extension of C_{K_{0}}, and let π denote a uniformizer of O. Write Γ^{K} for C_{K}⊗_{C_{K_{0}}} O; we write Γ for short if K is to be understood, as it will usually be in this chapter.

Definition 2.2.3. By a Frobenius lift on Γ, we mean any endomorphism σ : Γ → Γ lifting the absolute q-power Frobenius on K. Given σ, we may form the completion of the direct limit

Γ^{K} →^{σ} Γ^{K} →^{σ} ⋯;  (2.2.4)

for K = K_{0}, this ring is a finite totally ramified extension of Γ^{K_{0}} = O, which we denote by O^{perf}. In general, if σ is a Frobenius lift on Γ^{K} which maps O into itself, we may identify the completed direct limit of (2.2.4) with W (K^{perf}, O^{perf}); we may thus use the induced embedding Γ^{K} ↪ W (K^{perf}, O^{perf}) to define v_{n}, v_{n,r}, w_{r}, w on Γ.

Remark 2.2.5. In [19], a Frobenius lift is assumed to be a power of a p-power Frobenius lift, but all calculations therein work in this slightly less restrictive setting.

Convention 2.2.6. For the remainder of the chapter, assume chosen and fixed a Frobenius lift σ on Γ which carries O into itself.

Definition 2.2.7. Define the levelwise topology on Γ by declaring that a sequence {x_{l}}_{l=0}^{∞} converges to zero if and only if for each n, v_{n}(x_{l}) → ∞ as l → ∞. This topology is coarser than the usual π-adic topology.

Definition 2.2.8. For L/K finite separable, we may view Γ^{L} as a finite unramified extension of Γ^{K}, and σ extends uniquely to Γ^{L}; if L/K is Galois, then Gal(L/K) acts on Γ^{L} fixing Γ^{K}. More generally, we say L/K is pseudo-finite separable if L = M^{1/q^{n}} for some M/K finite separable and some nonnegative integer n; in this case, we define Γ^{L} to be a copy of Γ^{M} viewed as a Γ^{M}-algebra via σ^{n}. In particular, we have a unique extension of v_{K} to L, under which L is complete, and we have a distinguished extension of σ to Γ^{L} (but only because we built the choice of σ into the definition of Γ^{L}).

Remark 2.2.9. One can establish a rather strong functoriality for the formation of the Γ^{L}, as in [19, Section 2.2]. One of the simplifications introduced here is to avoid having to elaborate upon this.

Definition 2.2.10. For r > 0, put Γ_{r} = Γ ∩ W_{r}(K^{perf}, O). We say Γ has enough r-units if every nonzero element of K can be lifted to a unit of Γ_{r}. We say Γ has enough units if Γ has enough r-units for some r > 0.

Remark 2.2.11. (a) If K is perfect, then Γ has enough r-units for any r > 0, because a nonzero Teichmüller element is a unit in every Γ_{r}.

(b) If Γ^{K} has enough r-units, then Γ^{K^{1/q}} has enough qr-units, and vice versa.

Lemma 2.2.12. Suppose that Γ^{K} has enough units, and let L be a pseudo-finite separable extension of K. Then Γ^{L} has enough units.

Proof. It is enough to check the case when L is actually finite separable. Put d = [L : K]. Apply the primitive element theorem to produce \overline{x} ∈ L which generates L over K, and apply Lemma 2.1.12 to produce x ∈ Γ_{con}^{L} lifting \overline{x}. Recall that any two Banach norms on a finite dimensional vector space over a complete normed field are equivalent [32, Proposition 4.13]. In particular, if we let v_{L} denote the unique extension of v_{K} to L, then there exists a constant a > 0 such that whenever \overline{y} ∈ L and \overline{c_{0}}, …, \overline{c_{d−1}} ∈ K satisfy \overline{y} = ∑_{i=0}^{d−1} \overline{c_{i}}\overline{x}^{i}, we have v_{L}(\overline{y}) ≤ min_{i}\{v_{K}(\overline{c_{i}}\overline{x}^{i})\} + a. Pick r > 0 such that Γ^{K} has enough r-units and x is a unit in Γ_{r}^{L}, and choose s > 0 such that 1 − s/r > sa. Given \overline{y} ∈ L, lift each \overline{c_{i}} to either zero or a unit in Γ_{r}, and set y = ∑_{i=0}^{d−1} c_{i}x^{i}. Then for all n ≥ 0,

v_{n,r}(y) ≥ min_{i}{v_{n,r}(c_{i}x^{i})}

≥ min_{i}\{rv_{K}(\overline{c_{i}}) + riv_{L}(\overline{x})\}

≥ rv_{L}(\overline{y}) − ra.

In particular, v_{n,s}(y) > v_{0,s}(y) for n > 0, so y is a unit in Γ_{s}^{L}. Since s does not depend on y, we conclude that Γ^{L} has enough s-units, as desired.

Definition 2.2.13. Suppose that Γ has enough units. Define Γ_{con} = ∪_{r>0}Γ_{r} = Γ ∩ W_{con}(K, O); then Γ_{con} is again a discrete valuation ring with maximal ideal generated by π. Although Γ_{con} is not complete, it is henselian thanks to Lemma 2.1.12. For L/K pseudo-finite separable, we may view Γ_{con}^{L} as an extension of Γ_{con}^{K}, which is finite unramified if L/K is finite separable.

Remark 2.2.14. Remember that v_{K} is allowed to be trivial, in which case the distinction between Γ and Γ_{con} collapses.

Proposition 2.2.15. Let L be a finite separable extension of K. Then for any x ∈ Γ_{con}^{L} such that \overline{x} generates L over K, we have Γ_{con}^{L} ≅ Γ_{con}^{K}[x]/(P (x)), where P (x) denotes the minimal polynomial of x.

Proof. Straightforward.

Convention 2.2.16. For L the completed perfect closure or algebraic closure of K, we replace the superscript L by “perf” or “alg”, respectively, writing Γ^{perf} or Γ^{alg} for Γ^{L} and so forth. (Recall that these are obtained by embedding Γ^{K} into W (K^{perf}, O) via σ, and then embedding the latter into W (L, O) via Witt vector functoriality.) Beware that this convention disagrees with a convention of [19], in which Γ^{alg} = W (K^{alg}, O), without the completion; we will comment further on this discrepancy in Remark 2.4.13.

The next assertions are essentially [13, Proposition 8.1], only cast a bit more generally; compare also [20, Proposition 4.1].

Definition 2.2.17. By a valuation p-basis of K, we mean a subset S ⊂ K such that the set U of monomials in S of degree < p in each factor (and degree 0 in almost all factors) is a valuation basis of K over K^{p}. That is, each \overline{x} ∈ K has a unique expression of the form ∑_{\overline{u}∈U} \overline{c_{u}}\overline{u}, with each \overline{c_{u}} ∈ K^{p} and almost all zero, and one has

v_{K}(\overline{x}) = min_{\overline{u}∈U}\{v_{K}(\overline{c_{u}}\overline{u})\}.

Example 2.2.18. For example, K = k((t)) admits a valuation p-basis consisting of t plus a p-basis of k over k^{p}. In a similar vein, if [v(K^{∗}) : v((K^{p})^{∗})] = [K : K^{p}] < ∞, then one can choose a valuation p-basis for K by selecting elements of K^{∗} whose images under v generate v(K^{∗})/v((K^{p})^{∗}). (See also the criterion of [27, Chapter 9].)

Lemma 2.2.19. Suppose that Γ has enough units and that K admits a valuation p-basis S. Then there exists a Γ-linear map f : Γ^{perf} → Γ sectioning the inclusion Γ → Γ^{perf}, which maps Γ_{con}^{perf} to Γ_{con}.

Proof. Choose r > 0 such that Γ has enough r-units, and, for each \overline{s} ∈ S, choose a unit s of Γ_{r} lifting \overline{s}. Put U_{0} = \{1\}. For n a positive integer, let U_{n} be the set of products

∏_{\overline{s}∈S}(s^{e_{s}})^{σ^{−n}}

in which each e_{s} ∈ {0, …, q^{n} − 1}, all but finitely many e_{s} are zero (so the product makes sense), and the e_{s} are not all divisible by q. Put V_{n} = U_{0} ∪ ⋯ ∪ U_{n}; then the reductions of V_{n} form a basis of K^{q^{−n}} over K. We can thus write each element of Γ^{σ^{−n}} uniquely as a sum ∑_{u∈V_{n}} x_{u}u, with each x_{u} ∈ Γ and for any integer m > 0, only finitely many of the x_{u} nonzero modulo π^{m}. Define the map f_{n} : Γ^{σ^{−n}} → Γ sending x = ∑_{u∈V_{n}} x_{u}u to x_{1}. Note that each element of each U_{n} is a unit in (Γ^{σ^{−n}})_{r}. Since S is a valuation p-basis, it follows (by induction on m) that if we write x = ∑_{u∈V_{n}} x_{u}u, then

min_{j≤m}{v_{j,r}(x)} = min_{u∈V_{n}} min_{j≤m}{v_{j,r}(x_{u}u)}.

Hence for any r′ ∈ (0, r], f_{n} sends (Γ^{σ^{−n}})_{r′} to Γ_{r′}. That means in particular that the f_{n} fit together to give a function f that extends by continuity to all of Γ^{perf}, sections the map Γ → Γ^{perf}, and carries Γ_{con}^{perf} to Γ_{con}.

Remark 2.2.20. It is not clear to us whether it should be possible to loosen the restriction that K must have a valuation p-basis, e.g., by imitating the proof strategy of Lemma 2.2.12.

Proposition 2.2.21. Suppose that Γ has enough units and that K admits a valuation p-basis. Let μ : Γ ⊗_{Γ_{con}} Γ_{con}^{alg} → Γ^{alg} denote the multiplication map, so that μ(x ⊗ y) = xy.

(a) If x_{1}, …, x_{n} ∈ Γ are linearly independent over Γ_{con}, and μ(∑_{i=1}^{n} x_{i} ⊗ y_{i}) = 0, then y_{i} = 0 for i = 1, …, n.

(b) If x_{1}, …, x_{n} ∈ Γ are linearly independent over Γ_{con}, and μ(∑_{i=1}^{n} x_{i} ⊗ y_{i}) ∈ Γ, then y_{i} ∈ Γ_{con} for i = 1, …, n.

(c) The map μ is injective.

Proof. (a) Suppose the contrary; choose a counterexample with n minimal. We may assume without loss of generality that w(y_{1}) = min_{i}{w(y_{i})}; we may then divide through by y_{1} to reduce to the case y_{1} = 1, where we will work hereafter.

Any g ∈ Gal(K^{alg}/K^{perf}) extends uniquely to an automorphism of Γ^{alg} over Γ^{perf}, and to an automorphism of Γ_{con}^{alg} over Γ_{con}^{perf}. Then

0 = ∑_{i=1}^{n} x_{i}y_{i} = ∑_{i=1}^{n} x_{i}y_{i}^{g} = ∑_{i=2}^{n} x_{i}(y_{i}^{g} − y_{i});

by the minimality of n, we have y_{i}^{g} = y_{i} for i = 2, …, n. Since this is true for any g, we have y_{i} ∈ Γ_{con}^{perf} for each i.

Let f be the map from Lemma 2.2.19; then

0 = ∑ x_{i}y_{i} = f (∑ x_{i}y_{i}) = ∑ x_{i}f(y_{i}) = ∑ x_{i}(y_{i} − f(y_{i})),

so again y_{i} = f(y_{i}) for i = 2, …, n. Hence x_{1} = − ∑_{i=2}^{n} x_{i}y_{i}, contradicting the linear independence of the x_{i} over Γ_{con}.

(b) For g as in (a), we have 0 = ∑ x_{i}(y_{i}^{g} − y_{i}); by (a), we have y_{i}^{g} = y_{i} for all i and g, so y_{i} ∈ Γ_{con}^{perf}. Now 0 = ∑ x_{i}(y_{i} − f(y_{i})), so y_{i} = f(y_{i}) ∈ Γ_{con}.

(c) Suppose on the contrary that ∑_{i=1}^{n} x_{i} ⊗y_{i} ≠ 0 but ∑_{i=1}^{n} x_{i}y_{i} = 0; choose such a counterexample with n minimal. By (a), the x_{i} must be linearly dependent over Γ_{con}; without loss of generality, suppose we can write x_{1} = ∑_{i=2}^{n} c_{i}x_{i} with c_{i} ∈ Γ_{con}. Then ∑_{i=1}^{n} x_{i} ⊗ y_{i} = ∑_{i=2}^{n} x_{i} ⊗ (y_{i} + c_{i}) is a counterexample with only n − 1 terms, contradicting the minimality of n.

2.3 Relation to the Robba ring

We now recall how the constructions in the previous section relate to the usual Robba ring.

Convention 2.3.1. Throughout this section, assume that K = k((t)) and K_{0} = k; we may then describe Γ as the ring of formal Laurent series ∑_{i∈ℤ} c_{i}u^{i} with each c_{i} ∈ O, and w(c_{i}) → ∞ as i → −∞. Suppose further that the Frobenius lift is given by ∑ c_{i}u^{i} ↦ ∑ c_{i}^{σ}(u^{σ})^{i}, where u^{σ} = ∑ a_{i}u^{i} with lim inf_{i→−∞} w(a_{i})/(−i) > 0.

Definition 2.3.2. Define the naïve partial valuations v_{n}^{naive} on Γ by the formula

v_{n}^{naive} (∑ c_{i}u^{i}) = min{i : w(c_{i}) ≤ n}.

These functions satisfy some identities analogous to those in Definition 2.1.5:

v_{n}^{naive}(x + y) ≥ min{v_{n}^{naive}(x), v_{n}^{naive}(y)}  (x, y ∈ Γ[π^{−1}], n ∈ ℤ)
v_{n}^{naive}(xy) ≥ min_{m≤n}{v_{m}^{naive}(x) + v_{n−m}^{naive}(y)}  (x, y ∈ Γ[π^{−1}], n ∈ ℤ).

Again, equality holds in each case if the minimum on the right side is achieved exactly once. Put

v_{n,r}^{naive}(x) = rv_{n}^{naive}(x) + n.

For r > 0, let Γ_{r}^{naive} be the set of x ∈ Γ such that v_{n,r}^{naive}(x) → ∞ as n → ∞. Define the map w_{r}^{naive} on Γ_{r}^{naive} by

w_{r}^{naive}(x) = min_{n}{v_{n,r}^{naive}(x)};

then w_{r}^{naive} is a valuation on Γ_{r}^{naive} by the same argument as in Lemma 2.1.7. Put

Γ_{con}^{naive} = ∪_{r>0}Γ_{r}^{naive}.

By the hypothesis on the Frobenius lift, we can choose r > 0 such that u^{σ}/u^{q} is a unit in Γ_{r}^{naive}.

Lemma 2.3.3. For r > 0 such that u^{σ}/u^{q} is a unit in Γ_{r}^{naive}, and s ∈ (0, qr], we have

min_{j≤n}{v_{j,s}^{naive}(x)} = min_{j≤n}{v_{j,s/q}^{naive}(x^{σ})}  (2.3.4)

for each n ≥ 0 and each x ∈ Γ.

Proof. The hypothesis ensures that (2.3.4) holds for x = u^{i} for any i ∈ ℤ and any n. For general x, write x = ∑_{i} c_{i}u^{i}; then on one hand,

min_{j≤n}{v_{j,s/q}^{naive}(x^{σ})} ≥ min_{i∈ℤ}{min_{j≤n}{v_{j,s/q}^{naive}(c_{i}^{σ}(u^{σ})^{i})}}

= min_{i∈ℤ}{min_{j≤n}{v_{j,s}^{naive}(c_{i}u^{i})}}

= min_{j≤n}{v_{j,s}^{naive}(x)}.

On the other hand, if we take the smallest j achieving the minimum on the left side of (2.3.4), then the minimum of v_{j,s}^{naive}(c_{i}u^{i}) is achieved by a unique integer i. Hence the one inequality in the previous sequence is actually an equality.

Lemma 2.3.5. For r > 0 such that u^{σ}/u^{q} is a unit in Γ_{r}^{naive}, and s ∈ (0, qr], we have

min_{j≤n}{v_{j,s}(x)} = min_{j≤n}{v_{j,s}^{naive}(x)}  (2.3.6)

for each n ≥ 0 and each x ∈ Γ. In particular, Γ_{s}^{naive} = Γ_{s}, and w_{s}(x) = w_{s}^{naive}(x) for all x ∈ Γ_{s}.

Proof. Write x = ∑_{i=0}^{∞}[\overline{x_{i}}]π^{i} with each \overline{x_{i}} ∈ K^{perf}. Choose an integer l such that \overline{x_{i}}^{q^{l}} ∈ K for i = 0, …, n, and write \overline{x_{i}}^{q^{l}} = ∑_{h∈ℤ} \overline{c_{hi}}t^{h} with \overline{c_{hi}} ∈ k. Choose c_{hi} ∈ O lifting \overline{c_{hi}}, with c_{hi} = 0 whenever \overline{c_{hi}} = 0, and put y_{i} = ∑_{h} c_{hi}u^{h}. Pick an integer m > n, and define

x′ = ∑_{i=0}^{n} y_{i}^{q^{m}}(π^{i})^{σ^{l+m}};

then w(x′ − x^{σ^{l+m}}) > n. Hence for j ≤ n, v_{j}(x′) = v_{j}(x^{σ^{l+m}}) = q^{l+m}v_{j}(x) and v_{j}^{naive}(x′) = v_{j}^{naive}(x^{σ^{l+m}}). From the way we chose the y_{i}, we have

v_{j}^{naive}(y_{i}^{q^{m}}(π^{i})^{σ^{l+m}}) = q^{l+m}v_{0}(\overline{x_{i}}) (j ≥ i).

It follows that v_{j}^{naive}(x′) = q^{l+m}v_{j}(x) for j ≤ n; that is, we have v_{j}^{naive}(x^{σ^{l+m}}) = q^{l+m}v_{j}(x) for j ≤ n. In particular, we have

min_{j≤n}{v_{j,s}(x)} = min_{j≤n}{v_{j,s/q^{l+m}}^{naive}(x^{σ^{l+m}})}.

By Lemma 2.3.3, this yields the desired result. (Compare [19, Lemmas 3.6 and 3.7].)

Corollary 2.3.7. For r > 0 such that u^{σ}/u^{q} is a unit in Γ_{r}^{naive}, Γ has enough qr-units, and Γ_{con} = Γ_{con}^{naive}.

Remark 2.3.8. The ring Γ_{r}^{naive}[π^{−1}] is the ring of bounded rigid analytic functions on the annulus |π|^{r} ≤ |u| < 1, and the valuation w_{s}^{naive} is the supremum norm on the circle |u| = |π|^{s}. This geometric interpretation motivates the subsequent constructions, and so is worth keeping in mind; indeed, much of the treatment of analytic rings in the rest of this chapter is modeled on the treatment of rings of functions on annuli given by Lazard [28], and our results generalize some of the results in [28] (given Remark 2.3.9 below).

Remark 2.3.9. In the context of this section, the ring Γ_{an,con} is what is usually called the Robba ring over K. The point of view of [19], maintained here, is that the Robba ring should always be viewed as coming with the “equipment” of a Frobenius lift σ; this seems to be the most convenient angle from which to approach σ-modules. However, when discussing a statement about Γ_{an,con} that depends only on its underlying topological ring (e.g., the Bézout property, as in Theorem 2.9.6), one is free to use any Frobenius, and so it is sometimes convenient to use a “standard” Frobenius lift, under which u^{σ} = u^{q} and v_{n}^{naive} = v_{n} for all n. In general, however, one cannot get away with only standard Frobenius lifts because the property of standardness is not preserved by passing to Γ_{an,con}^{L} for L a finite separable extension of k((t)).

Remark 2.3.10. It would be desirable to be able to have it possible for Γ_{r}^{naive} to be the ring of rigid analytic functions on an annulus over a p-adic field whose valuation is not discrete (e.g., the completed algebraic closure ℂ_{p} of ℚ_{p}), since the results of Lazard we are analogizing hold in that context. However, this seems rather difficult to accommodate in the formalism developed above; for instance, the v_{n} cannot be described in terms of Teichmüller elements, so an axiomatic characterization is probably needed. There are additional roadblocks later in the story; we will flag some of these as we go along.

Remark 2.3.11. One can carry out an analogous comparison between naïve and true partial valuations when K is the completion of k(x_{1}, …, x_{n}) for a “monomial” valuation, in which v(x_{1}), …, v(x_{n}) are linearly independent over ℚ; this gives additional examples in which the hypothesis “Γ has enough units” can be checked, and hence additional examples in which the framework of this paper applies. See [25] for details.

2.4 Analytic rings

We now proceed roughly as in [19, Section 3.3]; however, we will postpone certain “reality checks” on the definitions until the next section.

Convention 2.4.1. Throughout this section, and for the rest of the chapter, assume that the field K is complete with respect to the valuation v_{K}, and that Γ^{K} has enough r_{0}-units for some fixed r_{0} > 0. Note that the assumption that K is complete ensures that Γ_{r} is complete under w_{r} for any r ∈ [0, r_{0}).

Definition 2.4.2. Let I be a subinterval of [0, r_{0}) bounded away from r_{0}, i.e., I ⊆ [0, r] for some r < r_{0}. Let Γ_{I} be the Fréchet completion of Γ_{r_{0}}[π^{−1}] for the valuations w_{s} for s ∈ I; note that the functions v_{n}, v_{n,s}, w_{s} extend to Γ_{I} by continuity, and that σ extends to a map σ : Γ_{I} → Γ_{q^{−1}I}. For I ⊆ J subintervals of [0, r_{0}) bounded away from 0, we have a natural map Γ_{J} → Γ_{I}; this map is injective with dense image. For I = [0, s], note that Γ_{I} = Γ_{s}[π^{−1}]. For I = (0, s], we write Γ_{an,s} for Γ_{I}.

Remark 2.4.3. In the context of Section 2.3, Γ_{I} is the ring of rigid analytic functions on the subspace of the open unit disc defined by the condition log_{|π|} |u| ∈ I; compare Remark 2.3.8.

Definition 2.4.4. For I a subinterval of [0, r_{0}) bounded away from r_{0}, and for x ∈ Γ_{I} nonzero, define the Newton polygon of x to be the lower convex hull of the set of points (v_{n}(x), n), minus any segment the negative of whose slope is not in I. Define the slopes of x to be the negations of the slopes of the Newton polygon of x. Define the multiplicity of s ∈ (0, r] as a slope of x to be the difference in vertical coordinates between the endpoints of the segment of the Newton polygon of x of slope −s, or 0 if no such segment exists. If x has only finitely many slopes, define the total multiplicity of x to be the sum of the multiplicities of all slopes of x. If x has only one slope, we say x is pure of that slope.

Remark 2.4.5. The analogous definition of total multiplicity for Γ_{r}^{naive} counts the total number of zeroes (with multiplicities) that a function has in the annulus |π|^{r} ≤ |u| < 1.

Remark 2.4.6. Note that the multiplicity of any given slope is always finite. More generally, for any closed subinterval I = [r′, r] of [0, r_{0}), the total multiplicity of any x ∈ Γ_{I} is finite. Explicitly, the total multiplicity equals i − j, where i is the largest n achieving min_{n}{v_{n,r}(x)} and j is the smallest n achieving min_{n}{v_{n,r′}(x)}. In particular, if x ∈ Γ_{an,r}, the slopes of x form a sequence decreasing to zero.

Lemma 2.4.7. For x, y ∈ Γ_{I} nonzero, the multiplicity of each s ∈ I as a slope of xy is the sum of the multiplicities of s as a slope of x and of y. In particular, Γ_{I} is an integral domain.

Proof. For x, y ∈ Γ_{r}[π^{−1}], this follows at once from Lemma 2.1.7. In the general case, note that the conclusion of Lemma 2.1.7 still holds, by approximating x and y suitably well by elements of Γ_{r}[π^{−1}].

Definition 2.4.8. Let Γ_{an,con}^{K} be the union of the Γ_{an,r}^{K} over all r ∈ (0, r_{0}); this ring is an integral domain by Lemma 2.4.7. Remember that we are allowing v_{K} to be trivial, in which case Γ_{an,con} = Γ_{con}[π^{−1}] = Γ[π^{−1}].

Example 2.4.9. In the context of Section 2.3, the ring Γ_{an,con} consists of formal Laurent series ∑_{n∈ℤ} c_{n}u^{n} with each c_{n} ∈ O[π^{−1}], lim inf_{n→−∞} w(c_{n})/(−n) > 0, and lim inf_{n→∞} w(c_{n})/n ≥ 0. The latter is none other than the Robba ring over O[π^{−1}].

We make a few observations about finite extensions of Γ_{an,con}.

Proposition 2.4.10. Let L be a finite separable extension of K. Then the multiplication map

μ : Γ_{an,con}^{K} ⊗_{Γ_{con}^{K}} Γ_{con}^{L} → Γ_{an,con}^{L}

is an isomorphism. More precisely, for any x ∈ Γ_{con}^{L} such that \overline{x} generates L over K, we have Γ_{an,con}^{L} ≅ Γ_{an,con}^{K}[x]/(P (x)).

Proof. For s > 0 sufficiently small, we have Γ_{s}^{L} ≅ Γ_{s}^{K}[x]/(P (x)) by Lemma 2.1.12, from which the claim follows.

Corollary 2.4.11. Let L be a finite Galois extension of K. Then the fixed subring of Frac Γ_{an,con}^{L} under the action of G = Gal(L/K) is equal to Frac Γ_{an,con}^{K}.

Proof. By Proposition 2.4.10, the fixed subring of Γ_{an,con}^{L} under the action of G is equal to Γ_{an,con}^{K}. Given x/y ∈ Frac Γ_{an,con}^{L} fixed under G, put x′ = ∏_{g∈G} x^{g}; since x′ is G-invariant, we have x′ ∈ Γ_{an,con}^{K}. Put y′ = x′y/x ∈ Γ_{an,con}^{L}; then x′/y′ = x/y, and both x′ and x′/y′ are G-invariant, so y′ is as well. Thus x/y ∈ Frac Γ_{an,con}^{K}, as desired.

Lemma 2.4.12. Let I be a subinterval of (0, r_{0}) bounded away from r_{0}. Then the union ∪Γ_{I}^{L}, taken over all pseudo-finite separable extensions L of K, is dense in Γ_{I}^{alg}.

Proof. Let M be the algebraic closure (not completed) of K. Then ∪Γ^{L} is clearly dense in Γ^{M} for the p-adic topology. By Remark 2.2.11 and Lemma 2.2.12, the set of pseudo-finite separable extensions L such that Γ^{L} has enough r_{0}-units is cofinal. Hence the set U of x ∈ ∪Γ_{r_{0}}^{L} with w_{r_{0}}(x) ≥ 0 is dense in the set V of x ∈ Γ_{r_{0}}^{M} with w_{r_{0}}(x) ≥ 0 for the p-adic topology. On these sets, the topology induced on U or V by any one w_{s} with s ∈ (0, r_{0}) is coarser than the p-adic topology. Thus U is also dense in V for the Fréchet topology induced by the w_{s} for s ∈ I. It follows that ∪Γ_{I}^{L} is dense in Γ_{I}^{M}; however, the condition that 0 ∉ I ensures that Γ_{I}^{M} = Γ_{I}^{alg}, so we have the desired result.

Remark 2.4.13. Recall that in [19] (contrary to our present Convention 2.2.16), the residue field of Γ^{alg} is the algebraic closure of K, rather than the completion thereof. However, the definition of Γ_{an,con}^{alg} comes out the same, and our convention here makes a few statements a bit easier to make. For instance, in the notation of [19], an element x of Γ_{an,con}^{alg} can satisfy v_{n}(x) = ∞ for all n < 0 without belonging to Γ_{con}^{alg}. (Thanks to Francesco Baldassarri for suggesting this change.)

2.5 Reality checks

Before proceeding further, we must make some tedious but necessary “reality checks” concerning the analytic rings. This is most easily done for K perfect, where elements of Γ_{I} have canonical decompositions (related to the “strong semiunit decompositions” of [19, Proposition 3.14].)

Definition 2.5.1. For K perfect, define the functions f_{n} : Γ[π^{−1}] → K for n ∈ ℤ by the formula x = ∑_{n∈ℤ}[f_{n}(x)]π^{n}, where the brackets again denote Teichmüller lifts. Then

v_{n}(x) = min_{m≤n}{v_{K}(f_{m}(x))} ≤ v_{K}(f_{n}(x)),

which implies that f_{n} extends uniquely to a continuous function f_{n} : Γ_{I} → K for any subinterval I ⊆ [0, ∞), and that the sum ∑_{n∈ℤ}[f_{n}(x)]π^{n} converges to x in Γ_{I}. We call this sum the Teichmüller presentation of x. Let x_{+}, x_{−}, x_{0} be the sums of [f_{n}(x)]π^{n} over those n for which v_{K}(f_{n}(x)) is positive, negative, or zero; we call the presentation x = x_{+}+x_{−}+x_{0} the plus-minus-zero presentation of x.

From the existence of Teichm̈uller presentations, it is obvious that for instance, if x ∈ Γ_{an,r} satisfies v_{n}(x) = ∞ for all n < 0, then x ∈ Γ_{r}. In order to make such statements evident in case K is not perfect, we need an approximation of the same technique.

Definition 2.5.2. Define a semiunit to be an element of Γ_{r_{0}} which is either zero or a unit. For I ⊆ [0, r_{0}) bounded away from r_{0} and x ∈ Γ_{I}, a semiunit presentation of x (over Γ_{I}) is a convergent sum x = ∑_{i∈ℤ} u_{i}π^{i}, in which each u_{i} is a semiunit.

Lemma 2.5.3. Suppose that u_{0}, u_{1}, … are semiunits.

(a) For each i ∈ ℤ and r ∈ (0, r_{0}),

w_{r}(u_{i}π^{i}) ≥ min_{n≤i} {v_{n,r} (∑_{j=0}^{i} u_{j}π^{j})} .

(b) Suppose that ∑_{i=0}^{∞} u_{i}π^{i} converges π-adically to some x such that for some r ∈ (0, r_{0}), v_{n,r}(x) → ∞ as n → ∞. Then w_{r}(u_{i}π^{i}) → ∞ as i → ∞, so that ∑_{i} u_{i}π^{i} is a semiunit presentation of x over Γ_{r}.

Proof. (a) The inequality is evident for i = 0; we prove the general claim by induction on i. If w_{r}(u_{i}π^{i}) ≥ w_{r}(u_{j}π^{j}) for some j < i, then the induction hypothesis yields the claim. Otherwise, w_{r}(u_{i}π^{i}) < w_{r}(∑_{j<i} u_{j}π^{j}), so v_{n,r}(∑_{j=0}^{i} u_{j}π^{j}) = v_{n,r}(u_{i}π^{i}), again yielding the claim.

(b) Choose r′ ∈ (r, r_{0}); we can then apply (a) to deduce that

w_{r′}(u_{i}π^{i}) ≥ min_{n≤i}{v_{n,r′}(x)}

= min_{n≤i}{(r′/r)v_{n,r}(x) + (1 − r′/r)n}.

It follows that

w_{r}(u_{i}π^{i}) ≥ min_{n≤i}{v_{n,r}(x) + (r/r′ − 1)n} + (1 − r/r′)i

= min_{n≤i}{v_{n,r}(x) + (1 − r/r′)(i − n)}.

Since v_{n,r}(x) → ∞ as n → ∞, the right side tends to ∞ as n → ∞.

Lemma 2.5.4. Given a subinterval I of [0, r_{0}) bounded away from r_{0}, and r ∈ I, suppose that x ∈ Γ_{[r,r]} has the property that for any s ∈ I, v_{n,s}(x) → ∞ as n → ±∞. Suppose also that ∑_{i} u_{i}π^{i} is a semiunit presentation of x over Γ_{[r,r]}. Then ∑_{i} u_{i}π^{i} converges in Γ_{I}; in particular, x ∈ Γ_{I}.

Proof. By applying Lemma 2.5.3(a) to ∑_{i=−N}^{N} u_{i}π^{i} and using continuity, we deduce that w_{r}(u_{i}π^{i}) ≥ min_{n≤i}{v_{n,r}(x)}. For s ∈ I with s ≥ r, we have w_{s}(u_{i}π^{i}) ≥ (s/r)w_{r}(u_{i}π^{i}) + (s/r − 1)(−i), so w_{s}(u_{i}π^{i}) → ∞ as i → −∞. On the other hand, for s ∈ I with s < r, we have

w_{s}(u_{i}π^{i}) = (s/r)w_{r}(u_{i}π^{i}) + (1 − s/r)i

≥ (s/r) min_{n≤i}{v_{n,r}(x)} + (1 − s/r)i

= (s/r) min_{n≤i}{(r/s)v_{n,s}(x) + (1 − r/s)n} + (1 − s/r)i

= min_{n≤i}{v_{n,s}(x) + (s/r − 1)(n − i)}

≥ min_{n≤i}{v_{n,s}(x)};

by the hypothesis that v_{n,s}(x) → ∞ as n → ±∞, we have w_{s}(u_{i}π^{i}) → ∞ as i → −∞ also in this case. We conclude that ∑_{i<0} u_{i}π^{i} converges in Γ_{I}; put y = x − ∑_{i<0} u_{i}π^{i}. Then ∑_{i=0}^{∞} u_{i}π^{i} converges to y under w_{r}, hence also π-adically. By Lemma 2.5.3(b), ∑_{i=0}^{∞} u_{i}π^{i} converges in Γ_{r}, so we have x ∈ Γ_{I}, as desired.

One then has the following variant of [19, Proposition 3.14].

Proposition 2.5.5. For I a subinterval of [0, r_{0}) bounded away from r_{0}, every x ∈ Γ_{I} admits a semiunit presentation.

Proof. We first verify that for r ∈ (0, r_{0}), every element of Γ_{r} admits a semiunit presentation. Given x ∈ Γ_{r}, we can construct a sum ∑_{i} u_{i}π^{i} converging π-adically to x, in which each u_{i} is a semiunit. By Lemma 2.5.3(b), this sum actually converges under w_{s} for each s ∈ [0, r], hence yields a semiunit presentation. We now proceed to the general case; by Lemma 2.5.4, it is enough to treat the case I = [r, r]. Choose a sum ∑_{i=0}^{∞} x_{i} converging to x in Γ_{[r,r]}, with each x_{i} ∈ Γ_{r}[π^{−1}]. We define elements y_{il} ∈ Γ_{r}[π^{−1}] for i ∈ ℤ and l ≥ 0, such that for each l, there are only finitely many i with y_{il} ≠ 0, as follows. By the vanishing condition on the y_{il}, x_{0} + ⋯ + x_{l} − ∑_{j<l} ∑_{i} y_{ij}π^{i} belongs to Γ_{r}[π^{−1}] and so admits a semiunit presentation ∑_{i} u_{i}π^{i} by virtue of the previous paragraph. For each i with w_{r}(u_{i}π^{i}) < w_{r}(x_{l+1}) (of which there are only finitely many), put y_{il} = u_{i}; for all other i, put y_{il} = 0. Then

w_{r} (x_{0} + ⋯ + x_{l} − ∑_{j≤l} ∑_{i} y_{ij}π^{i}) ≥ w_{r}(x_{l+1}).

In particular, the doubly infinite sum ∑_{i,l} y_{il}π^{i} converges to x under w_{r}. If we set z_{i} = ∑_{l} y_{il}, then the sum ∑_{i} z_{i}π^{i} converges to x under w_{r}. Note that whenever y_{il} ≠ 0, w_{r}(x_{l}) ≤ w_{r}(y_{il}π^{i}) by Lemma 2.5.3, whereas w_{r}(y_{il}π^{i}) < w_{r}(x_{l+1}) by construction. Thus for any fixed i, the values of w_{r}(y_{il}π^{i}), taken over all l such that y_{il} ≠ 0, form a strictly increasing sequence. Since each such y_{il} is a unit in Γ_{r_{0}}, we have w_{r_{0}}(y_{il}π^{i}) = (r_{0}/r)w_{r}(y_{il}π^{i}) + (1 − r_{0}/r)i; hence the values of w_{r_{0}}(y_{il}π^{i}) also form an increasing sequence. Consequently, the sum ∑_{l} y_{il} converges in Γ_{r_{0}} (not just under w_{r}) and its limit z_{i} is a semiunit. Thus ∑_{i} z_{i}π^{i} is a semiunit presentation of x over Γ_{[r,r]}, as desired.

Corollary 2.5.6. For r ∈ (0, r_{0}) and x ∈ Γ_{[r,r]}, we have x ∈ Γ_{r} if and only if v_{n}(x) = ∞ for all n < 0.

Proof. If x ∈ Γ_{r}, then v_{n}(x) = ∞ for all n < 0. Conversely, suppose that v_{n}(x) = ∞ for all n < 0. Apply Proposition 2.5.5 to produce a semiunit presentation x = ∑_{i} u_{i}π^{i}. Suppose there exists j < 0 such that u_{j} ≠ 0; pick such a j minimizing w_{r}(u_{j}π^{j}). Then v_{j,n}(x) = w_{r}(u_{j}π^{j}) ≠ ∞, contrary to assumption. Hence u_{j} = 0 for j < 0, and so x = ∑_{i=0}^{∞} u_{i}π^{i} ∈ Γ_{r}.

Corollary 2.5.7. Let I ⊆ J be subintervals of [0, r_{0}) bounded away from r_{0}. Suppose x ∈ Γ_{I} has the property that for each s ∈ J, v_{n,s}(x) → ∞ as n → ±∞. Then x ∈ Γ_{J}.

Proof. Produce a semiunit presentation of x over Γ_{J} using Proposition 2.5.5, then apply Lemma 2.5.4.

The numerical criterion provided by Corollary 2.5.7 in turn implies a number of results that are evident in the case of K perfect (using Teichmüller presentations).

Corollary 2.5.8. For K ⊆ K′ an extension of complete fields such that Γ^{K} and Γ^{K′} have enough r_{0}-units, and I ⊆ J ⊆ [0, r_{0}) bounded away from r_{0}, we have

Γ_{I}^{K} ∩ Γ_{J}^{K′} = Γ_{J}^{K}.

Corollary 2.5.9. Let I = [a, b] and J = [c, d] be subintervals of [0, r_{0}) bounded away from r_{0} with a ≤ c ≤ b ≤ d. Then the intersection of Γ_{I} and Γ_{J} within Γ_{I∩J} is equal to Γ_{I∪J}. Moreover, any x ∈ Γ_{I∩J} with w_{s}(x) > 0 for s ∈ I ∩ J can be written as y + z with y ∈ Γ_{I}, z ∈ Γ_{J}, and

w_{s}(y) ≥ (s/c)w_{c}(x)  (s ∈ [a, c])
w_{s}(z) ≥ (s/b)w_{b}(x)  (s ∈ [b, d])
min{w_{s}(y), w_{s}(z)} ≥ w_{s}(x)  (s ∈ [c, b]).

Proof. The first assertion follows from Corollary 2.5.7. For the second assertion, apply Proposition 2.5.5 to obtain a semiunit presentation x = ∑ u_{i}π^{i}. Put y = ∑_{i≤0} u_{i}π^{i} and z = ∑_{i>0} u_{i}π^{i}; these satisfy the claimed inequalities.

Remark 2.5.10. The notion of a semiunit presentation is similar to that of a “semiunit decomposition” as in [19], but somewhat less intricate. In any case, we will have only limited direct use for semiunit presentations; we will mostly exploit them indirectly, via their role in proving Lemma 2.5.11 below.

Lemma 2.5.11. Let I be a closed subinterval of [0, r] for some r ∈ (0, r_{0}), and suppose x ∈ Γ_{I}. Then there exists y ∈ Γ_{r} such that

w_{s}(x − y) ≥ min_{n<0}{v_{n,s}(x)} (s ∈ I).

Proof. Apply Proposition 2.5.5 to produce a semiunit presentation ∑_{i} u_{i}π^{i} of x. Then we can choose m > 0 such that w_{s}(u_{i}π^{i}) > min_{n<0}{v_{n,s}(x)} for s ∈ I and i > m. Put y = ∑_{i=0}^{m} u_{i}π^{i}; then the desired inequality follows from Lemma 2.5.3(a).

Corollary 2.5.12. A nonzero element x of Γ_{I} is a unit in Γ_{I} if and only if it has no slopes; if I = (0, r], this happens if and only if x is a unit in Γ_{r}[π^{−1}].

Proof. If x is a unit in Γ_{I}, it has no slopes by Lemma 2.4.7. Conversely, suppose that x has no slopes; then there exists a single m which minimizes v_{m,s}(x) for all s ∈ I. Without loss of generality we may assume that m = 0; we may then apply Lemma 2.5.11 to produce y ∈ Γ_{r} such that w_{s}(x − y) ≥ min_{n<0}{v_{n,s}(x)} for all s ∈ I. Since Γ has enough r-units, we can choose a unit z ∈ Γ_{r} such that w(y − z) > 0; then w_{s}(1 − xz^{−1}) > 0 for all s ∈ I. Hence the series ∑_{i=0}^{∞}(1 − xz^{−1})^{i} converges in Γ_{I}, and its limit u satisfies uxz^{−1} = 1. This proves that x is a unit. In case I = (0, r], x has no slopes if and only if there is a unique m which minimizes v_{m,s}(x) for all s ∈ (0, r]; this is only possible if v_{n}(x) = ∞ for n < m. By Corollary 2.5.6, this implies x ∈ Γ_{r}[π^{−1}]; by the same argument, x^{−1} ∈ Γ_{r}[π^{−1}].

2.6 Principality

In Remark 2.3.8, the annulus of which Γ_{r}^{naive} is the rigid of rigid analytic functions is affinoid (in the sense of Berkovich in case the endpoints are not rational) and one-dimensional, and so Γ_{r}^{naive} is a principal ideal domain. This can be established more generally. Before proceeding further, we mention a useful “positioning lemma”, which is analogous to but not identical with [19, Lemma 3.24].

Lemma 2.6.1. For r ∈ (0, r_{0}) and x ∈ Γ_{[r,r]} nonzero, there exists a unit u ∈ Γ_{r_{0}} and an integer i such that, if we write y = uπ^{i}x, then:

(a) w_{r}(y) = 0;

(b) v_{0}(y − 1) > 0;

(c) v_{n,r}(y) > 0 for n < 0.

Proof. Define i to be the largest integer minimizing v_{−i,r}(x). Apply Lemma 2.5.11 to find z ∈ Γ_{r} such that w_{r}(π^{i}x − z) ≥ min_{n<0}{v_{n,r}(π^{i}x)}. Since Γ has enough r_{0}-units, we can choose a unit u of Γ_{r_{0}} such that u^{−1} ≡ z (mod π); then u and i have the desired properties.

Definition 2.6.2. For x ∈ Γ_{r} nonzero, define the height of x as the largest n such that w_{r}(x) = v_{n,r}(x); it can also be described as the p-adic valuation of x plus the total multiplicity of x. By convention, we say 0 has height −∞.

Lemma 2.6.3 (Division algorithm). For r ∈ (0, r_{0}) and x, y ∈ Γ_{r} with x nonzero, there exists z ∈ Γ_{r} such that y − z is divisible by x, and z has height less than that of x. Moreover, we can ensure that w_{r}(z) ≥ w_{r}(y).

Proof. Let m be the height of x. Apply Proposition 2.5.5 to choose a semiunit presentation ∑_{i=0}^{∞} u_{i}π^{i} of x, and put x′ = x − ∑_{i=0}^{m−1} u_{i}π^{i}; then x′π^{−m} is a unit in Γ_{r}, and by Lemma 2.5.3,

w_{r}(x − x′) ≥ w_{r}(x) + (1 − r/r_{0}).

Define a sequence {y_{l}}_{l=0}^{∞} as follows. Put y_{0} = y. Given y_{l} with y_{l} − y divisible by x and w_{r}(y_{l}) ≥ w_{r}(y), if y_{l} has height less than m, we may take z = y_{l} and be done with the proof of the lemma. So we may assume that y_{l} has height at least m, which means that min_{n}{v_{n,r}(y_{l})} is achieved by at least one n ≥ m. Pick y′_{l} ∈ Γ_{r_{0}} with w_{r}(y′_{l} −y_{l}) ≥ w_{r}(y_{l})+(1−r/r_{0}), and apply Lemma 2.5.11 to y′_{l}π^{−m} to produce z_{l} ∈ Γ_{r_{0}} such that w_{r_{0}}(z_{l}−y′_{l}π^{−m}) ≥ min_{n<0}{v_{n,r_{0}}(y′_{l}π^{−m})}. Put

y_{l+1} = y_{l} − z_{l}(π^{m}/x′)x
= (y_{l} − y′_{l}) + (y′_{l} − z_{l}π^{m}) + z_{l}π^{m}(1 − x′/x).

By construction, we have w_{r}(y_{l} − y′_{l}) ≥ w_{r}(y_{l}) + (1 − r/r_{0}) and w_{r}(z_{l}π^{m}(1 − x′/x)) ≥ w_{r}(y_{l}) + (1 − r/r_{0}). Moreover, for n ≥ m, we have

v_{n,r}(y′_{l} − z_{l}π^{m}) = (r/r_{0})v_{n,r_{0}}(y′_{l} − z_{l}π^{m}) + (1 − r/r_{0})n

≥ (r/r_{0})w_{r_{0}}(y′_{l} − z_{l}π^{m}) + (1 − r/r_{0})m

≥ (r/r_{0}) min_{j<m}{v_{j,r_{0}}(y′_{l})} + (1 − r/r_{0})m

= min_{j<m}{v_{j,r}(y′_{l}) + (1 − r/r_{0})(m − j)}

≥ min_{j<m}{v_{j,r}(y′_{l})} + (1 − r/r_{0})

≥ w_{r}(y′_{l}) + (1 − r/r_{0}) = w_{r}(y_{l}) + (1 − r/r_{0}).

It follows that for n ≥ m, we have v_{n,r}(y_{l+1}) ≥ w_{r}(y_{l}) + (1 − r/r_{0}). We may assume that y_{l+1} also has height at least m, in which case w_{r}(y_{l+1}) ≥ w_{r}(y_{l}) + (1 − r/r_{0}). Hence (unless the process stops at some finite l, in which case we already know that we win) the y_{l} converge to zero under w_{r}, and

y = x ∑_{l=0}^{∞}(y_{l} − y_{l+1})/x ∈ Γ_{r}

is divisible by x, so we may take z = 0.

Remark 2.6.4. Note how we used the fact that Γ^{K} has enough r_{0}-units, not just enough r-units. Also, note that the discreteness of the valuation on K was essential to ensuring that the sequence {y_{l}} converges to zero.

This division algorithm has the usual consequence.

Proposition 2.6.5. For r ∈ (0, r_{0}), Γ_{r} is a principal ideal domain.

Proof. Let J be a nonzero ideal of Γ_{r}, and pick x ∈ J of minimal height. Then for any y ∈ J, apply Lemma 2.6.3 to produce z of height less than x with y − z divisible by x. Then z ∈ J, so we must have z = 0 by the minimality in the choice of x. In other words, every y ∈ J is divisible by x, as claimed.

Remark 2.6.6. Here is one of the roadblocks mentioned in Remark 2.3.10: if O is not discretely valued, then it is not even a PID itself, so the analogue of Γ_{r} cannot be one either.

To extend Proposition 2.6.5 to more Γ_{I}, we use the following factorization lemma (compare [19, Lemma 3.25]). We will refine this lemma a bit later; see Lemma 2.9.1.

Lemma 2.6.7. For I = [r′, r] ⊆ [0, r_{0}) and x ∈ Γ_{I}, there exists a unit u of Γ_{I} such that ux ∈ Γ_{r}, and all of the slopes of ux in [0, r] belong to I.

Proof. By applying Lemma 2.6.1, we may reduce to the case where w_{r′}(x) = 0, v_{0}(x − 1) > 0, and v_{n,r′}(x) > 0 for n < 0; then for n < 0, we must have v_{n}(x) > 0 and so v_{n,s}(x) > 0 for all s ∈ I. Put

c = min_{s∈I}{min_{n≤0}{v_{n,s}(x − 1)}} > 0.

Define the sequence u_{0}, u_{1}, … of units of Γ_{I} as follows. First set u_{0} = 1. Given u_{l} such that min_{n≤0}{v_{n,s}(u_{l}x − 1)} ≥ c for all s ∈ I, apply Lemma 2.5.11 to produce y_{l} ∈ Γ_{r} such that w_{s}(y_{l} − u_{l}x) ≥ min_{n<0}{v_{n,s}(u_{l}x)} for all s ∈ I. We may thus take u_{l+1} = u_{l}(1 − y_{l} + u_{l}x), because w_{s}(y_{l} − u_{l}x) ≥ c; moreover, for n < 0,

v_{n,r′}(u_{l+1}x) = v_{n,r′}(y_{l} − u_{l+1}x)

= v_{n,r′}((y_{l} − u_{l}x)(1 − u_{l}x))

≥ min_{m}{v_{m,r′}(y_{l} − u_{l}x) + v_{n−m,r′}(1 − u_{l}x)}.

This last minimum is at least min_{n<0}{v_{n,r′}(u_{l}x)}. Moreover, if it is ever less than min_{n<0}{v_{n,r′}(u_{l}x)} + c, then the smallest value of n achieving min_{n}{v_{n,r′}(u_{l+1}x)} is strictly greater than the smallest value of n achieving min_{n}{v_{n,r′}(u_{l}x)} (since in that case, terms in the last minimum above with m ≤ 0 cannot affect the minimum of the left side over all n < 0).

In other words, for every l, there exists l′ > l such that

min_{n<0}{v_{n,r′}(u_{l′}x)} ≥ min_{n<0}{v_{n,r′}(u_{l}x)} + c.

Hence in case s = r′, we have min_{n<0}{v_{n,s}(u_{l}x)} → ∞ as l → ∞; consequently, the same also holds for s ∈ I. It follows that the sequence {u_{l}} converges to a limit u ∈ Γ_{I}, and that v_{n,s}(ux) = ∞ for n < 0, so that ux ∈ Γ_{r} by Corollary 2.5.6. Moreover, by construction, min_{n}{v_{n,r′}(ux)} is achieved by n = 0, so all of the slopes of ux are at least r′.

Proposition 2.6.8. Let I be a closed subinterval of [0, r_{0}). Then Γ_{I} is a principal ideal domain.

Proof. Put I = [r′, r], and let J be a nonzero ideal of Γ_{I}. By Lemma 2.6.7, each element x of J can be written (nonuniquely) as a unit u of Γ_{I} times an element y of Γ_{r}. Let J′ be the ideal of Γ_{r} generated by all such y; by Proposition 2.6.8, J′ is principal, generated by some z. Since J′ ⊆ J ∩ Γ_{r}, we have z ∈ J; on the other hand, each x ∈ J has the form uy with u ∈ Γ_{I} and y ∈ Γ_{r}, and y is a multiple of z in Γ_{r}, so x is a multiple of z in Γ_{I}. Hence z generates J, as desired.

Remark 2.6.9. Proposition 2.6.8 generalizes Lazard’s [28, Corollaire de Proposition 4].

2.7 Matrix approximations and factorizations

We need a matrix approximation lemma similar to [19, Lemma 6.2]; it is in some sense a matricial analogue of Lemma 2.6.1.

Lemma 2.7.1. Let I be a closed subinterval of [0, r] for some r ∈ (0, r_{0}), and let M be an invertible n × n matrix over Γ_{I}. Then there exists an invertible n × n matrix U over Γ_{r}[π^{−1}] such that w_{s}(MU − I_{n}) > 0 for s ∈ I. Moreover, if w_{s}(det(M) − 1) > 0, we can ensure that det(U) = 1.

Proof. By applying Lemma 2.6.1 to det(M) (and then multiplying a single row of U by the resulting unit), we can ensure that w_{s}(det(M) − 1) > 0 for s ∈ I. With this extra hypothesis, we proceed by induction on n. Let C_{i} denote the cofactor of M_{ni} in M, so that det(M) = ∑_{i=1}^{n} C_{i}M_{ni}, and in fact C_{i} = (M^{−1})_{in} det(M). Put α_{i} = det(M)^{−1}M_{ni}, so that ∑_{i=1}^{n} α_{i}C_{i} = 1. Choose β_{1}, …, β_{n−1}, β′_{n} ∈ Γ_{r}[π^{−1}] such that for s ∈ I and i = 1, …, n − 1,

w_{s}(β_{i} − α_{i}) > −w_{s}(C_{i}) w_{s}(β′_{n} − α_{n}) > −w_{s}(C_{n}).

Note that for c ∈ O with w(c) sufficiently large, β_{n} = β′_{n} + c satisfies w_{s}(β_{n} − α_{n}) > −w_{s}(C_{n}) for s ∈ I. Moreover, by Proposition 2.6.8, we can find γ ∈ Γ_{r}[π^{−1}] generating the ideal generated by β_{1}, …, β_{n−1}; then the β′_{n} + c are pairwise coprime for different c ∈ O, so only finitely many of them can have a nontrivial common factor with γ. In particular, for w(c) sufficiently large, β_{1}, …, β_{n} generate the unit ideal in Γ_{r}[π^{−1}]. With β_{1}, …, β_{n} so chosen, we can choose a matrix A over Γ_{r}[π^{−1}] of determinant 1 such that A_{ni} = β_{i} for i = 1, …, n (because Γ_{r}[π^{−1}] is a PID, again by Proposition 2.6.8). Put M′ = MA^{−1}, and let C′_{n} be the cofactor of M′_{nn} in M′. Then

C′_{n} = (AM^{−1})_{nn} det(M)

= ∑_{i=1}^{n} A_{ni}(M^{−1})_{in} det(M) = ∑_{i=1}^{n} β_{i}C_{i},

so that

C′_{n} = 1 + ∑_{i=1}^{n}(β_{i} − α_{i})C_{i}

and so w_{s}(C′_{n} − 1) > 0 for s ∈ I. In particular, C′_{n} is a unit in Γ_{I}. Apply the induction hypothesis to the upper left (n − 1) × (n − 1) submatrix of M′, and extend the resulting (n − 1) × (n − 1) matrix V to an n × n matrix by setting V_{ni} = V_{in} = 0 for i = 1, …, n − 1 and V_{nn} = 1. Then we have det(M′V) = det(M), so w_{s}(det(M′V) − 1) > 0 for s ∈ I, and

w_{s}((M′V − I_{n})_{ij}) > 0 (i = 1, …, n − 1; j = 1, …, n − 1; s ∈ I).

We now perform an “approximate Gaussian elimination” over Γ_{I} to transform M′V into a new matrix N with w_{s}(N − I_{n}) > 0 for s ∈ I. First, define a sequence of matrices {X^{(h)}}_{h=0}^{∞} by X^{(0)} = M′V and

X_{ij}^{(h+1)} = \begin{cases}X_{ij}^{(h)} & i < n \\ X_{nj}^{(h)} − ∑_{m=1}^{n−1} X_{nm}^{(h)}X_{mj}^{(h)} & i = n; \end{cases}

note that X^{(h+1)} is obtained from X^{(h)} by subtracting X_{nm}^{(h)} times the m-th row from the n-th row for m = 1, …, n−1 in succession. At each step, for each s ∈ I, min_{1≤j≤n−1}{w_{s}(X_{nj}^{(h)})} increases by at least min_{1≤i,j≤n−1}{w_{s}((M′V − I_{n})_{ij})}; the latter is bounded away from zero over all s ∈ I, because I is closed and w_{s}(x) is a continuous function of s. Thus for h sufficiently large, we have

w_{s}(X_{nj}^{(h)}) > max{0, max_{1≤i≤n−1}{−w_{s}(X_{in}^{(h)})}} (s ∈ I; j = 1, …, n − 1).

Pick such an h and set X = X^{(h)}; note that det(X) = det(M′V), so w_{s}(det(X) − 1) > 0 for s ∈ I. For s ∈ I,

w_{s}((X − I_{n})_{ij}) > 0  (i = 1, …, n; j = 1, …, n − 1)
w_{s}(X_{in}X_{nj}) > 0  (i = 1, …, n − 1; j = 1, …, n − 1)

and hence also w_{s}(X_{nn} − 1) > 0.

Next, we perform “approximate backsubstitution”. Define a sequence of matrices {W^{(h)}}_{h=0}^{∞} by setting W^{(0)} = X and

W_{ij}^{(h+1)} = \begin{cases}W_{ij}^{(h)} − W_{in}^{(h)}W_{nj}^{(h)} & i < n \\ W_{ij}^{(h)} & i = n; \end{cases}

note that W^{(h+1)} is obtained from W^{(h)} by subtracting W_{in}^{(h)} times the n-th row from the i-th row for i = 1, …, n − 1. At each step, for s ∈ I, w_{s}(W_{in}^{(h)}) increases by at least w_{s}(X_{nn} − 1); again, the latter is bounded away from zero over all s ∈ I because I is closed and w_{s}(x) is continuous in s. Thus for h sufficiently large,

w_{s}(W_{in}^{(h)}) > 0 (s ∈ I; 1 ≤ i ≤ n − 1).

Pick such an h and set W = W_{h}; then w_{s}(W − I_{n}) > 0 for s ∈ I. (Note that the inequality w_{s}(X_{in}X_{nj}) > 0 for i = 1, …, n − 1 and j = 1, …, n − 1 ensures that the second set of row operations does not disturb the fact that w_{s}(W_{ij}^{(h)}) > 0 for i = 1, …, n − 1 and j = 1, …, n − 1.) To conclude, note that by construction, (M′V)^{−1}W is a product of elementary matrices over Γ_{I}, each consisting of the diagonal matrix plus one off-diagonal entry. By suitably approximating the off-diagonal entry of each matrix in the product by an element of Γ_{r}, we get an invertible matrix Y over Γ_{r} such that w_{s}(M′V Y − I_{n}) > 0 for s ∈ I. We may thus take U = A^{−1}V Y to obtain the desired result.

We also need a factorization lemma in the manner of [19, Lemma 6.4].

Lemma 2.7.2. Let I = [a, b] and J = [c, d] be subintervals of [0, r_{0}) bounded away from r_{0}, with a ≤ c ≤ b ≤ d, and let M be an n × n matrix over Γ_{I∩J} with w_{s}(M − I_{n}) > 0 for s ∈ I ∩ J. Then there exist invertible n × n matrices U over Γ_{I} and V over Γ_{J} such that M = UV.

Proof. We construct sequences of matrices U_{l} and V_{l} over Γ_{I} and Γ_{J}, respectively, with

w_{s}(U_{l} − I_{n}) ≥ (s/c)w_{c}(M − I_{n})  (s ∈ [a, c])
w_{s}(V_{l} − I_{n}) ≥ (s/b)w_{b}(M − I_{n})  (s ∈ [b, d])
min{w_{s}(U_{l} − I_{n}), w_{s}(V_{l} − I_{n})} ≥ w_{s}(M − I_{n})  (s ∈ [c, b])
w_{s}(U_{l}^{−1}MV_{l}^{−1} − I_{n}) ≥ 2^{l}w_{s}(M − I_{n})  (s ∈ [c, b]),

as follows. Start with U_{0} = V_{0} = I_{n}. Given U_{l}, V_{l}, put M_{l} = U_{l}^{−1}MV_{l}^{−1}. Apply Corollary 2.5.9 to split M_{l} − I_{n} = Y_{l} + Z_{l} with Y_{l} defined over Γ_{I}, Z_{l} defined over Γ_{J}, and

w_{s}(Y_{l}) ≥ (s/c)w_{c}(M_{l} − I_{n}) ≥ (s/c)w_{c}(M − I_{n})  (s ∈ [a, c])
w_{s}(Z_{l}) ≥ (s/b)w_{b}(M_{l} − I_{n}) ≥ (s/b)w_{b}(M − I_{n})  (s ∈ [b, d])
min{w_{s}(Y_{l}), w_{s}(Z_{l})} ≥ w_{s}(M_{l} − I_{n}) ≥ 2^{l}w_{s}(M − I_{n})  (s ∈ [c, b]).

Put U_{l+1} = U_{l}(I+Y_{l}) and V_{l+1} = (I+Z_{l})V_{l}; then one calculates that w_{s}(M_{l+1}− I_{n}) ≥ 2^{l+1}w_{s}(M − I_{n}) for s ∈ [c, b]. We deduce that the sequences {U_{l}} and {V_{l}} each converge under w_{s} for s ∈ [c, b], and the limits U and V satisfy min{w_{s}(U −I_{n}), w_{s}(V −I_{n})} ≥ w_{s}(M −I_{n}) for s ∈ [c, b], and M = UV. However, the subset x ∈ Γ_{I} on which

w_{s}(x) ≥ \begin{cases}(s/c)w_{c}(M − I_{n}) & s ∈ [a, c] \\ w_{s}(M − I_{n}) & s ∈ [c, b] \end{cases}

is complete under any one w_{s}, so U has entries in Γ_{I} and w_{s}(U − I_{n}) ≥ (s/c)w_{c}(M − I_{n}) for s ∈ [a, c]. Similarly, V has entries in Γ_{J} and w_{s}(V − I_{n}) ≥ (s/b)w_{b}(M − I_{n}) for s ∈ [b, d]. In particular, U and V are invertible over Γ_{I} and Γ_{J}, and M = UV, yielding the desired factorization.

2.8 Vector bundles

Over an open rigid analytic annulus, one specifies a vector bundle by specifying a vector bundle (necessarily freely generated by global sections) on each closed subannulus and providing glueing data; if the field of coefficients is spherically complete, it can be shown that the result is again freely generated by global sections. Here we generalize the discretely valued case of this result to analytic rings. (For rank 1, the annulus statement can be extracted from results of [28]; the general case can be found in [21, Theorem 3.4.3]. In any case, it follows from our Theorem 2.8.4 below.)

Definition 2.8.1. Let I be a subinterval of [0, r_{0}) bounded away from r_{0}, and let S be a collection of closed subintervals of I closed under finite intersections, whose union is all of I. Define an S-vector bundle over Γ_{I} to be a collection consisting of one finite free Γ_{J}-module M_{J} for each J ∈ S, plus isomorphisms

ι_{J_{1},J_{2}} : M_{J_{1}} ⊗_{Γ_{J_{1}}} Γ_{J_{2}} ≅ M_{J_{2}}

whenever J_{2} ⊆ J_{1}, satisfying the compatibility condition ι_{J_{2},J_{3}} ∘ ι_{J_{1},J_{2}} = ι_{J_{1},J_{3}} whenever J_{3} ⊆ J_{2} ⊆ J_{1}. These may be viewed as forming a category in which a morphism between the collections {M_{J}} and {N_{J}} consists of a collection of morphisms M_{J} → N_{J} of Γ_{J}-modules which commute with the isomorphisms ι_{J_{1},J_{2}}.

This definition obeys the analogue of the usual glueing property for coherent sheaves on an affinoid space (i.e., the theorem of Kiehl-Tate).

Lemma 2.8.2. Let I be a subinterval of [0, r_{0}) bounded away from r_{0}, and let S_{1} ⊆ S_{2} be two collections of closed subintervals of I as in Definition 2.8.1. Then the natural functor from the category of S_{2}-vector bundles over Γ_{I} to S_{1}-vector bundles over Γ_{I} is an equivalence.

Proof. We define a quasi-inverse functor as follows. Given J ∈ S_{2}, by compactness we can choose J_{1}, …, J_{m} ∈ S_{1} with J ⊆ J′ = J_{1} ∪ ⋯ ∪ J_{m}; it is enough to consider the case where m = 2 and J_{1} ∩ J_{2} ≠ ∅, as we can repeat the construction to treat the general case. Define M_{J′} to be the Γ_{J′}-submodule of M_{J_{1}} ⊕ M_{J_{2}} consisting of those pairs (v_{1}, v_{2}) such that

ι_{J_{1},J_{1}∩J_{2}}(v_{1}) = ι_{J_{2},J_{1}∩J_{2}}(v_{2}).

Let v_{1}, …, v_{n} be a basis of M_{J_{1}} and let w_{1}, …, w_{n} be a basis of M_{J_{2}}. Then there is an invertible n × n matrix A over M_{J_{1}∩J_{2}} given by w_{j} = ∑_{i} A_{ij}v_{i}. By Lemma 2.7.2, A can be factored as UV, where U is invertible over Γ_{J_{1}} and V is invertible over Γ_{J_{2}}. Set

e_{j} = (∑_{i} U_{ij}v_{i}, ∑_{i}(V^{−1})_{ij}w_{i});

then e_{1}, …, e_{n} form a basis of M_{J′}, since the first components form a basis of M_{J_{1}}, the second components form a basis of M_{J_{2}}, and the intersection of Γ_{J_{1}} and Γ_{J_{2}} within Γ_{J_{1}∩J_{2}} equals Γ_{J_{1}∪J_{2}} (by Corollary 2.5.9). In particular, the natural maps M_{J′} ⊗_{Γ_{J′}} Γ_{J_{i}} → M_{J_{i}} for i = 1, 2 are isomorphisms. We may thus set M_{J} = M_{J′} ⊗_{Γ_{J′}} Γ_{J}.

Definition 2.8.3. By Lemma 2.8.2, the category of S-vector bundles over Γ_{I} is canonically independent of the choice of S. We thus refer to its elements simply as vector bundles over Γ_{I}.

It follows that for I closed, any vector bundle over Γ_{I} is represented by a free module; a key result for us is that one has a similar result over Γ_{an,r}.

Theorem 2.8.4. For r ∈ (0, r_{0}), the natural functor from finite free Γ_{an,r}- modules to vector bundles over Γ_{an,r} = Γ_{(0,r]} is an equivalence.

Proof. To produce a quasi-inverse functor, let J_{1} ⊆ J_{2} ⊆ ⋯ be an increasing sequence of closed intervals with right endpoints r, whose union is (0, r]; for ease of notation, write Γ_{i} for Γ_{J_{i}}. We can specify a vector bundle over Γ_{i} by specifying a finite free Γ_{i}-module E_{i} for each i, plus identifications E_{i+1} ⊗_{Γ_{i+1}} Γ_{i} ≅ E_{i}. Choose a basis v_{1,1}, …, v_{1,n} of E_{1}. Given a basis v_{i,1}, …, v_{i,n} of E_{i}, we choose a basis v_{i+1,1}, …, v_{i+1,n} of E_{i+1} as follows. Pick any basis e_{1}, …, e_{n} of E_{i+1}, and define an invertible n × n matrix M_{i} over Γ_{i} by writing e_{l} = ∑_{j}(M_{i})_{jl}v_{i,j}. Apply Lemma 2.7.1 to produce an invertible n × n matrix U_{i} over Γ_{r} such that w_{s}(M_{i}U_{i} −I_{n}) > 0 for s ∈ J_{i}. Apply Lemma 2.5.11 to produce an n×n matrix V_{i} over Γ_{r} with w_{s}(M_{i}U_{i}−I_{n}−V_{i}) ≥ min_{m<0}{v_{m,s}(M_{i}U_{i}−I_{n})} for s ∈ J_{i}; then w_{r}(V_{i}) > 0, so I_{n} + V_{i} is invertible over Γ_{r}. Put W_{i} = M_{i}U_{i}(I_{n} + V_{i})^{−1}, and define v_{i+1,1}, …, v_{i+1,n} by v_{i+1,l} = ∑_{j}(W_{i})_{jl}v_{i,j}; these form another basis of E_{i+1} because we changed basis over Γ_{r}.

If we write J_{i} = [r_{i}, r], then for any fixed s ∈ (0, r], we have

w_{s}(W_{i} − I_{n}) = w_{s}((M_{i}U_{i} − I_{n} − V_{i})(I_{n} + V_{i})^{−1})

≥ min_{m<0}{v_{m,s}(M_{i}U_{i} − I_{n})}

= min_{m<0}{(s/r_{i})v_{m,r_{i}}(M_{i}U_{i} − I_{n}) + (1 − s/r_{i})m}

≥ min_{m<0}{v_{m,r_{i}}(M_{i}U_{i} − I_{n})} + (s/r_{i} − 1)

> (s/r_{i} − 1),

which tends to ∞ as i → ∞. Thus the product W_{1}W_{2} ⋯ converges to an invertible matrix W over Γ_{an,r}, and the basis e_{1}, …, e_{n} of E_{1} defined by

e_{l} = ∑_{j} W_{jl}v_{1,j}

actually forms a basis of each E_{i}. Hence the original vector bundle can be reconstructed from the free Γ_{an,r}-module generated by e_{1}, …, e_{n}; this yields the desired quasi-inverse.

Corollary 2.8.5. For r ∈ (0, r_{0}), let M be a finite free Γ_{an,r}-module. Then every closed submodule of M is free; in particular, every closed ideal of Γ_{an,r} is principal.

Proof. A submodule is closed if and only if it gives rise to a sub-vector bundle of the vector bundle associated to M; thus the claim follows from Theorem 2.8.4.

Remark 2.8.6. One might expect that more generally every vector bundle over Γ_{I} is represented by a finite free Γ_{I}-module; we did not verify this.

2.9 The Bézout property

One pleasant consequence of Theorem 2.8.4 is the fact that the ring Γ_{an,r} has the Bézout property, as we verify in this section. We start by refining the conclusion of Lemma 2.6.7 (again, compare [19, Lemma 3.25]).

Lemma 2.9.1. For r, s, s′ ∈ (0, r_{0}) with s′ < s < r, and f ∈ Γ_{[s′,r]}, there exists g ∈ Γ_{r}[π^{−1}] with the following properties.

(a) The ideals generated by f and g in Γ_{[s,r]} coincide.

(b) The slopes of g in [0, r] are all contained in [s, r].

Moreover, any such g also has the following property.

(c) f is divisible by g in Γ_{[s′,r]}.

Proof. By Lemma 2.6.7, we can find a unit u of Γ_{[s′,r]} such that uf ∈ Γ_{r}[π^{−1}] and the slopes of uf in [0, r] are all contained in [s, r]. We may thus take g = uf to obtain at least one g ∈ Γ_{r}[π^{−1}] satisfying (a) and (b); hereafter, we let g be any element of Γ_{r}[π^{−1}] satisfying (a) and (b). Then the multiplicity of each element of [s, r] as a slope of g is equal to its multiplicity as a slope of f. Since Γ_{r}[π^{−1}] is a PID by Proposition 2.6.8, we can find an element h ∈ Γ_{r}[π^{−1}] generating the ideal generated by uf and g in Γ_{r}[π^{−1}]; in particular, the multiplicity of each element of [s, r] as a slope of h is less than or equal to its multiplicity as a slope of g. However, h must also generate the ideal generated by f and g in Γ_{[s,r]}, which is generated already by f alone; in particular, the multiplicity of each element of [s, r] as a slope of f is equal to its multiplicity as a slope of h. We conclude that each element of [s, r] occurs as a slope of f, g, h all with the same multiplicity. Since g only has slopes in [s, r], g/h must be a unit in Γ_{r}[π^{−1}]; hence uf is already divisible by g in Γ_{r}[π^{−1}], so f is divisible by g in Γ_{[s′,r]} as desired.

Lemma 2.9.2. Given r ∈ (0, r_{0}) and x ∈ Γ_{r}[π^{−1}] with greatest slope s_{0} < r, choose r′ ∈ (s_{0}, r). Then for any y ∈ Γ_{r}[π^{−1}] and any c > 0, there exists z ∈ Γ_{r}[π^{−1}] with y −z divisible by x in Γ_{an,r}, such that w_{s}(z) > c for s ∈ [r′, r].

Proof. As in the proof of Lemma 2.6.1, we can find a unit u ∈ Γ_{r} and an integer i such that min_{n}{v_{n,r}(uxπ^{i})} is achieved by n = 0 but not by any n > 0, and that v_{0}(uxπ^{i} − 1) > 0. Since s_{0} < r, in fact min_{n}{v_{n,r}(uxπ^{i})} is only achieved by n = 0, so w_{r}(uxπ^{i} − 1) > 0. Similarly, w_{s}(uxπ^{i} − 1) > 0 for s ∈ [r′, r]; since [r′, r] is a closed interval, we can choose d > 0 such that w_{s}(uxπ^{i} − 1) ≥ d for s ∈ [r′, r]. Now simply take

z = y(1 − uxπ^{i})^{N}

for some integer N with Nd + w_{s}(y) > c for s ∈ [r′, r].

We next introduce a “principal parts lemma” (compare [19, Lemma 3.31]).

Lemma 2.9.3. For r ∈ (0, r_{0}), let I_{1} ⊂ I_{2} ⊂ ⋯ be an increasing sequence of closed subintervals of (0, r] with right endpoints r, whose union is all of (0, r], and put Γ_{i} = Γ_{I_{i}}. Given f ∈ Γ_{an,r} and g_{i} ∈ Γ_{i} such that for each i, g_{i+1} − g_{i} is divisible by f in Γ_{i}, there exists g ∈ Γ_{an,r} such that for each i, g −g_{i} is divisible by f in Γ_{i}.

Proof. Apply Lemma 2.9.1 to produce f_{i} ∈ Γ_{r} dividing f in Γ_{an,r}, such that f and f_{i} generate the same ideal in Γ_{i}, f_{i} only has slopes in I_{i}, and f/f_{i} has no slopes in I_{i}; put f_{0} = 1. By Lemma 2.9.1 again (with s′ varying), f_{i} is divisible by f_{i−1} in Γ_{an,r}, hence also in Γ_{r}[π^{−1}]; put h_{i} = f_{i}/f_{i−1} ∈ Γ_{r}[π^{−1}] and h_{0} = 1. Set x_{0} = 0. Given x_{i} ∈ Γ_{r}[π^{−1}] with x_{i} − g_{i} divisible by f_{i} in Γ_{i}, note that the ideal generated by h_{i+1} and f_{i} in Γ_{r}[π^{−1}] is principal by Proposition 2.6.8. Moreover, any generator has no slopes by Lemma 2.4.7 and so must be a unit in Γ_{r}[π^{−1}] by Corollary 2.5.12. That is, we can find a_{i+1}, b_{i+1} ∈ Γ_{r}[π^{−1}] with a_{i+1}h_{i+1} + b_{i+1}f_{i} = 1. Moreover, by applying Lemma 2.9.2, we may choose a_{i+1}, b_{i+1} with w_{s}(b_{i+1}(g_{i+1} − x_{i})f_{i}) ≥ i for s ∈ I_{i}. (More precisely, apply Lemma 2.9.2 with the roles of x and y therein played by h_{i+1} and b_{i+1}, respectively; this is valid because h_{i+1} has greatest slope less than any element of I_{i}.) Now put x_{i+1} = x_{i} + b_{i+1}(g_{i+1} − x_{i})f_{i}; then x_{i+1} − g_{i} is divisible by f_{i} in Γ_{i}, as then is x_{i+1} − g_{i+1} since g_{i+1} − g_{i} is divisible by f_{i} in Γ_{i}. By Lemma 2.9.1, x_{i+1} − g_{i+1} is divisible by f_{i} also in Γ_{i+1}. Since x_{i+1} − g_{i+1} is also divisible by h_{i+1} in Γ_{i+1}, x_{i+1} − g_{i+1} is divisible by f_{i+1} in Γ_{i+1}. For any given s, we have w_{s}(x_{i+1} − x_{i}) ≥ i for i large, so the x_{i} converge to a limit g in Γ_{an,r}. Since g − g_{i} is divisible by f_{i} in Γ_{i}, it is also divisible by f in Γ_{i}. This yields the desired result.

Remark 2.9.4. The use of the f_{i} in the proof of Lemma 2.9.3 is analogous to the use of “slope factorizations” in [19]. Slope factorizations (convergent products of pure elements converging to a specified element of Γ_{an,r}), which are inspired by the comparable construction in [28], will not be used explicitly here; see [19, Lemma 3.26] for their construction.

We are finally ready to analyze the Bézout property.

Definition 2.9.5. A Bézout ring/domain is a ring/domain in which every finitely generated ideal is principal. Such rings look like principal ideal rings from the point of view of finitely generated modules. For instance:

• Every n-tuple of elements of a Bézout domain which generate the unit ideal is unimodular, i.e., it occurs as the first row of a matrix of determinant 1 [19, Lemma 2.3]. (Beware that [19, Lemma 2.3] is stated for a Bézout ring but is only valid for a Bézout domain.)

• The saturated span of any subset of a finite free module over a Bézout domain is a direct summand [19, Lemma 2.4].

• Every finitely generated locally free module over a Bézout domain is free [19, Proposition 2.5], as is every finitely presented torsion-free module [12, Proposition 4.9].

• Any finitely generated submodule of a finite free module over a Bézout ring is free (straightforward).

Theorem 2.9.6. For r ∈ (0, r_{0}), the ring Γ_{an,r} is a Bézout domain (as then is Γ_{an,con}). More precisely, if J is an ideal of Γ_{an,r}, the following are equivalent.

(a) The ideal J is closed.

(b) The ideal J is finitely generated.

(c) The ideal J is principal.

Proof. Clearly (c) implies both (a) and (b). Also, (a) implies (c) by Theorem 2.8.4. It thus suffices to show that (b) implies (a); by induction, it is enough to check in case J is generated by two nonzero elements x, y. Moreover, we may form the closure of J, find a generator z, and then divide x and y by z; in other words, we may assume that 1 is in the closure of J, and then what we are to show is that 1 ∈ J. Let I_{1} ⊂ I_{2} ⊂ ⋯ be an increasing sequence of closed subintervals of (0, r], with right endpoints r, whose union is all of (0, r]. Then x and y generate the unit ideal in Γ_{i} = Γ_{I_{i}} for each i; that is, we can choose a_{i}, b_{i} ∈ Γ_{i} with a_{i}x + b_{i}y = 1. Note that b_{i+1} − b_{i} is divisible by x in Γ_{i}; by Lemma 2.9.3, we can choose b ∈ Γ_{an,r} with b − b_{i} divisible by x in Γ_{i} for each i. Then by − 1 is divisible by x in each Γ_{i}, hence also in Γ_{an,r} (by Corollary 2.5.7); that is, x and y generate the unit ideal in Γ_{an,r}, as desired. We have thus shown that (a), (b), (c) are equivalent, proving that Γ_{an,r} is a Bézout ring. Since Γ_{an,con} is the union of the Γ_{an,r} for r ∈ (0, r_{0}), it is also a Bézout ring because any finitely generated ideal is generated by elements of some Γ_{an,r}.

Remark 2.9.7. In Lazard’s theory (in which Γ_{I} becomes the ring of rigid analytic functions on the annulus log_{|π|} |u| ∈ I), the implication (b) ⟹ (a) is [28, Proposition 11], and holds without restriction on the coefficient field. The implication (a) ⟹ (b) is equivalent to spherical completeness of the coefficient field [28, Théorème 2]; however, the analogue here would probably require K also to be spherically complete (compare Remark 2.8.6), which is an undesirable restriction. For instance, it would complicate the process of descending the slope filtration in Chapter 6.

3 σ-modules

We now introduce modules equipped with a semilinear endomorphism (σmodules) and study their properties, specifically over Γ_{an,con}. In order to highlight the parallels between this theory and the theory of stable vector bundles (see for instance [33]), we have shaped our presentation along the lines of that of Hartl and Pink [17]; they study vector bundles with a Frobenius structure on a punctured disc over a complete nonarchimedean field of equal characteristic p, and prove results very similar to our results over Γ_{an,con}^{alg}. Beware that our overall sign convention is “arithmetic” and not “geometric”; it thus agrees with the sign conventions of [18] (and of [19]), but disagrees with the sign convention of [17] and with the usual convention in the vector bundle setting.

Remark 3.0.1. We retain all notation from Chapter 2, except that we redefine the term “slope”; see Definition 3.4.1. In particular, K is a field complete with respect to the valuation v_{K}, and Γ^{K} is assumed to have enough r_{0}-units for some r_{0} > 0. Remember that v_{K} is allowed to be trivial unless otherwise specified; that means any result about Γ_{an,con} also applies to Γ[π^{−1}], unless its statement explicitly requires v_{K} to be nontrivial.

3.1 σ-modules

Definition 3.1.1. For a ring R containing O in which π is not a zero divisor, equipped with a ring endomorphism σ, a σ-module over a ring R is a finite locally free R-module M equipped with a map F : σ^{∗}M → M (the Frobenius action) which becomes an isomorphism after inverting π. (Here σ^{∗}M = M ⊗_{R,σ} R; that is, view R as a module over itself via σ, and tensor it with M over R.) We can view M as a left module over the twisted polynomial ring R{σ}; we can also view F as a σ-linear additive endomorphism of M. A homomorphism of σmodules is a module homomorphism equivariant with respect to the Frobenius actions.

Remark 3.1.2. We will mostly consider σ-modules over Bézout rings like Γ_{an,con}, in which case there is no harm in replacing “locally free” by “free” in the definition of a σ-module.

Remark 3.1.3. The category of σ-modules is typically not abelian (unless v_{K} is trivial), because we cannot form cokernels thanks to the requirement that the underlying modules be locally free.

Remark 3.1.4. For any positive integer a, σ^{a} is also a Frobenius lift, so we may speak of σ^{a}-modules. This will be relevant when we want to perform “restriction of scalars” in Section 3.2. However, there is no loss of generality in stating definitions and theorems in the case a = 1, i.e., for σ-modules.

Definition 3.1.5. Given a σ-module M of rank n and an integer c (which must be nonnegative if π^{−1} ∉ R), define the twist M(c) of M by c to be the module M with the Frobenius action multiplied by π^{c}. (Beware that this definition reflects an earlier choice of normalization, as in Remark 2.1.11, and a choice of a sign convention.) If π is invertible in R, define the dual M^{∨} of M to be the σ-module Hom_{R}(M, R) ≅ (∧^{n−1}M) ⊗ (∧^{n}M)^{⊗−1} and the internal hom of M, N as M^{∨} ⊗ N.

Definition 3.1.6. Given a σ-module M over a ring R, let H^{0}(M) and H^{1}(M) denote the kernel and cokernel, respectively, of the map F −1 on M; note that if N is another σ-module, then there is a natural bilinear map H^{0}(M)×H^{1}(N) → H^{1}(M ⊗ N). Given two σ-modules M_{1} and M_{2} over R, put Ext(M_{1}, M_{2}) = H^{1}(M_{1}^{∨} ⊗ M_{2}); a standard homological calculation (as in [17, Proposition 2.4]) shows that Ext(M_{1}, M_{2}) coincides with the Yoneda Ext^{1} in this category. That is, Ext(M_{1}, M_{2}) classifies short exact sequences 0 → M_{2} → M → M_{1} → 0 of σ-modules over R, up to isomorphisms

\xymatrix{0 \ar[r] & M_{2} \ar[r] \ar[d] & M \ar[r] \ar[d] & M_{1} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & M_{2} \ar[r] & M \ar[r] & M_{1} \ar[r] & 0}

which induce the identity maps on M_{1} and M_{2}.

3.2 Restriction of Frobenius

We now introduce two functors analogous to those induced by the “finite maps” in [17, Section 7]. Beware that the analogy is not perfect; see Remark 3.2.2.

Definition 3.2.1. Fix a ring R equipped with an endomorphism σ. For a a positive integer, let [a] : R{σ^{a}} → R{σ} be the natural inclusion homomorphism of twisted polynomial rings. Define the a-pushforward functor [a]_{∗}, from σ-modules to σ^{a}-modules, to be the restriction functor along [a]. Define the a-pullback functor [a]^{∗}, from σ^{a}-modules to σ-modules, to be the extension of scalars functor

M ↦ R{σ} ⊗_{R{σ^{a}}} M.

Note that [a]^{∗} and [a]_{∗} are left and right adjoints of each other. Also, [a]^{∗}[a′]^{∗} = [aa′]^{∗} and [a]_{∗}[a′]_{∗} = [aa′]_{∗}. Furthermore, [a]_{∗}(M(c)) = ([a]_{∗}M)(ac).

Remark 3.2.2. There are some discrepancies in the analogy with [17], due to the fact that there the corresponding map [a] is actually a homomorphism of the underlying ring, rather than a change of Frobenius. The result is that some (but not all!) of the properties of the pullback and pushforward are swapped between here and [17]. For an example of this mismatch in action, see Proposition 3.4.4.

Remark 3.2.3. The functors [a] will ultimately serve to rescale the slopes of a σ-module; using them makes it possible to avoid the reliance in [19, Chapter 4] on making extensions of O. Among other things, this lets us get away with normalizing w in terms of the choice of O, since we will not have to change that choice at any point except in Lemma 5.2.4.

Lemma 3.2.4. For any positive integer a and any integer c, [a]_{∗}[a]^{∗}(R(c)) ≅ R(c)^{⊕a}.

Proof. We can write

[a]_{∗}[a]^{∗}(R(c)) ≅ ⊕_{i=0}^{a−1}{σ^{i}}(R(c)),

where on the right side R{σ^{a}} acts separately on each factor. Hence the claim follows. (Compare [17, Proposition 7.4].)

Lemma 3.2.5. Suppose that the residue field of O contains an algebraic closure of 𝔽_{q}. For i a positive integer, let L_{i} be the fixed field of O[π^{−1}] under σ^{i}.

(a) For any σ-module M over Γ_{an,con} and any positive integer a, H^{0}([a]_{∗}M) = H^{0}(M) ⊗_{L_{1}} L_{a}.

(b) For any σ-modules M and N over Γ_{an,con} and any positive integer a, M ≅ N if and only if [a]_{∗}M ≅ [a]_{∗}N.

Proof. (a) It suffices to show that H^{0}([a]_{∗}M) admits a basis invariant under the induced action of σ. Since σ generates Gal(L_{a}/L_{1}), this follows from Hilbert’s Theorem 90.

(b) A morphism [a]_{∗}M → [a]_{∗}N corresponds to an element of V = H^{0}(([a]_{∗}M)^{∨} ⊗ [a]_{∗}N) ≅ H^{0}([a]_{∗}(M^{∨} ⊗ N)), which by (a) coincides with H^{0}(M^{∨} ⊗ N) ⊗_{L_{1}} L_{a}. If there is an isomorphism [a]_{∗}M ≅ [a]_{∗}N, then the determinant locus on V is not all of V; hence the same is true on H^{0}(M^{∨} ⊗ N). We can thus find an F -invariant element of M^{∨} ⊗ N corresponding to an isomorphism M ≅ N. (Compare [17, Propositions 7.3 and 7.5].)

3.3 σ-modules of rank 1

In this section, we analyze some σ-modules of rank 1 over Γ_{an,con}; this amounts to solving some simple equations involving σ, as in [19, Proposition 3.19] (compare also [17, Propositions 3.1 and 3.3]).

Definition 3.3.1. Define the twisted powers π^{{m}} of π by the two-way recurrence

π^{{0}} = 1, π^{{m+1}} = (π^{{m}})^{σ}π.

First, we give a classification result.

Proposition 3.3.2. Let M be a σ-module of rank 1 over R, for R one of Γ_{con}^{alg}, Γ_{con}^{alg}[π^{−1}], Γ_{an,con}^{alg}. Then there exists a unique integer n, which is nonnegative if R = Γ_{con}^{alg}, such that M ≅ R(n).

Proof. Let v be a generator of M, and write F v = xv. Then x must be a unit in R, so by Corollary 2.5.12, x ∈ Γ_{con}^{alg}[π^{−1}] and so w(x) is defined. If M ≅ R(n), we must then have n = w(x); hence n is unique if it exists. Since the residue field of Γ_{con}^{alg} is algebraically closed, we can find a unit u in Γ_{con}^{alg} such that u^{σ}π^{−n}x ≡ u (mod π). Choose r > 0 such that w_{r}(u^{σ}π^{−n}x/u − 1) > 0; then there exists a unit y ∈ Γ^{alg} with u^{σ}π^{−n}x/u = y^{σ}/y, and a direct calculation (by induction on m) shows that v_{m,r}(y) > 0 for all m > 0. (For details, see the proof of Lemma 5.4.1.) Hence y is a unit in Γ_{con}^{alg}, and so w = (u/y)v is a generator of M satisfying F w = π^{n}w. Thus there exists an isomorphism M ≅ R(n).

We next compute some instances of H^{0}.

Lemma 3.3.3. Let n be a nonnegative integer. If x ∈ Γ_{an,con} and x − π^{n}x^{σ} ∈ Γ_{con}[π^{−1}], then x ∈ Γ_{con}[π^{−1}].

Proof. Suppose the contrary; put y = x−π^{n}x^{σ}. We can find m with v_{m}(y) = ∞ and 0 < v_{m}(x) < ∞, since both hold for m sufficiently small by Corollary 2.5.6. Then

v_{m}(x) > q^{−1}v_{m}(x) = q^{−1}v_{m}(π^{n}x^{σ}) = q^{−1}v_{m−n}(x^{σ}) = v_{m−n}(x) ≥ v_{m}(x), contradiction.

Proposition 3.3.4. Let n be an integer.

(a) If n = 0, then H^{0}(Γ_{con}[π^{−1}](n)) = H^{0}(Γ_{an,con}(n)) ≠ 0; moreover, any nonzero element of H^{0}(Γ_{an,con}) is a unit in Γ_{an,con}.

(b) If n > 0, then H^{0}(Γ_{an,con}(n)) = 0.

(c1) If n < 0 and v_{K} is trivial, then H^{0}(Γ_{an,con}(n)) = 0.

(c2) If n < 0, v_{K} is nontrivial, and K is perfect, then H^{0}(Γ_{an,con}(n)) ≠ 0.

Proof. The group H^{0}(R(n)) consists of those x ∈ R with

π^{n}x^{σ} = x,  (3.3.5)

so our assertions are all really about the solvability of this equation.

(a) If n = 0, then H^{0}(Γ_{con}[π^{−1}](n)) = H^{0}(Γ_{an,con}(n)) by Lemma 3.3.3, and the former equals the fixed field of O[π^{−1}] under σ.

(b) By Lemma 3.3.3, any solution x of (3.3.5) over Γ_{an,con} actually belongs to Γ_{con}[π^{−1}]. In particular, w(x) = w(x^{σ}) is well-defined. But (3.3.5) yields w(x^{σ}) + n = w(x), which for n > 0 forces x = 0.

(c1) If n < 0 and v_{K} is trivial, then w(x) = w(x^{σ}) is well-defined, but (3.3.5) yields w(x^{σ}) + n = w(x), which forces x = 0.

(c2) If n < 0, v_{K} is nontrivial, and K is perfect, we may pick \overline{u} ∈ K with v_{K}(\overline{u}) > 0 (since v_{K} is nontrivial), and then set x to be the limit of the convergent series

∑_{m∈ℤ}(π^{\{m\}})^{n}[\overline{u}^{q^{m}}]

to obtain a nonzero solution of (3.3.5).

Remark 3.3.6. If n < 0, v_{K} is nontrivial, and K is not perfect, then the size of H^{0}(Γ_{an,con}(n)) depends on the particular choice of the Frobenius lift σ on Γ_{an,con}. For instance, in the notation of Section 2.3, if σ is a so-called “standard Frobenius lift” sending u to u^{q}, then any solution x = ∑ x_{i}u^{i} of (3.3.5) must have x_{i} = 0 whenever i is not divisible by q. By the same token, x_{i} = 0 whenever i is not divisible by q^{2}, or by q^{3}, and so on; hence we must have x ∈ O[π^{−1}], which as in (b) above is impossible for n > 0. On the other hand, if u^{σ} = (u + 1)^{q} − 1, then x = log(1 + u) ∈ Γ_{an,con} satisfies x^{σ} = qx; indeed, the existence of such an x is a backbone of the theory of (Φ, Γ)-modules associated to p-adic Galois representations, as in [4].

We next consider H^{1}.

Proposition 3.3.7. Let n be an integer.

(a) If n = 0 and K is separably closed, then H^{1}(Γ_{con}[π^{−1}](n)) = H^{1}(Γ_{an,con}(n)) = 0.

(b1) If n ≥ 0, then the map H^{1}(Γ_{con}[π^{−1}](n)) → H^{1}(Γ_{an,con}(n)) is injective.

(b2) If n > 0 and v_{K} is trivial, then H^{1}(Γ_{an,con}(n)) = 0.

(b3) If n > 0, v_{K} is nontrivial, and K is perfect, then H^{1}(Γ_{an,con}(n)) ≠ 0, with a nonzero element given by [\overline{x}] for any \overline{x} ∈ K with v_{K}(\overline{x}) < 0.

(c) If n < 0 and K is perfect, then H^{1}(Γ_{con}[π^{−1}](n)) = H^{1}(Γ_{an,con}(n)) = 0.

Proof. The group H^{1}(R(n)) consists of the quotient of the additive group of R by the subgroup of those x ∈ R for which the equation

x = y − π^{n}y^{σ}  (3.3.8)

has a solution y ∈ R, so our assertions are all really about the solvability of this equation.

(a) If n = 0 and K is separably closed, then for each x ∈ Γ, there exists y ∈ Γ such that x ≡ y − y^{q} (mod π). By iterating this construction, we can produce for any x ∈ Γ[π^{−1}] an element y ∈ Γ[π^{−1}] satisfying (3.3.8), such that

v_{m}(y) ≥ min{v_{m}(x), v_{m}(x)/q};

in particular, if x ∈ Γ_{con}[π^{−1}], then y ∈ Γ_{con}[π^{−1}]. Moreover, given x ∈ Γ_{an,con}, we can write x as a convergent series of elements of Γ_{con}[π^{−1}], and thus produce a solution of (3.3.8). Hence H^{1}(Γ_{con}(n)) = H^{1}(Γ_{an,con}(n)) = 0.

(b1) This follows at once from Lemma 3.3.3.

(b2) If n > 0 and v_{K} is trivial, then for any x ∈ Γ_{an,con}, the series

y = ∑_{m=0}^{∞}(π^{{m}})^{n}x^{σ^{m}}

converges π-adically to a solution of (3.3.8).

(b3) By (b1), it suffices to show that x = [\overline{x}] represents a nonzero element of H^{1}(Γ_{con}[π^{−1}](n)). By (b2), there exists a unique y ∈ Γ[π^{−1}] satisfying (3.3.8); however, we have v_{mn}(y) = q^{m}v_{K}(\overline{x}) for all m ≥ 0, and so y ∉ Γ_{con}[π^{−1}].

(c) Let x = ∑_{i}[\overline{x_{i}}]π^{i} be the Teichmüller presentation of x. Pick c > 0, and let z_{1} and z_{2} be the sums of [\overline{x_{i}}]π^{i} over those i for which v_{K}(\overline{x_{i}}) < c and v_{K}(\overline{x_{i}}) ≥ c, respectively. Then the sums

y_{1} = ∑_{m=0}^{∞} −(π^{{−m−1}})^{n}z_{1}^{σ^{−m−1}}

y_{2} = ∑_{m=0}^{∞}(π^{{m}})^{n}z_{2}^{σ^{m}}

converge to solutions of z_{1} = y_{1} − π^{n}y_{1}^{σ} and z_{2} = y_{2} − π^{n}y_{2}^{σ}, respectively. Hence y = y_{1} + y_{2} is a solution of (3.3.8). Moreover, if x ∈ Γ_{con}[π^{−1}], then \overline{x_{i}} = 0 for i sufficiently small, so we can choose c to ensure z_{2} = 0; then y = y_{1} ∈ Γ_{con}[π^{−1}].

3.4 Stability and semistability

As in [17], we can set up a formal analogy between the study of σ-modules over Γ_{an,con} and the study of stability of vector bundles.

Definition 3.4.1. For M a σ-module of rank 1 over Γ_{an,con} generated by some v, define the degree of M, denoted deg(M), to be the unique integer n such that M ⊗ Γ_{an,con}^{alg} ≅ Γ_{an,con}^{alg}(n), as provided by Proposition 3.3.2; concretely, deg(M) is the valuation of the unit via which F acts on a generator of M. For a σ-module M over Γ_{an,con} of rank n, define deg(M) = deg(∧^{n}M). Define μ(M) = deg(M)/ rank(M); we refer to μ(M) as the slope of M (or as the weight of M, per the terminology of [17, Section 6]).

Lemma 3.4.2. Let M be a σ-module over Γ_{an,con}, and let N be a σ-submodule of M with rank(M) = rank(N). Then deg(N) ≥ deg(M), with equality if and only if M = N; moreover, equality must hold if v_{K} is trivial.

Proof. By taking exterior powers, it suffices to check this for rank M = rank N = 1; also, there is no harm in assuming that K is algebraically closed. By Proposition 3.3.2, M ≅ Γ_{an,con}^{alg}(c) and N ≅ Γ_{an,con}^{alg}(d) for some integers c, d. By twisting, we may reduce to the case d = 0. Then by Proposition 3.3.4, we have 0 ≥ c, with equality forced if v_{K} is trivial; moreover, if c = 0, then N contains a generator which also belongs to H^{0}(M). But every nonzero element of the latter also generates M, so c = 0 implies M = N. (Compare [17, Proposition 6.2].)

Lemma 3.4.3. If 0 → M_{1} → M → M_{2} → 0 is a short exact sequence of σ-modules over Γ_{an,con}, then deg(M) = deg(M_{1}) + deg(M_{2}).

Proof. Put n_{1} = rank(M_{1}) and n_{2} = rank(M_{2}). Then the claim follows from the existence of the isomorphism

∧^{n_{1}+n_{2}}M ≅ (∧^{n_{1}}M_{1}) ⊗ (∧^{n_{2}}M_{2})

of σ-modules.

Proposition 3.4.4. Let a be a positive integer, let M be a σ-module over Γ_{an,con}, and let N be a σ^{a}-module over Γ_{an,con}.

(a) deg([a]_{∗}M) = a deg(M) and deg([a]^{∗}N) = deg(N).

(b) rank([a]_{∗}M) = rank(M) and rank([a]^{∗}N) = a rank(N).

(c) μ([a]_{∗}M) = aμ(M) and μ([a]^{∗}N) = \frac{1}{a}μ(N).

Proof. Straightforward (compare [17, Proposition 7.1], but note that the roles of the pullback and pushforward are interchanged here).

Definition 3.4.5. We say a σ-module M over Γ_{an,con} is semistable if μ(M) ≤ μ(N) for any nonzero σ-submodule N of M. We say M is stable if μ(M) < μ(N) for any nonzero proper σ-submodule N of M. Note that the direct sum of semistable σ-modules of the same slope is also semistable. By Proposition 3.4.4, for any positive integer a, if [a]_{∗}M is (semi)stable, then M is (semi)stable.

Remark 3.4.6. As noted earlier, the inequalities are reversed from the usual definitions of stability and semistability for vector bundles, because of an overall choice of sign convention.

Remark 3.4.7. Beware that this use of the term “semistable” is only distantly related to its use to describe p-adic Galois representations!

Lemma 3.4.8. For any integer c and any positive integer n, the σ-module Γ_{an,con}(c)^{⊕n} is semistable of slope c.

Proof. There is no harm in assuming that K is algebraically closed, and that v_{K} is nontrivial. Let N be a nonzero σ-submodule of M of rank d′ and degree c′. Then

Γ_{an,con}^{alg}(c′) ≅ ∧^{d′}N ⊆ ∧^{d′}Γ_{an,con}^{alg}(c)^{⊕n} ≅ Γ_{an,con}^{alg}(cd′)^{⊕\binom{n}{d′}}.

In particular, H^{0}(Γ_{an,con}^{alg}(cd′ − c′)) ≠ 0; by Proposition 3.3.4(b), this implies c′ ≥ cd′, yielding semistability. (Compare [17, Proposition 6.3(b)].)

Lemma 3.4.9. For any positive integer a and any integer c, the σ-module [a]^{∗}(Γ_{an,con}(c)) over Γ_{an,con} is semistable of rank a, degree c, and slope c/a. Moreover, if a and c are coprime, then [a]^{∗}(Γ_{an,con}(c)) is stable.

Proof. By Lemma 3.2.4 and Lemma 3.4.8, [a]_{∗}[a]^{∗}(Γ_{an,con}(c)) is semistable of rank a and slope c; as noted in Definition 3.4.5, it follows that [a]^{∗}(Γ_{an,con}(c)) is semistable. Let M be a nonzero σ-submodule of [a]^{∗}(Γ_{an,con}(c)). If a and c are coprime, then deg(M) = (c/a) rank(M) is an integer; since rank(M) ≤ a, this is only possible for rank(M) = a. But then Lemma 3.4.2 implies that μ(M) > c/a unless M = [a]^{∗}(Γ_{an,con}(c)). We conclude that [a]^{∗}(Γ_{an,con}(c)) is stable. (Compare [17, Proposition 8.2].)

3.5 Harder-Narasimhan filtrations

Using the notions of degree and slope, we can make the usual formal construction of Harder-Narasimhan filtrations, with its usual properties.

Definition 3.5.1. Given a multiset S of n real numbers, define the Newton polygon of S to be the graph of the piecewise linear function on [0, n] sending 0 to 0, whose slope on [i − 1, i] is the i-th smallest element of S; we refer to the point on the graph corresponding to the image of n as the endpoint of the polygon. Conversely, given such a graph, define its slope multiset to be the slopes of the piecewise linear function on [i − 1, i] for i = 1, …, n. We say that the Newton polygon of S lies above the Newton polygon of S′ if no vertex of the polygon of S lies below the polygon of S′, and the two polygons have the same endpoint.

Definition 3.5.2. Let M be a σ-module over Γ_{an,con}. A semistable filtration of M is an exhaustive filtration 0 = M_{0} ⊂ M_{1} ⊂ ⋯ ⊂ M_{l} = M of M by saturated σ-submodules, such that each successive quotient M_{i}/M_{i−1} is semistable of some slope s_{i}. A Harder-Narasimhan filtration (or HN-filtration) of M is a semistable filtration with s_{1} < ⋯ < s_{l}. An HN-filtration is unique if it exists, as M_{1} can then be characterized as the unique maximal σ-submodule of M of minimal slope, and so on.

Definition 3.5.3. Let M be a σ-module over Γ_{an,con}. Given a semistable filtration 0 = M_{0} ⊂ M_{1} ⊂ ⋯ ⊂ M_{l} = M of M, form the multiset consisting of, for i = 1, …, l, the slope μ(M_{i}/M_{i−1}) with multiplicity rank(M_{i}/M_{i−1}). We call this the slope multiset of the filtration, and we call the associated Newton polygon the slope polygon of the filtration. If M admits a Harder-Narasimhan filtration, we refer to the slope multiset as the Harder-Narasimhan slope multiset (or HN-slope multiset) of M, and to the Newton polygon as the Harder-Narasimhan polygon (or HN-polygon) of M.

Proposition 3.5.4. Let M be a σ-module over Γ_{an,con} admitting an HNfiltration. Then the HN-polygon lies above the slope polygon of any semistable filtration of M.

Proof. Let 0 = M_{0} ⊂ M_{1} ⊂ ⋯ ⊂ M_{l} = M be an HN-filtration, and let 0 = M′_{0} ⊂ M′_{1} ⊂ ⋯ ⊂ M′_{m} = M be a semistable filtration. To prove the inequality, it suffices to prove that for each of i = 1, …, l, we can choose rank(M/M_{i}) slopes from the slope multiset of the semistable filtration whose sum is greater than or equal to the sum of the greatest rank(M/M_{i}) HN-slopes of M; note that the latter is just deg(M/M_{i}). For l = 1, …, m, put

d_{l} = rank(M′_{l} + M_{i}) − rank(M′_{l−1} + M_{i}) ≤ rank(M′_{l}/M′_{l−1}).

Since M′_{l}/M′_{l−1} is semistable and (M′_{l} + M_{i})/(M′_{l−1} + M_{i}) is a quotient of M′_{l}/M′_{l−1}, we have

μ(M′_{l}/M′_{l−1}) ≥ μ((M′_{l} + M_{i})/(M′_{l−1} + M_{i})).

However, we also have

∑_{i=1}^{l} d_{l}μ((M′_{l} + M_{i})/(M′_{l−1} + M_{i})) = deg(M/M_{i}),

yielding the desired inequality.

Remark 3.5.5. We will ultimately prove that every σ-module over Γ_{an,con} admits a Harder-Narasimhan filtration (Proposition 4.2.5), and that the successive quotients become isomorphic over Γ_{an,con}^{alg} to direct sums of “standard” σ-modules of the right slope (Theorems 6.3.3 and 6.4.1). Using the formalism of Harder-Narasimhan filtrations makes it a bit more convenient to articulate the proofs of these assertions.

3.6 Descending subobjects

We will ultimately be showing that the formation of a Harder-Narasimhan filtration of a σ-module over Γ_{an,con} commutes with base change. In order to prove this sort of statement, it will be useful to have a bit of terminology.

Definition 3.6.1. Given an injection R ↪ S of integral domains equipped with compatible endomorphisms σ, a σ-module M over R, and a saturated σ-submodule N_{S} of M_{S} = M ⊗_{R} S, we say that N_{S} descends to R if there is a saturated σ-submodule N of M such that N_{S} = N ⊗_{R} S; note that N is unique if it exists, because it can be characterized as M ∩ N_{S}. Likewise, given a filtration of M_{S}, we say the filtration descends to R if it is induced by a filtration of M.

The following lemma lets us reduce most descent questions to consideration of submodules of rank 1.

Lemma 3.6.2. With notation as in Definition 3.6.1, suppose that R is a Bézout domain, and put d = rank N_{S}. Then N_{S} descends to R if and only if ∧^{d}N_{S} ⊆ (∧^{d}M)_{S} descends to R.

Proof. If N_{S} = N ⊗_{R} S, then ∧^{d}N_{S} = (∧^{d}N) ⊗_{R} S descends to R. Conversely, if ∧^{d}N_{S} = (N′)⊗_{R}S, let N be the σ-submodule of M consisting of those v ∈ M such that v ∧ w = 0 for all w ∈ N′. Then N is saturated and N ⊗_{R} S = N_{S}, since N is defined by linear conditions which in M_{S} cut out precisely N_{S}. Since R is a Bézout domain, this suffices to ensure that N is free; since N is visibly stable under F, N is in fact a σ-submodule of M, and so N_{S} descends to R.

4 Slope filtrations of σ-modules

In this chapter, we give a classification of σ-modules over Γ_{an,con}^{alg}, as in [19, Chapter 4]. However, this presentation looks somewhat different, mainly because of the formalism introduced in the previous chapter. We also have integrated into a single presentation the cases where v_{K} is nontrivial and where v_{K} is trivial; these are presented separately in [19] (in Chapters 4 and 5 respectively). These two cases do have different flavors, which we will point out as we go along. Beware that although we have mostly made the exposition self-contained, there remains one notable exception: we do not repeat the key calculation made in [19, Lemma 4.12]. See Lemma 4.3.3 for the relevance of this calculation.

Convention 4.0.1. To lighten the notational load, we write R for Γ_{an,con}^{alg}. Whenever working over R, we also make the harmless assumption that π is σ-invariant.

4.1 Standard σ-modules

Following [17, Section 8], we introduce the standard building blocks into which we will decompose σ-modules over R.

Definition 4.1.1. Let c, d be coprime integers with d > 0. Define the σ-module M_{c,d} = [d]^{∗}(R(c)) over R; that is, M_{c,d} is freely generated by e_{1}, …, e_{d} with

F e_{1} = e_{2}, …, F e_{d−1} = e_{d}, F e_{d} = π^{c}e_{1}.

This σ-module is stable of slope c/d by Lemma 3.4.9. We say a σ-module M is standard if it is isomorphic to some M_{c,d}; in that case, we say a basis e_{1}, …, e_{d} as above is a standard basis of M.

Lemma 4.1.2. (a) M_{c,d} ⊗ M_{c′,d′} ≅ M_{c′′,d′′}^{⊕dd′/d′′}, where c/d + c′/d′ = c′′/d′′ in lowest terms.

(b) M_{c,d}(c′) ≅ M_{c+c′d,d}.

(c) M_{c,d}^{∨} ≅ M_{−c,d}.

Proof. To verify (a), it is enough to do so after applying [dd′]_{∗} thanks to Lemma 3.2.5. Then the desired isomorphism follows from Lemma 3.2.4. Assertion (b) follows from (a), and (c) follows from the explicit description of M_{c,d} given above. (Compare [17, Proposition 8.3].)

Proposition 4.1.3. Let c, d be coprime integers with d > 0.

(a) The group H^{0}(M_{c,d}) is nonzero if and only if v_{K} is nontrivial and c/d ≤ 0, or v_{K} is trivial and c/d = 0.

(b) The group H^{1}(M_{c,d}) is nonzero if and only if v_{K} is nontrivial and c/d > 0.

Proof. These assertions follow from Propositions 3.3.4 and 3.3.7, plus the fact that H^{i}([d]^{∗}M) ≅ H^{i}(M) for i = 0, 1.

Corollary 4.1.4. Let c, c′, d, d′ be integers, with d, d′ positive and gcd(c, d) = gcd(c′, d′) = 1.

(a) We have Hom(M_{c′,d′}, M_{c,d}) ≠ 0 if and only if v_{K} is nontrivial and c′/d′ ≥ c/d, or v_{K} is trivial and c′/d′ = c/d.

(b) We have Ext(M_{c′,d′}, M_{c,d}) ≠ 0 if and only if v_{K} is nontrivial and c′/d′ < c/d.

Remark 4.1.5. One can show that End(M_{c,d}) is a division algebra (this can be deduced from the fact that M_{c,d} is stable) and even describe it explicitly, as in [17, Proposition 8.6]. For our purposes, it will be enough to check that End(M_{c,d}) is a division algebra after establishing the existence of Dieudonné-Manin decompositions; see Corollary 4.5.9.

4.2 Existence of eigenvectors

In classifying σ-modules over R, it is useful to employ the language of “eigenvectors”.

Definition 4.2.1. Let d be a positive integer. A d-eigenvector (or simply eigenvector if d = 1) of a σ-module M over R is a nonzero element v of M such that F^{d}v = π^{c}v for some integer c. We refer to the quotient c/d as the slope of v.

Proposition 4.2.2. Suppose that v_{K} is nontrivial. Then every nontrivial σmodule over R contains an eigenvector.

Proof. The calculation is basically that of [19, Proposition 4.8]: use the fact that F makes things with “very positive partial valuations” converge better whereas F^{−1} makes things with “very negative partial valuations” converge better. However, one can simplify the final analysis a bit, as is done in [17, Theorem 4.1]. We first set some notation as in [19, Proposition 4.8]. Let M be a nontrivial σ-module over R. Choose a basis e_{1}, …, e_{n} of M, and define the invertible n × n matrix A over R by the equation F e_{j} = ∑_{i} A_{ij}e_{i}. Choose r > 0 such that A and its inverse have entries in Γ_{an,r}^{alg}. Choose ϵ > 0, and choose an integer c with c ≤ min{w_{r}(A), w_{r}((A^{−1})^{σ^{−1}})} − ϵ. Choose an integer m > c such that the interval

(\frac{−c + m}{(q − 1)r} , \frac{q(c + m)}{(q − 1)r})

is nonempty (true for m sufficiently large), and choose d in the intersection of that interval with the image of v_{K}. (The choice of d is possible because v_{K} is nontrivial and the residue field of Γ_{con}^{alg} is algebraically closed, so the image of v_{K} contains a copy of ℚ and hence is dense in ℝ.) For an interval I ⊆ (0, r], define the functions a, b : Γ_{I} → Γ_{I} as follows. For x ∈ Γ_{I}, let ∑_{j∈ℤ}[\overline{x_{j}}]π^{j} be the Teichmüller presentation of x (as in Definition 2.5.1). Let a(x) and b(x) be the sums of [\overline{x_{j}}]π^{j} over those j with v_{K}(\overline{x_{j}}) < d and v_{K}(\overline{x_{j}}) ≥ d, respectively. We may think of a and b as splitting x into “negative” and “positive” terms. (This decomposition shares its canonicality with the corresponding decomposition in [17, Theorem 4.1] but not with the one in [19, Proposition 4.8].) For I ⊆ (0, r], let M_{I} denote the Γ_{I}-span of the e_{i}. For v ∈ M_{I}, write v = ∑_{i} x_{i}e_{i}, and for s ∈ I, define

v_{n,s}(v) = min_{i}{v_{n,s}(x_{i})}
w_{s}(v) = min_{i}{w_{s}(x_{i})}.

Put a(v) = ∑_{i} a(x_{i})e_{i} and b(v) = ∑_{i} b(x_{i})e_{i}; then by the choice of d, we have for v ∈ M_{(0,r]},

w_{r}(π^{m}F^{−1}(a(v))) ≥ w_{r}(a(v)) + ϵ
w_{r}(π^{−m}F (b(v))) ≥ w_{r}(b(v)) + ϵ.

Put

f(v) = π^{−m}b(v) − F^{−1}(a(v)).

If v ∈ M_{(0,r]} is such that w = F v − π^{m}v also lies in M_{(0,r]}, then

F (v + f(w)) − π^{m}(v + f(w))
= F (f(w)) − π^{m}f(w) + w

= F (π^{−m}b(w)) − a(w) − b(w) + π^{m}F^{−1}(a(w)) + w

= π^{−m}F (b(w)) + π^{m}F^{−1}(a(w))

lies in M_{(0,r]} as well, and

w_{r}(f(w)) ≥ w_{r}(π^{−m}w)
w_{r}(F (v + f(w)) − π^{m}(v + f(w))) ≥ w_{r}(w) + ϵ.

Now define a sequence \{v_{l}\}_{l=0}^{∞} in M_{(0,r]} as follows. Pick \overline{x} ∈ K with v_{K}(\overline{x}) = d, and set

v_{0} = π^{−m}[\overline{x}]e_{1} + [\overline{x}^{1/q}]F^{−1}e_{1}.

Given v_{l} ∈ M_{(0,r]}, set

w_{l} = F v_{l} − π^{m}v_{l}, v_{l+1} = v_{l} + f(w_{l}).

We calculated above that w_{l} ∈ M_{(0,r]} implies w_{l+1} ∈ M_{(0,r]}; since w_{0} ∈ M_{(0,r]} evidently, we have w_{l} ∈ M_{(0,r]} for all l, so v_{l+1} ∈ M_{(0,r]} and the iteration continues. Moreover,

w_{r}(v_{l+1} − v_{l}) ≥ w_{r}(π^{−m}w_{l})
w_{r}(w_{l}) = w_{r}(F (v_{l−1} + f(w)) − π^{m}(v_{l−1} + f(w)))
≥ w_{r}(w_{l−1}) + ϵ.

Hence w_{l} → 0 and f(w_{l}) → 0 under w_{r} as l → ∞, so the v_{l} converge to a limit v ∈ M_{[r,r]}. We now check that v ≠ 0. Since w_{r}(w_{l+1}) ≥ w_{r}(w_{l}) + ϵ for all l, we certainly have w_{r}(w_{l}) ≥ w_{r}(w_{0}) for all l, and hence w_{r}(v_{l+1} − v_{l}) ≥ w_{r}(π^{−m}w_{0}). To compute w_{r}(w_{0}), note that w_{r}(π^{−m}[\overline{x}]e_{1}) = dr − m, whereas

w_{r}([\overline{x}^{1/q}]F^{−1}e_{1}) ≥ dr/q + c
> dr − m

by the choice of d. Hence w_{r}(v_{0}) = dr − m. On the other hand,

w_{r}(π^{−m}w_{0}) = w_{r}(π^{−m}F v_{0} − v_{0})
= w_{r}(π^{−2m}[\overline{x}^{q}]F e_{1} − [\overline{x}^{1/q}]F^{−1}e_{1})
≥ min{dqr + c − 2m, dr/q + c}.

The second term is strictly greater than dr − m as above, while the first term equals dr − m plus dr(q − 1) + c − m, and the latter is positive again by the choice of d. Thus w_{r}(π^{−m}w_{0}) > w_{r}(v_{0}), and so w_{r}(v_{l+1} − v_{l}) > w_{r}(v_{0}); in particular, w_{r}(v) = w_{r}(v_{0}) and so v ≠ 0. Since the v_{l} converge to v in M_{[r,r]}, the F v_{l} converge to F v in M_{[r/q,r/q]}. On the other hand, F v_{l} = w_{l} + π^{m}v_{l}, so the F v_{l} converge to π^{m}v in M_{[r,r]}. In particular, the F v_{l} form a Cauchy sequence under w_{s} for s = r/q and s = r, hence also for all s ∈ [r/q, r], and the limit in M_{[r/q,r]} must equal both F v and π^{m}v. Therefore v ∈ M_{[r/q,r]} and F v = π^{m}v in M_{[r/q,r/q]}. But now by induction on i, v ∈ M_{[r/q^{i},r]} for all i, so v ∈ M_{(0,r]} ⊂ M is a nonzero eigenvector, as desired.

Lemma 4.2.3. Suppose that v_{K} is nontrivial. Let M be a σ-module of rank n over R.

(a) There exists an integer c_{0} such that for any integer c ≥ c_{0}, there exists an injection R(c)^{⊕n} ↪ M.

(b) There exists an integer c_{1} such that for any integer c ≤ c_{1}, there exists an injection M ↪ R(c)^{⊕n}.

Proof. By taking duals, we may reduce (b) to (a). We prove (a) by induction on n, with empty base case n = 0. By Proposition 4.2.2, there exists an eigenvector of M; the saturated span of this eigenvector is a rank 1 σ-submodule of M, necessarily isomorphic to some R(m) by Proposition 3.3.2. By the induction hypothesis, we can choose c_{0} ≥ m so that R(c_{0})^{n−1} injects into M/R(m). Let N be the preimage of R(c_{0})^{n−1} in M; then there exists an exact sequence

0 → R(m) → N → R(c_{0})^{n−1} → 0,

which splits by Corollary 4.1.4. Thus N ≅ R(m) ⊕ R(c_{0})^{n−1} ⊆ M, and R(m) contains a copy of R(c_{0}) by Corollary 4.1.4. This yields the desired result by Corollary 4.1.4 again.

Proposition 4.2.4. For any nonzero σ-module M over Γ_{an,con}, the slopes of all nonzero σ-submodules of N are bounded below. Moreover, there is a nonzero σ-submodule of N of minimal slope, and any such σ-submodule is semistable.

Proof. To check the first assertion, we may assume (by enlarging K as needed) that K is algebraically closed, so that Γ_{an,con} = R, and that v_{K} is nontrivial. By Lemma 4.2.3, there exists an injection M ↪ R(c)^{⊕n} for some c, where n = rank M. By Lemma 3.4.8, it follows that μ(N) ≥ c for any σ-submodule N of M, yielding the first assertion. As for the second assertion, the slopes of σ-submodules of M form a discrete subset of ℚ, because their denominators are bounded above by n. Hence this set has a least element, yielding the remaining assertions.

Proposition 4.2.5. Every σ-module over Γ_{an,con} admits a Harder-Narasimhan filtration.

Proof. Let M be a nontrivial σ-module over Γ_{an,con}. By Proposition 4.2.4, the set of slopes of nonzero σ-submodules of M has a least element s_{1}. Suppose that N_{1}, N_{2} are σ-submodules of M with μ(N_{1}) = μ(N_{2}) = s_{1}; then the internal sum N_{1} + N_{2} is a quotient of the direct sum N_{1} ⊕ N_{2}. By Proposition 4.2.4, each of N_{1} and N_{2} is semistable, as then is N_{1} ⊕ N_{2}. Hence μ(N_{1} + N_{2}) ≤ s_{1}; by the minimality of s_{1}, we have μ(N_{1} + N_{2}) = s_{1}. Consequently, the set of σ-submodules of M of slope s_{1} has a maximal element M_{1}. Repeating this argument with M replaced by M/M_{1}, and so on, yields a Harder-Narasimhan filtration.

Remark 4.2.6. Running this argument is a severe obstacle to working with spherically complete coefficients (as suggested by Remark 2.3.10), as it is no longer clear that there exists a minimum slope among the σ-submodules of a given σ-module.

Remark 4.2.7. It may be possible to simplify the calculations in this section by using Lemma 6.1.1; we have not looked thoroughly into this possibility.

4.3 More eigenvectors

We now give a crucial refinement of the conclusion of Proposition 4.2.2 by extracting an eigenvector of a specific slope in a key situation. This is essentially [19, Proposition 4.15]; compare also [17, Proposition 9.1]. Beware that we are omitting one particularly unpleasant part of the calculation; see Lemma 4.3.3 below. We start by identifying H^{0}(R(−1)).

Lemma 4.3.1. The map

\overline{y} ↦ ∑_{i∈ℤ}[\overline{y}^{q^{−i}}]π^{i}

induces a bijection 𝔪_{K} → H^{0}(R(−1)), where 𝔪_{K} denotes the subset of K on which v_{K} is positive.

Proof. On one hand, for v_{K}(\overline{y}) > 0, the sum y = ∑_{i∈ℤ}[\overline{y}^{q^{−i}}]π^{i} converges and satisfies y^{σ} = πy. Conversely, if y^{σ} = πy, comparing the Teichmüller presentations of y^{σ} and of πy forces y to assume the desired form.

We next give a “positioning argument” for elements of H^{1}(R(m)), following [19, Lemmas 4.13 and 4.14].

Lemma 4.3.2. For m a positive integer, every nonzero element of H^{1}(R(m)) is represented by some x ∈ Γ_{con}^{alg} with v_{n}(x) = v_{m−1}(x) for n ≥ m. Moreover, we can ensure that for each n ≥ 0, either v_{n}(x) = ∞ or v_{n}(x) < 0.

Proof. We first verify that each element of H^{1}(R(m)) is represented by an element of Γ_{con}^{alg}[π^{−1}]. If v_{n}(x) ≥ 0 for all n ∈ ℤ, then the sum y = ∑_{i=0}^{∞} x^{σ^{i}}π^{mi} converges in R and satisfies y − π^{m}y^{σ} = x, so x represents the zero class in H^{1}(R(m)). In other words, if x ∈ R has plus-minus-zero representation x_{+} + x_{−} + x_{0}, then x and x_{−} represent the same class in H^{1}(R(m)), and visibly x_{−} ∈ Γ_{con}^{alg}[π^{−1}]. We next verify that each element of H^{1}(R(m)) is represented by an element x of Γ_{con}^{alg}[π^{−1}] with inf_{n}\{v_{n}(x)\} > −∞. Given any x ∈ Γ_{con}^{alg}[π^{−1}], let x = ∑_{i}[\overline{x_{i}}]π^{i} be the Teichmüller representation of x. For each i, let c_{i} be the smallest nonnegative integer such that q^{−c_{i}}v_{K}(\overline{x_{i}}) ≥ −1, and put

y_{i} = ∑_{j=1}^{c_{i}}[\overline{x_{i}}^{q^{−j}}]π^{i−mj};

since c_{i} grows only logarithmically in i, i − mc_{i} → ∞ and the sum ∑_{i} y_{i} converges π-adically. Moreover,

lim sup_{i→∞} min_{1≤j≤c_{i}}\{−q^{−j}v_{0}(\overline{x_{i}})/(i − mj)\}

is finite, because the same is true for each of q^{−j} (clear), −v_{0}(\overline{x_{i}})/i (by the definition of R), and i/(i − mj) (because c_{i} grows logarithmically in i). Hence the sum y = ∑_{i} y_{i} actually converges in R; if we set x′ = x − π^{m}y^{σ} + y, then x and x′ represent the same class in H^{1}(R(m)). However,

x′ = ∑_{i}[\overline{x_{i}}^{q^{−c_{i}}}]π^{i−mc_{i}}

satisfies v_{n}(x′) ≥ −1 for all n. Since x and π^{m}x^{σ} represent the same element of H^{1}(R(m)), we can also say that each element of H^{1}(R(m)) is represented by an element x of Γ_{con}^{alg} with inf_{n}{v_{n}(x)} > −∞. Given such an x, put h = inf_{n}{v_{n}(x)}; we attempt to construct a sequence x_{0}, x_{1}, … with the following properties:

(a) x_{0} = x;

(b) each x_{l} generates the same element of H^{1}(R(m)) as does x;

(c) x_{l} ≡ 0 (mod π^{lm});

(d) for each n and l, v_{n}(x_{l}) ≥ h.

We do this as follows. Given x_{l}, let ∑_{i=lm}^{∞}[\overline{x_{l,i}}]π^{i} be the Teichmüller presentation of x_{l}, and put

u_{l} = ∑_{i=lm}^{lm+m−1}[\overline{x_{l,i}}]π^{i}.

If v_{lm+m−1}(x_{l}) ≥ h/q, put x_{l+1} = x_{l} − u_{l} + π^{m}u_{l}^{σ}; otherwise, leave x_{l+1} undefined. If x_{l} is defined for each l, then set y = ∑_{l=0}^{∞} u_{l}; this sum converges in R, and its limit satisfies x − y + π^{m}y^{σ} = 0. Hence in this case, x represents the trivial class in H^{1}(R(m)). On the other hand, if we are able to define x_{l} but not x_{l+1}, put

y = x_{l} − u_{l}
z = π^{−(l+1)m}y^{σ^{−l−1}} + π^{−lm}u_{l}^{σ^{−l}};

then x and z represent the same class in H^{1}(R(m)). Moreover, h/q > v_{n}(u) and v_{n}(y) ≥ h for all n, so hq^{−l−1} > v_{n}(π^{−lm}u_{l}^{σ^{−l}}) and v_{n}(π^{−(l+1)m}y^{σ^{−l−1}}) ≥ hq^{−l−1}; consequently v_{n}(z) = v_{m−1}(z) for n ≥ m. In addition, we can pass from z to the minus part z_{−} of its plus-minus-zero representation (since z and z_{−} represent the same class in H^{1}(R(m)), as noted above) to ensure that v_{n}(z_{−}) is either infinite or negative for all n ≥ 0. We conclude that every nonzero class in H^{1}(R(m)) has a representative of the desired form.

Lemma 4.3.3. Let d be a positive integer. For any x ∈ H^{1}([d]^{∗}(R(d+1))), there exists y ∈ H^{0}(R(−1)) nonzero such that x and y pair to zero in H^{1}(R(−1) ⊗ [d]^{∗}(R(d + 1))) = H^{1}([d]^{∗}(R(1))).

Proof. Identify H^{1}([d]^{∗}(R(d + 1))) and H^{1}([d]^{∗}(R(1))) with H^{1}(R(d + 1)) and H^{1}(R(1)), respectively, where the modules in the latter cases are σ^{d}-modules. Then the question can be stated as follows: for any x ∈ R, there exist y, z ∈ R with y nonzero, such that

y^{σ} = πy, xy = z − πz^{σ^{d}}.

By Lemma 4.3.2, we may assume that x ∈ Γ_{con}^{alg} and that v_{n}(x) = v_{d}(x) < 0 for n > d. In this case, the claim follows from a rather involved calculation [19, Lemma 4.12] which we will not repeat here. (For a closely related calculation, see [17, Proposition 9.5].)

Proposition 4.3.4. Assume that v_{K} is nontrivial. For d a positive integer, suppose that

0 → M_{1,d} → M → M_{−1,1} → 0

is a short exact sequence of σ-modules over R. Then M contains an eigenvector of slope 0.

Proof. The short exact sequence corresponds to a class in

Ext(M_{−1,1}, M_{1,d}) ≅ H^{1}(M_{1,1} ⊗ M_{1,d})
≅ H^{1}(M_{d+1,d})
≅ H^{1}([d]^{∗}(R(d + 1))).

From the snake lemma, we obtain an exact sequence

H^{0}(M) → H^{0}(M_{−1,1}) = H^{0}(R(−1)) → H^{1}(M_{1,d}) = H^{1}([d]^{∗}R(1)),

in which the second map (the connecting homomorphism) coincides with the pairing with the given class in H^{1}([d]^{∗}(R(d + 1))). By Lemma 4.3.3, this homomorphism is not injective; hence H^{0}(M) ≠ 0, as desired.

Corollary 4.3.5. Assume that v_{K} is nontrivial. For d a positive integer, suppose that

0 → M_{1,1} → M → M_{−1,d} → 0

is a short exact sequence of σ-modules over R. Then M^{∨} contains an eigenvector of slope 0.

Proof. Dualize Proposition 4.3.4.

Corollary 4.3.6. Assume that v_{K} is nontrivial. For c, c′, c′′ integers with c + c′ ≤ 2c′′, suppose that

0 → M_{c,1} → M → M_{c′,1} → 0

is a short exact sequence of σ-modules over R. Then M contains an eigenvector of slope c′′.

Proof. By twisting, we may reduce to the case c′′ = 0. If c ≤ 0, then already M_{c,1} contains an eigenvector of slope 0 by Corollary 4.1.4, so we may assume c ≥ 0. Since c + c′ ≤ 0, by Corollary 4.1.4, we can find a copy of M_{−c,1} within M_{c′,1}; taking the preimage of M_{−c,1} within M allows us to reduce to the case c′ = −c. We treat the cases c ≥ 0 by induction, with base case c = 0 already treated. If c > 0, then twisting yields an exact sequence

0 → M_{c−1,1} → M(−1) → M_{−c−1,1} → 0.

By Corollary 4.1.4, we can choose a submodule of M_{−c−1,1} isomorphic to M_{−c+1,1}; let N be its inverse image in M(−1). Applying the induction hypothesis to the sequence

0 → M_{c−1,1} → N → M_{−c+1,1} → 0

yields an eigenvector of N of slope 0, and hence an eigenvector of M of slope 1. Let P be the saturated span of that eigenvector; it is isomorphic to M_{m,1} for some m by Proposition 3.3.2, and we must have m ≤ 1 by Corollary 4.1.4. If m ≤ 0, M has an eigenvector of slope 0, so suppose instead that m = 1. We then have an exact sequence

0 → P ≅ M_{1,1} → M → M/P → 0

in which M/P, which has rank 1 and degree −1 (by Lemma 3.4.3), is isomorphic to M_{−1,1} by Proposition 3.3.2. Applying Proposition 4.3.4 now yields the desired result. (Compare [17, Corollary 9.2].)

4.4 Existence of standard submodules

We now run the induction setup of [17, Theorem 11.1] to produce standard submodules of a σ-module of small slope.

Definition 4.4.1. For a given integer n ≥ 1, let (A_{n}), (B_{n}) denote the following statements about n.

(A_{n}) Let a be any positive integer, and let M be any σ^{a}-module over R. If rank(M) ≤ n and deg(M) ≤ 0, then M contains an eigenvector of slope 0.

(B_{n}) Let a be any positive integer, and let M be any σ^{a}-module over R. If rank(M) ≤ n, then M contains a saturated σ^{a}-submodule which is standard of slope ≤ μ(M).

Note that if v_{K} is nontrivial, both (A_{1}) and (B_{1}) hold thanks to Proposition 3.3.2.

Lemma 4.4.2. Assume that v_{K} is nontrivial. For n ≥ 2, if (A_{n−1}) and (B_{n−1}) hold, then (A_{n}) holds.

Proof. It suffices to show that if M is a σ^{a}-module over R with rank(M) = n and deg(M) ≤ 0, then M contains an eigenvector of slope 0; after twisting, we may reduce to the case 1 − n ≤ deg(M) ≤ 0. Suppose on the contrary that no such eigenvector exists; by Corollary 4.1.4, M then contains no eigenvector of any nonpositive slope. On the other hand, by Proposition 4.2.2, M contains an eigenvector; in particular, M contains a saturated σ^{a}-submodule of rank 1. By Proposition 3.3.2, we thus have an exact sequence

0 → M_{c,1} → M → N → 0

for some integer c, which by hypothesis must be positive. Choose c as small as possible; then deg(N) = deg(M) − c ≤ 0 by Lemma 3.4.3, so by (A_{n−1}), N contains an eigenvector of slope 0. That is, N contains a σ^{a}- submodule isomorphic to M_{0,1}; let M′ be the preimage in M of that submodule. We then have an exact sequence

0 → M_{c,1} → M′ → M_{0,1} → 0.

By Corollary 4.3.6, M′ contains an eigenvector of slope ⌈c/2⌉. By the minimality of the choice of c, we must have c ≤ ⌈c/2⌉, or c = 1. Put c′ = deg(M) − 1 = deg(N), so that c′ < 0. By (B_{n−1}), N contains a saturated σ^{a}-submodule P which is standard of slope ≤ c′/(n − 1); let P′ be the preimage in M of that σ^{a}-submodule. If rank(P) < n − 1, then deg(P′) ≤ 1 + rank(P)(c′/(n − 1)) < 1; since deg(P′) is an integer, we have deg(P′) ≤ 0. By (A_{n−1}), P′ contains an eigenvector of slope 0, contradicting our hypothesis. If rank(P) = n − 1, we have an exact sequence

0 → M_{1,1} → P′ → P ≅ M_{c′′,n−1} → 0

for some c′′ ≤ −1. By Corollary 4.1.4, there is a nonzero homomorphism M_{−1,n−1} → M_{c′′,n−1}; if its image has rank < n − 1, then again (A_{n−1}) forces P′ to contain an eigenvector of slope 0, contrary to assumption. Hence M_{c′′,n−1} contains a copy of M_{−1,n−1}; choose such a copy and let P′′ be its inverse image in P′. By Corollary 4.3.5, (P′′)^{∨} contains an eigenvector of slope 0, and hence a primitive eigenvector of slope at most 0; this eigenvector corresponds to a rank n − 1 submodule of P′′ of slope at most 0. By (A_{n−1}), P′′ contains an eigenvector of slope 0, contradicting our hypothesis. In any case, our hypothesis that M contains no eigenvector of slope 0 has been contradicted, yielding the desired result.

Lemma 4.4.3. Assume that v_{K} is nontrivial. For n ≥ 2, if (A_{n}) and (B_{n−1}) hold, then (B_{n}) holds.

Proof. Let M be a σ^{a}-module of rank n and degree c. Put b = n/ gcd(n, c); then by (A_{n}) applied after twisting, [b]_{∗}M (which has rank n and degree bc, by Proposition 3.4.4) has an eigenvector of slope bc/n. That is, M has a beigenvector v of slope c/n; this gives a nontrivial map f : M_{bc/n,b} → M sending a standard basis to v, F v, …, F^{b−1}v. Let N be the saturated span of the image of f, and put m = rank N. Then ∧^{m}N admits a b-eigenvector of slope cm/n, so by Corollary 4.1.4, the slope of N is at most c/n. If m < n, we may apply (B_{n−1}) to N to obtain the desired result. Suppose instead that m = n, which also implies b = n since necessarily m ≤ b ≤ n. Then the map f is injective, so its image has slope c/n. By Lemma 3.4.2, f must in fact be surjective; thus M ≅ M_{c,n}, as desired.

4.5 Dieudonné-Manin decompositions

Definition 4.5.1. A Dieudonné-Manin decomposition of a σ-module M over R is a direct sum decomposition M = ⊕_{i=1}^{m}M_{c_{i},d_{i}} of M into standard σsubmodules. The slope multiset of such a decomposition is the union of the multisets consisting of c_{i}/d_{i} with multiplicity d_{i} for i = 1, …, m.

Remark 4.5.2. If M admits a Dieudonné-Manin decomposition, then M admits a basis of n-eigenvectors for n = (rank M)!; more precisely, any basis of H^{0}([n]_{∗}M) over the fixed field of σ^{n} gives a basis of [n]_{∗}M over R. The slopes of these n-eigenvectors coincide with the slope multiset of the decomposition.

Proposition 4.5.3. Assume that v_{K} is nontrivial. Then every σ-module M over R admits a Dieudonné-Manin decomposition.

Proof. We first show that every semistable σ-module M over R is isomorphic to a direct sum of standard σ-submodules of slope μ(M). We see this by induction on rank(M); by Lemmas 4.4.2 and 4.4.3, we have (B_{n}) for all n, so M contains a saturated σ-submodule N which is standard of some slope ≤ μ(M). Since M has been assumed semistable, we have μ(N) ≥ μ(M); hence μ(N) = μ(M), and M/N is also semistable. By the induction hypothesis, M/N splits as a direct sum of standard σ-submodules of slope μ(M); then by Corollary 4.1.4, the exact sequence

0 → N → M → M/N → 0

splits. This yields the desired result. In the general case, by Proposition 4.2.5, M has an HN-filtration 0 = M_{0} ⊂ M_{1} ⊂ ⋯ ⊂ M_{l} = M, with each successive quotient M_{i}/M_{i−1} semistable of slope s_{i}, and s_{1} < ⋯ < s_{l}. By the above, M_{i}/M_{i−1} admits a Dieudonné-Manin decomposition with all slopes s_{i}; the filtration then splits thanks to Corollary 4.1.4. Hence M admits a Dieudonné-Manin decomposition.

We will use the case of v_{K} nontrivial to establish the existence of Dieudonné-Manin decompositions also when v_{K} is trivial.

Definition 4.5.4. For R a (commutative) ring, let R((t^{ℚ})) denote the Hahn-Mal’cev-Neumann algebra of generalized power series ∑_{i∈ℚ} c_{i}t^{i}, where each c_{i} ∈ R and the set of i ∈ ℚ with c_{i} ≠ 0 is well-ordered (has no infinite decreasing subsequence); these series form a ring under formal series multiplication, with a natural valuation v given by v(∑_{i} c_{i}t^{i}) = min{i : c_{i} ≠ 0}. For R an algebraically closed field, R((t^{ℚ})) is also algebraically closed; see [31, Chapter 13] for this and other properties of these algebras.

Lemma 4.5.5. Suppose that k is algebraically closed and that K = k((t^{ℚ})). Let M be a σ-module over O[π^{−1}] such that M ⊗ R admits a basis of eigenvectors. Then any such basis is a basis of M.

Proof. We may identify Γ^{K} with the π-adic completion of O((t^{ℚ})). In so doing, elements of R can be viewed as formal sums ∑_{i∈ℚ} c_{i}t^{i} with c_{i} ∈ O[π^{−1}]. Suppose v ∈ M ⊗ R nonzero satisfies F v = π^{m}v. We can then formally write v = ∑_{i∈ℚ} v_{i}t^{i} with v_{i} ∈ M, and then we have F v_{i} = π^{m}v_{qi} for each i. If v_{i} ≠ 0 for some i < 0, we then have v_{q^{l}i} = π^{−lm}F^{l}v_{i}, but this violates the convergence condition defining R. Hence v_{i} = 0 for i < 0. Let R^{+} be the subring of R consisting of series ∑_{i} c_{i}t^{i} with c_{i} = 0 for i < 0. Now if M ⊗R admits a basis of eigenvectors, then we have just shown that each basis element belongs to M ⊗ R^{+}, and likewise for the dual basis of M^{∨} ⊗ R^{+}. We can then reduce modulo the ideal of R^{+} consisting of series with constant coefficient zero, to produce a basis of eigenvectors of M.

Remark 4.5.6. Beware that in the proof of Lemma 4.5.5, there do exist nonzero eigenvectors in M ⊗ R^{+} with constant coefficient zero; however, these eigenvectors cannot be part of a basis.

Theorem 4.5.7. Let M be a σ-module over R.

(a) There exists a Dieudonné-Manin decomposition of M.

(b) For any Dieudonné-Manin decomposition M = ⊕_{j=1}^{m}M_{c_{j},d_{j}} of M, let s_{1} < ⋯ < s_{l} be the distinct elements of the slope multiset of the decomposition. For i = 1, …, l, let M_{i} be the direct sum of M_{c_{j},d_{j}} over all j for which c_{j}/d_{j} ≤ s_{i}. Then the filtration 0 ⊂ M_{1} ⊂ ⋯ ⊂ M_{l} = M coincides with the HN-filtration of M.

(c) The slope multiset of any Dieudonné-Manin decomposition of M consists of the HN-slopes of M. In particular, the slope multiset does not depend on the choice of the decomposition.

Proof. (a) For v_{K} nontrivial, this is Proposition 4.5.3, so we need only treat the case of v_{K} trivial. Another way to say this is every σ-module M over O[π^{−1}] is isomorphic to a direct sum of standard σ-submodules.

Set notation as in Lemma 4.5.5. By Proposition 4.5.3, M ⊗ R is isomorphic to a direct sum of standard σ-modules. That direct sum has the form N ⊗ R, where N is the corresponding direct sum of standard σ-modules over O[π^{−1}]. The isomorphism M ⊗ R → N ⊗ R corresponds to an element of H^{0}((M^{∨} ⊗ N) ⊗ R), which extends to a basis of eigenvectors of [n]_{∗}(M^{∨} ⊗ N) ⊗ R for some n. By Lemma 4.5.5, this basis consists of elements of [n]_{∗}(M^{∨} ⊗ N); hence M ≅ N, as desired.

(b) By Lemma 3.4.9, any standard σ-module is stable; hence a direct sum of standard σ-modules of a single slope is semistable. Thus the described filtration is indeed an HN-filtration.

(c) This follows from (b).

Remark 4.5.8. The case of v_{K} trivial in Theorem 4.5.7(a) is precisely the standard Dieudonné-Manin classification of σ-modules over a complete discretely valued field with algebraically closed residue field. It is more commonly derived on its own, as in [14], [29], [18], or [19, Theorem 5.6].

Corollary 4.5.9. For any coprime integers c, d with d > 0, End(M_{c,d}) is a division algebra.

Proof. Suppose ϕ ∈ End(M_{c,d}) is nonzero. Decompose im(ϕ) according to Theorem 4.5.7; then each standard summand of im(ϕ) must have slope ≤ c/d by Corollary 4.1.4. On the other hand, each summand is a σ-submodule of M_{c,d}, so must have slope ≥ c/d again by Corollary 4.1.4. Thus each standard summand of im(ϕ) must have slope exactly c/d. In particular, there can be only one such summand, it must have rank d, and by Lemma 3.4.2, im(ϕ) = M_{c,d}. Hence ϕ is surjective; since ϕ is a linear map between free modules of the same finite rank, it is also injective. We conclude that End(M_{c,d}) is indeed a division algebra, as desired.

Proposition 4.5.10. A σ-module M over R is semistable (resp. stable) if and only if M ≅ M_{c,d}^{⊕n} for some c, d, n (resp. M ≅ M_{c,d} for some c, d).

Proof. This is an immediate corollary of Theorem 4.5.7. (Compare [17, Corollary 11.6].)

Remark 4.5.11. By Theorem 4.5.7, every σ-module over R decomposes as a direct sum of semistable σ-modules, i.e., the Harder-Narasimhan filtration splits. However, when v_{K} is nontrivial, this decomposition/splitting is not canonical, so it does not make sense to try to prove any descent results for such decompositions. (When v_{K} is trivial, the splitting is unique by virtue of Corollary 4.1.4.) Of course, the number and type of summands in a Dieudonné-Manin decomposition are unique, since they are determined by the HN-polygon; indeed, they constitute complete invariants for isomorphism of σ-modules over R (compare [17, Corollary 11.8]).

Proposition 4.5.12. Let M be a σ-module over R, let M_{1} be the first step in the Harder-Narasimhan filtration, and put d = rank(M_{1}). Then ∧^{d}M_{1} is the first step in the Harder-Narasimhan filtration of ∧^{d}M.

Proof. Decompose M according to Theorem 4.5.7, so that M_{1} is the direct sum of the summands of minimum slope s_{1}. Take the d-th exterior power of this decomposition (i.e., apply the Künneth formula); by Lemma 4.1.2, the minimum slope among the new summands is ds_{1}, achieved only by ∧^{d}M_{1}.

Remark 4.5.13. More generally, the first step in the Harder-Narasimhan filtration of ∧^{i}M is ∧^{i}M_{j}, for the smallest j such that rank(M_{j}) ≥ i; the argument is similar.

Proposition 4.5.14. Let M be a σ-module over R, and let M ≅ ⊕_{i=1}^{l}M_{c_{i},d_{i}} be a Dieudonné-Manin decomposition of M.

(a) If v_{K} is nontrivial, then there exists a nonzero homomorphism f : M_{c,d} → M of σ-modules if and only if c/d ≥ min_{i}{c_{i}/d_{i}}, and there exists a nonzero homomorphism f : M → M_{c,d} of σ-modules if and only if c/d ≤ max_{i}{c_{i}/d_{i}}.

(b) If v_{K} is trivial, then there exists a nonzero homomorphism f : M_{c,d} → M or f : M → M_{c,d} of σ-modules if and only if c/d ∈ {c_{1}/d_{1}, …, c_{l}/d_{l}}.

Proof. Apply Corollary 4.1.4.

4.6 The calculus of slopes

Theorem 4.5.7 affords a number of consequences for the calculus of slopes.

Definition 4.6.1. Let M be a σ-module over Γ_{an,con}. Define the absolute HNslopes and absolute HN-polygon of M to be the HN-slopes and HN-polygon of M ⊗ R, and denote the latter by P (M). We say M is pure (or isoclinic) of slope s if the absolute HN-slopes of M are all equal to s. By Proposition 4.5.10, M is isoclinic if and only if M ⊗R is semistable. We use the adjective unit-root to mean “isoclinic of slope 0”.

Remark 4.6.2. We will show later (Theorem 6.4.1) that the HN-filtration of M ⊗ R coincides with the base extension of the HN-filtration of M, which will mean that the absolute HN-slopes of M coincide with the HN-slopes of M.

Proposition 4.6.3. Let M and M′ be σ-modules over Γ_{an,con}. Let c_{1}, …, c_{m} and c′_{1}, …, c′_{n} be the absolute HN-slopes of M and M′, respectively.

(a) The absolute HN-slopes of M ⊕ M′ are c_{1}, …, c_{m}, c′_{1}, …, c′_{n}.

(b) The absolute HN-slopes of M ⊗M′ are c_{i}c′_{j} for i = 1, …, m, j = 1, …, n.

(c) The absolute HN-slopes of ∧^{d}M are c_{i_{1}} + ⋯ + c_{i_{d}} for all 1 ≤ i_{1} < ⋯ < i_{d} ≤ m.

(d) The absolute HN-slopes of [a]_{∗}M are ac_{1}, …, ac_{m}.

(e) The absolute HN-slopes of M(b) are c_{1} + b, …, c_{m} + b.

Proof. There is no harm in tensoring up to R, or in applying [a]_{∗} for some positive integer a. In particular, using Theorem 4.5.7, we may reduce to the case where M and M′ admit bases of eigenvectors, whose slopes must be the c_{i} and the c′_{j}. Then we obtain bases of eigenvectors of M ⊕ M′, M ⊗ M′, ∧^{d}M, [a]_{∗}M, M(b), and thus may read off the claims.

Proposition 4.6.4. Let M_{1}, M_{2} be σ-modules over Γ_{an,con} such that each absolute HN-slope of M_{1} is less than each absolute HN-slope of M_{2}. Then Hom(M_{1}, M_{2}) = 0.

Proof. Tensor up to R, then apply Theorem 4.5.7 and Proposition 4.5.14.

4.7 Splitting exact sequences

As we have seen already (e.g., in Proposition 4.3.4), a short exact sequence of σ-modules over Γ_{an,con} may or may not split. Whether or not it splits depends very much on the Newton polygons involved. For starters, we have the following.

Definition 4.7.1. Given Newton polygons P_{1}, …, P_{m}, define the sum P_{1} + ⋯ + P_{m} of these polygons to be the Newton polygon whose slope multiset is the union of the slope multisets of P_{1}, …, P_{m}. Also, write P_{1} ≥ P_{2} to mean that P_{1} lies above P_{2}.

Proposition 4.7.2. Let 0 → M_{1} → M → M_{2} → 0 be an exact sequence of σ-modules over Γ_{an,con}.

(a) We have P (M) ≥ P (M_{1}) + P (M_{2}).

(b) We have P (M) = P (M_{1}) + P (M_{2}) if and only if the exact sequence splits over R.

Proof. For (a), note that from the HN-filtrations of M_{1} ⊗ R and M_{2} ⊗ R, we obtain a semistable filtration of M ⊗ R whose Newton polygon is P (M_{1}) + P (M_{2}). The claim now follows from Proposition 3.5.4. For (b), note that if the sequence splits, then P (M) = P (M_{1}) + P (M_{2}) by Proposition 4.6.3. Conversely, suppose that P (M) = P (M_{1}) + P (M_{2}); we prove by induction on rank that if Γ_{an,con} = R, then the exact sequence splits. Our base case is where M_{1} and M_{2} are standard. If μ(M_{1}) ≤ μ(M_{2}), then the exact sequence splits by Corollary 4.1.4, so assume that μ(M_{1}) > μ(M_{2}). By Theorem 4.5.7, we have an isomorphism M ≅ M_{1} ⊕ M_{2}, which gives a map M_{2} → M. By Corollary 4.5.9, the composition M_{2} → M → M_{2} is either zero or an isomorphism; in the former case, by exactness the image of M_{2} → M must land in M_{1} ⊆ M. But that violates Corollary 4.1.4, so the composition M_{2} → M → M_{2} is an isomorphism, and the exact sequence splits.

We next treat the case of M_{1} nonstandard. Apply Theorem 4.5.7 to obtain a decomposition M_{1} ≅ N ⊕ N′ with N standard. We have

P (M) ≥ P (N) + P (M/N)  [by (a)]

≥ P (N) + P (M_{1}/N) + P (M_{2}) [by (a)]

= P (M_{1}) + P (M_{2}) [because N is a summand of M_{1}] = P (M) [by hypothesis].

Hence all of the inequalities must be equalities; in particular, P (M/N) = P (M_{1}/N) + P (M_{2}). By the induction hypothesis, the exact sequence 0 → M_{1}/N → M/N → M_{2} → 0 splits; consequently, the exact sequence 0 → N′ → M → M/N′ → 0 splits. But we have an exact sequence 0 → N → M/N′ → M_{2} → 0 and as above, we have P (M/N′) = P (N) + P (M_{2}), so this sequence also splits by the induction hypothesis. This yields the claim. To conclude, note that the case of M_{2} nonstandard follows from the case of M_{1} nonstandard by taking duals. Hence we have covered all cases.

Corollary 4.7.3. Let 0 → M_{1} → M → M_{2} → 0 be an exact sequence of σ-modules over R, such that every slope of M_{1} is less than or equal to every slope of M_{2}. Then the exact sequence splits; in particular, the HN-multiset of M is the union of the HN-multiset of M_{1} and M_{2}.

Proof. With the assumption on the slopes, the filtration induced by the HNfiltrations of M_{1} and M_{2} becomes an HN-filtration after possibly removing one redundant step in the middle (in case the highest slope of M_{1} coincides with the lowest slope of M_{2}). Thus its Newton polygon coincides with the HN-polygon, so Proposition 4.7.2 yields the claim.

Corollary 4.7.4. Let 0 = M_{0} ⊂ M_{1} ⊂ ⋯ ⊂ M_{l} = M be a filtration of a σmodule M over R by saturated σ-submodules with isoclinic quotients. Suppose that the Newton polygon of the filtration coincides with the HN-polygon of M. Then the filtration splits.

Remark 4.7.5. In certain contexts, one can obtain stronger splitting theorems; for instance, the key step in [20] is a splitting theorem for σ-modules with connection over Γ_{con} (in the notation of Section 2.3).

5 Generic and special slope filtrations

Given a σ-module over Γ_{con}[π^{−1}], we have two paradigms for constructing slopes and HN-polygons: the “generic” paradigm, in which we pass to Γ[π^{−1}] as if v_{K} were trivial, and the “special” paradigm, in which we pass to Γ_{an,con}. (See Section 7.3 for an explanation of the use of these adjectives.) In this chapter, we compare these paradigms: our main results are that the special HN-polygon lies above the generic one (Proposition 5.5.1), and that when the two polygons coincide, one obtains a common HN-filtration over Γ_{con}^{alg}[π^{−1}] (Theorem 5.5.2). This last result is a key tool for constructing slope filtrations in general.

Convention 5.0.1. We continue to retain notations as in Chapter 2. We again point out that when working over Γ_{con}[π^{−1}], the adjective “generic” will imply passage to Γ[π^{−1}], while the adjective “special” will imply passage to Γ_{an,con}. We also abbreviate such expressions as “generic absolute HN-slopes” to “generic HN-slopes”. (Keep in mind that the modifier “absolute” will ultimately be rendered superfluous anyway by Theorem 6.4.1.)

5.1 Interlude: lattices

Besides descending subobjects, we will also have need to descend entire σmodules; this matter is naturally discussed in terms of lattices.

Definition 5.1.1. Let R ↪ S be an injection of domains, and let M be a finite locally free S-module. An R-lattice in M is an R-submodule N of M such that the induced map N ⊗_{R} S → M is a bijection. If M is a σ-module, an R-lattice in the category of σ-modules is a module-theoretic R-lattice which is stable under F.

The existence of a Γ-lattice for a σ-module defined over Γ[π^{−1}] is closely tied to nonnegativity of the slopes.

Proposition 5.1.2. Let M be a σ-module over Γ_{con}[π^{−1}] with nonnegative generic slopes. Then M contains an F -stable Γ_{con}-lattice N. Moreover, if the generic slopes of M are all zero, then N can be chosen so that F : σ^{∗}M → M is an isomorphism.

Proof. Put M′ = M ⊗ Γ^{alg}[π^{−1}]; then by Theorem 4.5.7, we can write M′ as a direct sum of standard submodules, whose slopes by hypothesis are nonnegative. From this presentation, we immediately obtain a Γ^{alg}-lattice of M′ (generated by standard basis vectors of the standard submodules); its intersection with M gives the desired lattice.

Proposition 5.1.2 also has the following converse.

Proposition 5.1.3. Let M be a σ-module over Γ. Then the generic HNslopes of M are all nonnegative; moreover, they are all zero if and only if F : σ^{∗}M → M is an isomorphism.

Proof. Let e_{1}, …, e_{n} be a basis of M, and define the n × n matrix A over Γ by F e_{j} = ∑_{i} A_{ij}e_{i}. Suppose v is an eigenvector of M, and write v = ∑_{i} c_{i}e_{i} and F v = ∑_{i} d_{i}e_{i}; then min_{i}{w(d_{i})} − min_{i}{w(c_{i})} is the slope of v. But w(A_{ij}) ≥ 0 for all i, j, so min_{i}{w(d_{i})} ≥ min_{i}{w(c_{i})}. This yields the first claim. For the second claim, note on one hand that if F : σ^{∗}M → M is an isomorphism, then M^{∨} is also a σ-module over Γ, and so the generic HN-slopes of both M and M^{∨} are nonnegative. Since these slopes are negatives of each other by Proposition 4.6.3, they must all be zero. On the other hand, if the generic HNslopes of M are all zero, then by Theorem 4.5.7, M ⊗ Γ^{alg}[π^{−1}] admits a basis e_{1}, …, e_{n} of eigenvectors of slope 0. Put M′ = M ⊗ Γ^{alg}; for i = 0, …, n, let M′_{i} be the intersection of M′ with the Γ^{alg}[π^{−1}]-span of e_{1}, …, e_{i}. Then each M′_{i} is F -stable; moreover, M′_{i}/M′_{i−1} is spanned by the image of π^{c_{i}}e_{i} for some c_{i}, and so F : σ^{∗}(M′_{i}/M′_{i−1}) → (M′_{i}/M′_{i−1}) is an isomorphism for each i. It follows that F : σ^{∗}M′ → M′ is an isomorphism, as then is F : σ^{∗}M → M.

Remark 5.1.4. The results in this section can also be proved using cyclic vectors, as in [19, Proposition 5.8]; compare Lemma 5.2.4 below.

5.2 The generic HN-filtration

Since the distinction between v_{K} trivial and nontrivial was not pronounced in the previous chapter, it is worth taking time out to clarify some phenomena specific to the “generic” (v_{K} trivial) setting.

Proposition 5.2.1. For any σ-module M over Γ^{alg}[π^{−1}], there is a unique decomposition M = P_{1} ⊕ ⋯ ⊕ P_{l}, where each P_{i} is isoclinic, and the generic slopes μ(P_{1}), …, μ(P_{l}) are all distinct.

Proof. The existence of such a decomposition follows from Theorem 4.5.7; the uniqueness follows from repeated application of Corollary 4.1.4.

Definition 5.2.2. Let M be a σ-module over Γ^{alg}[π^{−1}]. Define the slope decomposition of M to be the decomposition M = P_{1} ⊕ ⋯ ⊕ P_{l} given by Proposition 5.2.1.

For the rest of this section, we catalog some routine methods for identifying the generic slopes of a σ-module.

Definition 5.2.3. Let M be a σ-module over Γ[π^{−1}] of rank n. A cyclic vector of M is an element v ∈ M such that v, F v, ⋯, F^{n−1}v form a basis of M.

Lemma 5.2.4. Let M be a σ-module over Γ[π^{−1}] of rank n. Let v be a cyclic vector of M, and define a_{0}, …, a_{n−1} ∈ Γ[π^{−1}] by the equation

F^{n}v + a_{n−1}F^{n−1}v + ⋯ + a_{0}v = 0.

Then the generic HN-polygon of M coincides with the Newton polygon of the polynomial x^{n} + a_{n−1}x^{n−1} + ⋯ + a_{0}.

Proof. There is no harm in assuming that K is algebraically closed, that π is fixed by σ, or that O is large enough that the slopes of M are all integers. Then M admits a basis e_{1}, …, e_{n} with F e_{i} = π^{c_{i}}e_{i} for some integers c_{i}. Put v_{0} = v. Given v_{l}, write v_{l} = x_{l,1}e_{1} +⋯+x_{l,n}e_{n}, put b_{l} = π^{c_{l}}x_{l,l}^{σ}/x_{l,l}, and put v_{l+1} = F v_{l} − b_{l}v_{l}. Then v_{l} lies in the span of e_{l+1}, …, e_{n}; in particular, v_{n} = 0. We then have

(F − b_{n−1}) ⋯ (F − b_{1})(F − b_{0})v = 0; since v is a cyclic vector, there is a unique way to write F^{n}v as a linear combination of v, F v, ⋯, F^{n−1}v. Hence we have an equality of operators

(F − b_{n−1}) ⋯ (F − b_{1})(F − b_{0}) = F^{n} + a_{n−1}F^{n−1} + ⋯ + a_{0},

from which the equality of polygons may be read off directly.

Remark 5.2.5. One can turn Lemma 5.2.4 around and use it to prove the existence of Dieudonné-Manin decompositions in the case of v_{K} trivial; for instance, this is the approach in [19, Theorem 5.6]. One of the essential difficulties in [19] is that there is no analogous way to “read off” the HN-polygon of a σ-module over Γ_{an,con}; this forces the approach to constructing the slope filtration over Γ_{an,con} to be somewhat indirect.

Lemma 5.2.4 is sometimes inconvenient to apply, because the calculus of cyclic vectors is quite “nonlinear”. The following criterion will prove to be more useful for our purposes.

Lemma 5.2.6. Let M be a σ-module over Γ^{alg}[π^{−1}]. Suppose that there exists a basis e_{1}, …, e_{n} of M with the property that the matrix A given by F e_{j} = ∑_{i} A_{ij}e_{i} satisfies w(AD^{−1} − I_{n}) > 0 for some n × n diagonal matrix D over Γ^{alg}[π^{−1}]. Then the generic slopes of M are equal to the valuations of the diagonal entries of D. Moreover, there exists an invertible matrix U over Γ^{alg} with w(U − I_{n}) > 0, w(DU^{σ}D^{−1} − I_{n}) > 0, and U^{−1}AU^{σ} = D.

Proof. One can directly solve for U; see [19, Proposition 5.9] for this calculation. Note that it does not matter whether the D^{−1} appears to the left or to the right of A, as a change of basis will flip it over to the other side; the entries of D will get hit by σ or its inverse, but their valuations will not change. Alternatively, the existence of U also follows from Proposition 5.4.5 below.

Remark 5.2.7. Lemmas 5.2.4 and 5.2.6 suggest that one can read off the generic HN-polygon of a σ-module over Γ[π^{−1}] by computing the slopes of eigenvectors of a matrix via which F acts on some basis. This does not work in general, as observed by Katz [18].

5.3 Descending the generic HN-filtration

In the generic setting (v_{K} trivial), we have the following descent property for Harder-Narasimhan filtrations.

Proposition 5.3.1. Let M be a σ-module over Γ[π^{−1}]. Then the Harder-Narasimhan filtration of M, tensored up to Γ^{alg}[π^{−1}], gives the Harder-Narasimhan filtration of M ⊗ Γ^{alg}[π^{−1}].

Proof. One can prove this by Galois descent, as in [19, Proposition 5.10]; here is an alternate argument. It suffices to check that the first step M′_{1} of the Harder-Narasimhan filtration of M′ = M ⊗ Γ^{alg}[π^{−1}] descends to Γ[π^{−1}]; by Lemma 3.6.2 and Proposition 4.5.12, we may by taking exterior powers reduce to the case where M′_{1} has rank 1. In particular, the least slope s_{1} must be an integer. By twisting, we may assume that s_{1} = 0. By Proposition 5.1.2, we can find a σ-stable Γ-lattice N of M; put N′ = N ⊗ Γ^{alg}. Then N′ ∩ M′_{1} may be characterized as the set of limit points, for the π-adic topology, of sequences of the form {F^{l}v_{l}}_{l=0}^{∞} with v_{l} ∈ N′ for each l. (This may be verified on a basis of d-eigenvectors for appropriate d thanks to Theorem 4.5.7, where it is evident.) The characterization of N′ ∩ M′_{1} we just gave is linear, so it cuts out a rank one submodule of N already over Γ. This yields the desired result.

5.4 de Jong’s reverse filtration

We now consider the case of σ-modules over Γ_{con}[π^{−1}], in which case we have a “generic” HN-filtration defined over Γ^{alg}[π^{−1}], and a “special” HN-filtration defined over Γ_{an,con}^{alg}. These two filtrations are not directly comparable, because they live over incompatible overrings of Γ_{con}[π^{−1}]. To compare them, we must use a “reverse filtration” that meets both halfway; the construction is due to de Jong [13, Proposition 5.8], but our presentation follows [19, Proposition 5.11] (wherein it is called the “descending generic filtration”). We first need a lemma that descends some eigenvectors from Γ^{alg} to Γ_{con}^{alg}; besides de Jong’s [13, Proposition 5.8], this generalizes a lemma of Tsuzuki [34, 4.1.1].

Lemma 5.4.1. Let M be a σ-module over Γ_{con}^{alg}[π^{−1}] all of whose generic slopes are nonpositive. Let v ∈ M ⊗ Γ^{alg}[π^{−1}] be an eigenvector of slope 0. Then v ∈ M.

Proof. By Proposition 5.1.2, we can find an F -stable Γ_{con}^{alg}-lattice N^{∨} of M^{∨}; the dual lattice N is an F^{−1}-stable Γ_{con}^{alg}-lattice of M. Let e_{1}, …, e_{n} be a basis of N, define the n × n matrix A over Γ_{con}^{alg} by the equation F^{−1}e_{j} = ∑_{i} A_{ij}e_{i}, and choose r > 0 such that A is invertible over Γ_{r}^{alg}. Let v ∈ M ⊗ Γ^{alg}[π^{−1}] be an eigenvector of slope 0; in showing that v ∈ M, there is no harm in assuming (by multiplying by a power of π as needed) that v ∈ N ⊗ Γ^{alg}. Write v = ∑ x_{i}e_{i} with x_{i} ∈ Γ^{alg}. Then for each l ≥ 0, we have

min_{i} min_{m≤l}{v_{m,r}(x_{i})} ≥ w_{r}(A) + min_{i} min_{m≤l}{v_{m,r}(x_{i}^{σ^{−1}})}

≥ w_{r}(A) + q^{−1} min_{i} min_{m≤l}{v_{m,r}(x_{i})}.

It follows that

min_{i} min_{m≤l}{v_{m,r}(x_{i})} ≥ qw_{r}(A)/(q − 1),

and so x_{i} ∈ Γ_{con}^{alg}. Hence v ∈ M, as desired.

We now proceed to construct the reverse filtration.

Definition 5.4.2. Let M be a σ-module over Γ^{alg}[π^{−1}] with slope decomposition P_{1} ⊕ ⋯ ⊕ P_{l}, labeled so that μ(P_{1}) > ⋯ > μ(P_{l}). Define the reverse filtration of M as the semistable filtration 0 = M_{0} ⊂ M_{1} ⊂ ⋯ ⊂ M_{l} = M with M_{i} = P_{1} ⊕ ⋯ ⊕ P_{i} for i = 1, …, l. By construction, its Newton polygon coincides with the generic Newton polygon of M.

Proposition 5.4.3. Let M be a σ-module over Γ_{con}^{alg}[π^{−1}]. Then the reverse filtration of M ⊗ Γ^{alg}[π^{−1}] descends to Γ_{con}^{alg}[π^{−1}].

Proof. Put M′ = M ⊗ Γ^{alg}[π^{−1}], and let 0 = M′_{0} ⊂ M′_{1} ⊂ ⋯ ⊂ M′_{l} = M′ be the reverse filtration of M′. It suffices to show that M′_{1} descends to Γ_{con}^{alg}[π^{−1}]; by Lemma 3.6.2, we may reduce to the case where rank M′_{1} = 1 by passing from M to an exterior power. By twisting, we may then reduce to the case μ(M′_{1}) = 0. By Proposition 3.3.2, M′_{1} is then generated by an eigenvector of slope 0; by Lemma 5.4.1, that eigenvector belongs to M. Hence M′_{1} descends to M, proving the claim. (Compare [19, Proposition 5.11].)

Remark 5.4.4. The reverse filtration actually descends all the way to Γ_{con}[π^{−1}] whenever K is perfect, but we will not need this.

It may also be useful for some applications to have a quantitative version of Proposition 5.4.3.

Proposition 5.4.5. Let M be a σ-module over Γ_{con}^{alg}[π^{−1}]. Suppose that for some r > 0, there exists a basis e_{1}, …, e_{n} of M with the property that the n×n matrix A given by F e_{j} = ∑_{i} A_{ij}e_{i} has entries in Γ_{r}^{alg}[π^{−1}]. Suppose further that w(AD^{−1} − I_{n}) > 0 and w_{r}(AD^{−1} − I_{n}) > 0 for some n × n diagonal matrix D over O[π^{−1}], with w(D_{11}) ≥ ⋯ ≥ w(D_{nn}). Then there exists an invertible n × n matrix U over Γ_{qr}^{alg} with w(U − I_{n}) > 0, w_{r}(U − I_{n}) > 0, w(DU^{σ}D^{−1} − I_{n}) > 0, w_{r}(DU^{σ}D^{−1} − I_{n}) > 0, such that U^{−1}AU^{σ}D^{−1} − I_{n} is upper triangular nilpotent.

Proof. Put c_{0} = min{w(AD^{−1} − I_{n}), w_{r}(AD^{−1} − I_{n})} and U_{0} = I_{n}. Given U_{l}, put A_{l} = U_{l}^{−1}AU_{l}^{σ}, and write A_{l}D^{−1} − I_{n} = B_{l} + C_{l} with B_{l} upper triangular nilpotent and C_{l} lower triangular. Suppose that min{w(A_{l}D^{−1} − I_{n}), w_{r}(A_{l}D^{−1} − I_{n})} ≥ c_{0}; put c_{l} = min{w(C_{l}), w_{r}(C_{l})}. Choose a matrix X_{l} over Γ_{qr} with

C_{l} + X_{l} = DX_{l}^{σ}D^{−1},
min{w(X_{l}), w(DX_{l}^{σ}D^{−1})} ≥ w(C_{l}),
min{w_{r}(X_{l}), w_{r}(DX_{l}^{σ}D^{−1})} ≥ c_{l}.

(This amounts to solving a system of equations of the form c + x = λ^{−1}x^{σ} for λ ∈ O; the analysis is as in Proposition 3.3.7.) Put U_{l+1} = U_{l}(I_{n} + X_{l}). We then have

A_{l+1}D^{−1} = (I_{n} − X_{l} + X_{l}^{2} − ⋯)(I_{n} + B_{l} + C_{l})(I_{n} + DX_{l}^{σ}D^{−1}),

whence w(C_{l+1}) ≥ w(C_{l}) + c_{0} and w_{r}(C_{l+1}) ≥ c_{l} + c_{0}. Consequently c_{l} ≥ (l+1)c_{0}, so the U_{l} converge under w_{qr} to a limit U such that U^{−1}AU^{σ}D^{−1} −I_{n} is upper triangular nilpotent, as desired.

Remark 5.4.6. For more on de Jong’s original application of the reverse filtration, see Section 7.5.

5.5 Comparison of polygons

Using the reverse filtration, we obtain the fundamental comparison between the generic and special Newton polygons of a σ-module over Γ_{con}[π^{−1}].

Proposition 5.5.1. Let M be a σ-module over Γ_{con}[π^{−1}]. Then the special HN-polygon of M (i.e., the HN-polygon of M ⊗ Γ_{an,con}^{alg}) lies above the generic HN-polygon (i.e., the HN-polygon of M ⊗ Γ^{alg}[π^{−1}]).

Proof. Tensor up to Γ_{con}^{alg}[π^{−1}], then apply Proposition 3.5.4 to the reverse filtration (Proposition 5.4.3).

The case where the polygons coincide is especially pleasant, and will be crucial to our later results.

Theorem 5.5.2. Let M be a σ-module over Γ_{con}[π^{−1}] whose generic and special HN-polygons coincide. Then the generic and special HN-filtrations of M ⊗ Γ^{alg}[π^{−1}] and M ⊗ Γ_{an,con}^{alg}, respectively, are both obtained by base change from an exhaustive filtration of M.

Proof. It suffices to check that the first steps of the generic and special HNfiltrations descend and coincide; by Lemma 3.6.2, we may reduce to the case where the least slope of the common polygon occurs with multiplicity 1. Choose a basis e_{1}, …, e_{n} of M, let v ∈ M ⊗ Γ^{alg}[π^{−1}] be a generator of the first step of the HN-filtration of M ⊗ Γ^{alg}[π^{−1}], and write v = a_{1}v_{1} + ⋯ + a_{n}v_{n}. By Proposition 5.3.1, a_{i}/a_{j} ∈ Γ[π^{−1}] for any i, j with a_{j} ≠ 0. By Corollary 4.7.4, the reverse filtration splits over Γ_{an,con}^{alg}; by Proposition 3.3.7(b1), it also splits over Γ_{con}^{alg}[π^{−1}]. Hence a_{i}/a_{j} ∈ Γ_{con}^{alg}[π^{−1}] for any i, j with a_{j} ≠ 0. Since Γ[π^{−1}] ∩ Γ_{con}^{alg}[π^{−1}] = Γ_{con}[π^{−1}], we have a_{i}/a_{j} ∈ Γ_{con}[π^{−1}] for any i, j with a_{j} ≠ 0. Hence the first step of the generic HN-filtration descends to Γ_{con}[π^{−1}]; let M_{1} be the corresponding σ-submodule of M⊗Γ_{con}[π^{−1}]. Let M′_{1} be the first step of the HN-filtration of M ⊗ Γ_{an,con}^{alg}; then as in the proof of Proposition 4.2.5, M′_{1} is the maximal σ-submodule of M ⊗ Γ_{an,con}^{alg} of slope μ(M′_{1}) = μ(M_{1}). In particular, M_{1} ⊗ Γ_{an,con}^{alg} ⊆ M′_{1}, and by Lemma 3.4.2, M_{1} ⊗ Γ_{an,con}^{alg} and M′_{1} actually coincide. Hence the first step of the special HNfiltration is also a base extension of M_{1}. This yields the claim. (Compare [19, Proposition 5.16].)

Remark 5.5.3. The conditions of Theorem 5.5.2 may look restrictive, and indeed they are: many “natural” examples of σ-modules over Γ_{con}[π^{−1}] do not have the same special and generic HN-polygons (e.g., the example of Section 7.3). However, Theorem 5.5.2 represents a critical step in the descent process for slope filtrations, as it allows us to move information from the generic paradigm into the special paradigm: specifically, the descent from algebraically closed K down to general K is much easier in the generic setting. Of course, in order to use this link, we must be able to force ourselves into the setting of Theorem 5.5.2; this is done in the next chapter.

Note that Theorem 5.5.2 implies that the generic and special HN-polygons of M coincide if and only if the generic HN-filtration descends to Γ_{con}^{alg}[π^{−1}]. In some applications, it may be more useful to have a quantitative refinement of that statement; we can give one (in imitation of the proof of Proposition 4.3.4) as follows.

Proposition 5.5.4. Let M be a σ-module over Γ_{con}^{alg}[π^{−1}]. Suppose that for some r > 0, there exists a basis e_{1}, …, e_{n} of M with the property that the n×n matrix A given by F e_{j} = ∑_{i} A_{ij}e_{i} has entries in Γ_{r}^{alg}[π^{−1}]. Suppose further that w(AD^{−1} − I_{n}) > 0 and w_{r}(AD^{−1} − I_{n}) > 0 for some n × n diagonal matrix D over O[π^{−1}], with w(D_{11}) ≥ ⋯ ≥ w(D_{nn}). Let U be an invertible n × n matrix over Γ^{alg} such that w(U − I_{n}) > 0, w(DU^{σ}D^{−1} − I_{n}) > 0, and U^{−1}AU^{σ} = D (as in Lemma 5.2.6). Then the generic and special HN-polygons of M coincide if and only if

w_{r}(U − I_{n}) > 0,  w_{r}(DU^{σ}D^{−1} − I_{n}) > 0.  (5.5.5)

Proof. The conditions on U imply that M admits a basis of eigenvectors over Γ_{con}^{alg}[π^{−1}]; the slopes of these eigenvectors then give both the generic and special HN-slopes. Conversely, if the generic and special HN-polygons of M coincide, then U is invertible over Γ_{con}^{alg} by Theorem 5.5.2. It thus remains to prove that if U has entries in Γ_{con}^{alg}, then in fact (5.5.5) holds. We start with a series of reductions. By Proposition 5.4.5, we may reduce to the case where AD^{−1} − I_{n} is upper triangular nilpotent. Considering all matrices now as block matrices by grouping rows and columns where the diagonal entries of D have the same valuation, we see that the matrix U must now be block upper triangular, with diagonal blocks invertible over O. Neither this property nor (5.5.5) is disturbed by multiplying U by a block diagonal matrix over O, so we may reduce to the case where U is block upper triangular with identity matrices on the diagonal. Finally, note that at this point it suffices to check the case where n = 2, w(D_{11}) > w(D_{22}), and

AD^{−1} = (\begin{matrix}1 & a \\ 0 & 1 \end{matrix}) , U = (\begin{matrix}1 & u \\ 0 & 1 \end{matrix}) ,

as the general case follows by repeated application of this case. Put λ ∈ D_{11}D_{22}^{−1} ∈ πO. Then what we are to show is that given a ∈ Γ_{r}^{alg}, u ∈ Γ_{con}^{alg} with

w_{r}(a) > 0,  w(u) > 0,  λa + u = λu^{σ},  (5.5.6)

we must have u ∈ Γ_{r}^{alg} and w_{r}(u) > 0. Before verifying this, we make one further simplifying reduction, using the fact that there is no harm in replacing a by a − b + λ^{−1}b^{σ^{−1}} as long as w_{r}(b) > 0 and w_{r}(λ^{−1}b^{σ^{−1}}) > 0. Note that for b = π^{m}[\overline{x}] with \overline{x} ∈ K^{alg}, the condition that

w_{r}(b) ≤ w_{r}(λ^{−1}b^{σ^{−1}})

is equivalent to

v_{K}(\overline{x}) ≤ −\frac{q}{(q − 1)r} w(λ).

Moreover, this condition and the bound w_{r}(b) ≥ 0 together imply that w(λ^{−1}b^{σ^{−1}}) > 0. Now write a = ∑_{i=0}^{∞} π^{i}[\overline{a_{i}}], and put c = qw(λ)/(q − 1)r. For each i, let j_{i} be the smallest nonnegative integer such that q^{−j_{i}}v_{K}(\overline{a_{i}}) > −c. We may then replace a by

a′ = ∑_{i=0}^{∞} λ^{−1}λ^{−σ^{−1}} ⋯ λ^{−σ^{j_{i}−1}}(π^{i}[\overline{a_{i}}])^{σ^{−j_{i}}}

without disturbing the truth of (5.5.6). In particular, we may reduce to the case where v_{n}(a) > −c for all n ≥ 0. Under these conditions, put m = w(λ) > 0, and note that if v_{n}(u) ≤ −c/q for some u, then v_{n+m}(λu^{σ}) = qv_{n}(u) ≤ −c, so the equation λa + u = λu^{σ} implies v_{n+m}(u) = qv_{n}(u). By induction on l, we have

v_{n+lm}(u) = q^{l}v_{n}(u)

for all nonnegative integers l, but this contradicts the hypothesis that u ∈ Γ_{con}^{alg}. Consequently, we must have v_{n}(u) > −c/q for all n. Since −c/q ≥ −c(q − 1)/q = −w(λ)/r, the bound v_{n}(u) > −c/q implies that v_{n,r}(u) > 0 for n ≥ m. Since u ≡ 0 (mod λ), we have w_{r}(u) > 0, as desired.

6 Descents

We now show that the formation of the Harder-Narasimhan filtration commutes with base change, thus establishing the slope filtration theorem; the strategy is to show that a σ-module over Γ_{an,con} admits a model over Γ_{con} whose special and generic Newton polygons coincide, then invoke Theorem 5.5.2. The material here is derived from [19, Chapter 6], but our presentation here is much more streamlined.

6.1 A matrix lemma

The following lemma is analogous to [19, Proposition 6.8], but in our new approach, we can prove much less and still eventually get the desired conclusion; this simplifies the matrix calculation considerably.

Lemma 6.1.1. For r > 0, suppose that Γ = Γ^{K} contains enough r_{0}-units for some r_{0} > qr. Let D be an invertible n × n matrix over Γ_{[r,r]}, and put h = −w_{r}(D)−w_{r}(D^{−1}). Let A be an n×n matrix over Γ_{[r,r]} such that w_{r}(AD^{−1} − I_{n}) > h/(q−1). Then there exists an invertible n×n matrix U over Γ_{[r,qr]} such that U^{−1}AU^{σ}D^{−1} − I_{n} has entries in πΓ_{r} and w_{r}(U^{−1}AU^{σ}D^{−1} − I_{n}) > 0.

Proof. Put c_{0} = w_{r}(AD^{−1} − I_{n}) − h/(q − 1). Define sequences of matrices U_{0}, U_{1}, … and A_{0}, A_{1}, … as follows. Start with U_{0} = I_{n}. Given an invertible n×n matrix U_{l} over Γ_{[r,qr]}, put A_{l} = U_{l}^{−1}AU_{l}^{σ}. Suppose that w_{r}(A_{l}D^{−1}−I_{n}) ≥ c_{0} + h/(q − 1); put

c_{l} = min_{m≤0}{v_{m,r}(A_{l}D^{−1} − I_{n})} − h/(q − 1),

so that c_{l} ≥ c_{0} > 0. Let ∑ u_{ijlm}π^{m} be a semiunit presentation of (A_{l}D^{−1} − I_{n})_{ij}. Let X_{l} be the n × n matrix with (X_{l})_{ij} = ∑_{m≤0} u_{ijlm}π^{m}, and put U_{l+1} = U_{l}(I_{n} + X_{l}). By Lemma 2.5.3, for m ≤ 0 and s ∈ [r, qr],

w_{s}(u_{ijlm}π^{m}) ≥ (s/r)w_{r}(u_{ijlm}π^{m})

≥ (s/r) min_{k≤m}{v_{k,r}(A_{l}D^{−1} − I_{n})}

≥ (s/r)(c_{l} + h/(q − 1));

hence U_{l+1} is also invertible over Γ_{[r,qr]}. Moreover,

w_{r}(DX_{l}^{σ}D^{−1}) ≥ w_{r}(D) + w_{r}(X_{l}^{σ}) + w_{r}(D^{−1})
= w_{qr}(X_{l}) − h
≥ q(c_{l} + h/(q − 1)) − h
≥ qc_{l} + h/(q − 1).

Since

A_{l+1}D^{−1} = (I_{n} + X_{l})^{−1}(A_{l}D^{−1})(I_{n} + DX_{l}^{σ}D^{−1}),

we then have w_{r}(A_{l+1}D^{−1} −I_{n}) ≥ c_{0} +h/(q−1), so the iteration may continue. We now prove by induction that c_{l} ≥ (l + 1)c_{0} for l = 0, 1, …; this is clearly true for l = 0. Given the claim for l, write

(I_{n} + X_{l})^{−1}A_{l}D^{−1} =
I_{n} + (A_{l}D^{−1} − I_{n} − X_{l}) − X_{l}(A_{l}D^{−1} − I_{n}) + X_{l}^{2}(I_{n} + X_{l})^{−1}A_{l}D^{−1}.

Note that

v_{m}(A_{l}D^{−1} − I_{n} − X_{l}) = ∞  (m ≤ 0)
w_{r}(X_{l}(A_{l}D^{−1} − I)) ≥ (l + 2)c_{0} + 2h/(q − 1)
w_{r}(X_{l}^{2}(I_{n} + X_{l})^{−1}A_{l}D^{−1}) ≥ 2(l + 1)c_{0} + 2h/(q − 1).

Putting this together, this means

v_{m,r}((I_{n} + X_{l})^{−1}A_{l}D^{−1} − I_{n}) ≥ (l + 2)c_{0} + h/(q − 1) (m ≤ 0).

Since w_{r}(DX_{l}^{σ}D^{−1}) ≥ qc_{l} + h/(q − 1) ≥ (l + 2)c_{0} + h/(q − 1), we have v_{m,r}(A_{l+1}D^{−1} − I_{n}) ≥ (l + 2)c_{0} + h/(q − 1) for m ≤ 0, i.e., c_{l+1} ≥ (l + 2)c_{0}. Since w_{s}(X_{l}) ≥ (s/r)(c_{l} + h/(q − 1)) for s ∈ [r, qr], and c_{l} → ∞ as l → ∞, the sequence {U_{l}} converges to a limit U, which is an invertible n × n matrix over Γ_{[r,qr]} satisfying w_{r}(U^{−1}AU^{σ}D^{−1} − I_{n}) ≥ c_{0} + h/(q − 1) > 0. Moreover, by construction, we have v_{m}(U^{−1}AU^{σ}D^{−1} − I_{n}) = ∞ for m ≤ 0; by Corollary 2.5.6, U^{−1}AU^{σ}D^{−1} − I_{n} has entries in πΓ_{r}, as desired.

6.2 Good models of σ-modules

We now give a highly streamlined version of some arguments from [19, Chapter 6], that produce “good integral models” of σ-modules over Γ_{an,con}.

Lemma 6.2.1. In Lemma 6.1.1, suppose that A and D are both invertible over Γ_{an,r}. Then U is invertible over Γ_{an,qr}.

Proof. Put B = U^{−1}AU^{σ}D^{−1}, so that B is invertible over Γ_{r}. In the equation

U^{−1}AU^{σ} = BD,

the matrices A, U^{σ}, B, D are all invertible over Γ_{[r/q,r]}, so U must be as well. Since the entries of U and U^{−1} already belong to Γ_{[r,qr]}, they in fact belong to Γ_{[r/q,qr]} by Corollary 2.5.7. Repeating this argument, we see that U is invertible over Γ_{[r/q^{i},qr]} for all positive integers i, yielding the desired result.

Proposition 6.2.2. Let M be a σ-module over Γ_{an,con}^{K}. Then there exists a finite separable extension L of K and a σ-module N over Γ_{con}^{L}[π^{−1}] such that N ⊗ Γ_{an,con}^{L} ≅ M ⊗ Γ_{an,con}^{K} and the generic and special HN-polygons of N coincide.

Proof. Note that it is enough to do this with L pseudo-finite separable, since applying F allows to pass from L to L^{q}. Also, there is no harm in assuming that the slopes of M are integers, after applying [a]_{∗} as necessary. Let e_{1}, …, e_{n} be a basis of M, and define the invertible n × n matrix A over Γ_{an,con}^{K} by F e_{j} = ∑_{i} A_{ij}e_{i}. By Theorem 4.5.7, there exists an invertible n × n matrix U over Γ_{an,con}^{alg} and a diagonal n × n matrix D whose diagonal entries are powers of π, such that U^{−1}AU^{σ} = D. Put h = max_{i,j}{w(D_{ii})−w(D_{jj})} = −w(D) − w(D^{−1}). Choose r > 0 such that Γ has enough r_{0}-units for some r_{0} > qr, A is defined and invertible over Γ_{an,r}^{K}, and U is defined and invertible over Γ_{an,qr}^{alg}. Let M_{r} be the Γ_{an,qr}^{K}-span of the e_{i}. By Lemma 2.4.12, we can choose L pseudo-finite separable over K such that Γ^{L} has enough r_{0}-units, and such that there exists an n × n matrix V over Γ_{[r,qr]}^{L} with w_{s}(V − U) > −w_{s}(U^{−1}) + qh/(q − 1) for s ∈ [r, qr]. Since

V^{−1}AV^{σ}D^{−1} = (U^{−1}V)^{−1}D(U^{−1}V)^{σ}D^{−1},

it follows that w_{r}(V^{−1}AV^{σ}D^{−1} − I_{n}) > h/(q − 1). By Lemma 6.1.1, there exists an invertible n × n matrix W over Γ_{[r,qr]}^{L} for which the matrix B = (V W)^{−1}A(V W)^{σ} is such that BD^{−1} − I_{n} has entries in πΓ_{r}^{L} and w_{r}(BD^{−1} − I_{n}) > 0. By Lemma 6.2.1, the matrix V W is actually invertible over Γ_{an,qr}^{L}. Let N be the σ-module over Γ_{con}^{L}[π^{−1}]-module generated by v_{1}, …, v_{n} with Frobenius action defined by F v_{j} = ∑_{i} B_{ij}v_{i}; then by what we have just shown, N ⊗Γ_{an,con}^{L} ≅ M ⊗Γ_{an,con}^{L}. By Lemma 5.2.6, the generic HN-slopes of N are the w(D_{ii}), which by construction are also the special HN-slopes of N. (Compare [19, Proposition 6.9].)

Remark 6.2.3. It should also be possible to establish Proposition 6.2.2 without the finite extension L of K.

6.3 Isoclinic σ-modules

Before proceeding to the general descent problem for HN-filtrations, we analyze the isoclinic case.

Definition 6.3.1. Let M be a σ-module over Γ_{con}[π^{−1}]. We say that M is isoclinic (of slope μ(M)) if M ⊗ Γ^{alg}[π^{−1}] is isoclinic. By Proposition 5.5.1, this implies that M ⊗ Γ_{an,con}^{alg} is also isoclinic.

Proposition 6.3.2. Let M be a σ-module over Γ_{con}[π^{−1}] which is unit-root (isoclinic of slope 0). Then

H^{0}(M) = H^{0}(M ⊗ Γ[π^{−1}]) = H^{0}(M ⊗ Γ_{an,con}[π^{−1}]);

in particular, if K is algebraically closed, then M admits a basis of eigenvectors.

Proof. There is no harm (thanks to Corollary 2.5.8) in assuming from the outset that K is algebraically closed. In this case, the eigenvectors of M ⊗ Γ^{alg}[π^{−1}] all have slope 0 by Corollary 4.1.4; by Lemma 5.4.1, they all belong to M. Hence M admits a basis of eigenvectors; in particular, M ≅ M_{0,1}^{⊕n}. Then H^{0}(M) = H^{0}(M ⊗ Γ_{an,con}[π^{−1}]) by Proposition 3.3.4.

Theorem 6.3.3. (a) The base change functor, from isoclinic σ-modules over Γ_{con}[π^{−1}] of some prescribed slope s to isoclinic σ-modules over Γ[π^{−1}] of slope s, is fully faithful.

(b) The base change functor, from isoclinic σ-modules over Γ_{con}[π^{−1}] of slope s to isoclinic σ-modules over Γ_{an,con} of slope s, is an equivalence of categories.

In particular, any isoclinic σ-module over Γ_{con}^{alg}[π^{−1}] is isomorphic to a direct sum of standard σ-modules of the same slope (and hence is semistable by Proposition 4.5.10).

Proof. To see that the functors are fully faithful, let M and N be isoclinic σ-modules over Γ_{con}[π^{−1}] of the same slope s. Then M^{∨} ⊗ N is unit-root, and Hom(M, N) = H^{0}(M^{∨} ⊗ N), so the full faithfulness assertion follows from Proposition 6.3.2. To see that the functor in (b) is essentially surjective, apply Proposition 6.2.2 to produce an F -stable Γ_{con}^{L}[π^{−1}]-lattice N in M ⊗Γ_{an,con}^{L} for some finite separable extension L of K, which we may take to be Galois. Note that N is unique by full faithfulness of the base change functor (from Γ_{con}^{L}[π^{−1}] to Γ_{an,con}^{L}); hence the action of G = Gal(L/K) on M ⊗ Γ_{an,con}^{L} induces an action on N. By ordinary Galois descent, there is a unique Γ_{con}[π^{−1}]-lattice M_{b} of N such that N = M_{b} ⊗ Γ_{con}^{L}[π^{−1}]; because of the uniqueness, M_{b} is F -stable. This yields the desired result. (Compare [19, Theorem 6.10].)

Remark 6.3.4. The functor in (a) is not essentially surjective in general. For instance, if K = k((t)) with k perfect, M is an isoclinic σ-module over Γ_{con}, and b_{n} is the highest break of the representation of Gal(K^{sep}/K) on the images modulo π^{n} of the eigenvectors of M ⊗ Γ^{alg}, then b_{n}/n is bounded. By contrast, if M is an isoclinic σ-module over Γ, b_{n}/n need not be bounded.

Incidentally, Theorem 6.3.3 allows us to give a more succinct characterization of isoclinic σ-modules, which one could take as an alternate definition.

Proposition 6.3.5. Let c, d be integers with d > 0. Then a σ-module M over Γ_{an,con} is isoclinic of slope s = c/d if and only if M admits a Γ_{con}-lattice N such that π^{−c}F^{d} maps some (any) basis of N to another basis of N.

Proof. If such a lattice exists, then we may apply Proposition 5.1.3 to ([d]_{∗}M)(−c) to deduce that its generic HN-slopes are all zero. By Proposition 4.6.3, M is isoclinic of slope c/d. Conversely, suppose M is isoclinic of slope s; then ([d]_{∗}M)(−c) is isoclinic of slope 0. By Theorem 6.3.3, ([d]_{∗}M)(−c) admits a unique F -stable Γ_{con}[π^{−1}]- lattice N′. By Proposition 5.1.2, N′ in turn admits an F -stable Γ_{con}-lattice N; by Proposition 5.1.3, the Frobenius on N carries any basis to another basis.

6.4 Descent of the HN-filtration

At last, we are ready to establish the slope filtration theorem [19, Theorem 6.10].

Theorem 6.4.1. Let M be a σ-module over Γ_{an,con}. Then there exists a unique filtration 0 = M_{0} ⊂ M_{1} ⊂ ⋯ ⊂ M_{l} = M of M by saturated σ-submodules with the following properties.

(a) For i = 1, …, l, the quotient M_{i}/M_{i−1} is isoclinic of some slope s_{i}.

(b) s_{1} < ⋯ < s_{l}.

Moreover, this filtration coincides with the Harder-Narasimhan filtration of M.

Proof. Since isoclinic σ-modules are semistable by Theorem 6.3.3, any filtration as in (a) and (b) is a Harder-Narasimhan filtration. In particular, the filtration is unique if it exists. To prove existence, it suffices to show that the HN-filtration of M′ = M⊗Γ_{an,con}^{alg} descends to Γ_{an,con}. Let M′_{1} be the first step of that filtration; by Lemma 3.6.2, it is enough to check that M′_{1} descends to Γ_{an,con} in the case rank(M′_{1}) = 1. By Proposition 6.2.2, there exists a finite separable extension L of K and a σmodule N over Γ_{con}^{L}[π^{−1}] such that M ⊗ Γ_{an,con}^{L} ≅ N ⊗ Γ_{an,con}^{L}, and N has the same generic and special HN-polygons. Of course there is no harm in assuming L is Galois with Gal(L/K) = G. By Theorem 5.5.2, M′_{1} descends to Γ_{an,con}^{L}; let M_{1} be the descended submodule of M ⊗ Γ_{an,con}^{L}. Let e_{1}, …, e_{n} be a basis of M, let v be a generator of M_{1}, and write v = a_{1}e_{1}+ ⋯+a_{n}e_{n} with a_{i} ∈ Γ_{an,con}^{L}. Then for each i, j with a_{j} ≠ 0, a_{i}/a_{j} ∈ Frac Γ_{an,con}^{L} is invariant under G. By Corollary 2.4.11, a_{i}/a_{j} ∈ Frac Γ_{an,con}^{K} for each i, j with a_{j} ≠ 0; clearing denominators, we obtain a nonzero element of M_{1} ∩ M. Hence M_{1} descends to Γ_{an,con}, as desired. (Compare [19, Theorem 6.10].)

Corollary 6.4.2. For any extension K′ of K (complete with respect to a valuation v_{K′} extending v_{K}, such that Γ^{K′} has enough units), and any σ-module M over Γ_{an,con}^{K}, the HN-filtration of M ⊗ Γ_{an,con}^{K′} coincides with the result of tensoring the HN-filtration of M with Γ_{an,con}^{K′}. In other words, the formation of the HN-filtration commutes with base change.

Proof. The characterization of the HN-filtration given by Theorem 6.4.1 is stable under base change.

Corollary 6.4.3. A σ-module over Γ_{an,con} is semistable if and only if it is isoclinic.

Proof. If M is an isoclinic σ-module over Γ_{an,con}, then M is semistable by Proposition 4.5.10. Conversely, if M is not isoclinic, then by Theorem 6.4.1, M admits a nonzero σ-submodule M_{1} with μ(M_{1}) < μ(M), so M is not semistable.

7 Complements

7.1 Differentials and the slope filtration

The slope filtration turns out to behave nicely with respect to differentials; this is what allows the application to Crew’s conjecture.

Definition 7.1.1. Let S/R be an extension of topological rings. A module of continuous differentials is a topological S-module Ω_{S/R}^{1} equipped with a continuous R-linear derivation d : S → Ω_{S/R}^{1}, having the following universal property: for any topological S-module M equipped with a continuous Rlinear derivation D : S → M, there exists a unique morphism ϕ : Ω_{S/R}^{1} → M of topological S-modules such that D = ϕ ∘ d. Since the definition is via a universal property, the module of continuous differentials is unique up to unique isomorphism if it exists at all.

Convention 7.1.2. For the remainder of this section, assume that Γ_{con}, viewed as a topological O-algebra via the levelwise topology, has a module of continuous differentials Ω_{Γ_{con}/O}^{1} which is finite free over Γ_{con}. In this case, for any finite separable extension L of K, Ω_{Γ_{con}/O}^{1} ⊗_{Γ_{con}} Γ_{an,con}^{L} is a module of continuous differentials of Γ_{an,con}^{L} over O[π^{−1}].

Example 7.1.3. If K = k((t)) as in Section 2.3, we have a module of continuous differentials for Γ_{con} over O given by Ω_{Γ_{con}/O}^{1} = Γ_{con} dt with the formal derivation d sending ∑ c_{i}t^{i} to (∑ ic_{i}t^{i−1})dt.

Remark 7.1.4. Note that Ω_{Γ_{con}/O}^{1} may be viewed as a σ-module, via the action of dσ. Since dσ(ω) ≡ 0 (mod π) for any ω ∈ Ω_{Γ_{con}/O}^{1}, by Proposition 5.1.3 the generic slopes of Ω_{Γ_{con}/O}^{1} as a σ-module are all positive. By Proposition 5.5.1, the special slopes of Ω_{Γ_{an,con}/O}^{1} as a σ-module are also nonnegative.

Definition 7.1.5. For S = Γ_{con}, Γ_{con}[π^{−1}], Γ_{an,con}, define a ∇-module over S to be a finite free S-module equipped with an integrable connection ∇ : M → M ⊗ Ω_{S/O}^{1}. (Integrability here means that the composition of ∇ with the induced map M ⊗ Ω_{S/O}^{1} → M ⊗ ∧_{S}^{2}Ω_{S/O}^{1} is the zero map.) Define a (σ, ∇)- module over S to be a finite free S-module M equipped with the structures of both a σ-module and a ∇-module, which commute as in the following diagram:

\xymatrix{M \ar[r]^{∇} \ar[d]^{F} & M ⊗ Ω_{S/O}^{1} \ar[d]^{F ⊗dσ} \\ M \ar[r]^{∇} & M ⊗ Ω_{S/O}^{1}.}

Proposition 7.1.6. Let M be a (σ, ∇)-module over Γ_{an,con}. Then each step of the HN-filtration of M is a (σ, ∇)-submodule of M.

Proof. Let M_{1} be the first step of the HN-filtration, which is isoclinic by Theorem 6.4.1; it suffices to check that M_{1} is a (σ, ∇)-submodule of M. To simplify notation, write N for Ω_{Γ_{an,con}/O}^{1}. Then the map M_{1} → (M/M_{1}) ⊗ N induced by ∇ is a morphism of σ-modules, and the slopes of (M/M_{1})⊗N are all strictly greater than the slope of M_{1} by Remark 7.1.4. By Proposition 4.6.4, the map M_{1} → (M/M_{1}) ⊗ N must be zero; that is, M_{1} is a ∇-submodule of M. This proves the desired result.

Proposition 7.1.7. Let M be an isoclinic σ-module over Γ_{con}[π^{−1}] such that M ⊗ Γ_{an,con} admits the structure of a (σ, ∇)-module. Then M admits a corresponding structure of a (σ, ∇)-module.

Proof. Write N for Ω_{Γ_{con}[π^{−1}]/O}^{1}, so that ∇ induces an additive map M ⊗ Γ_{an,con} → (M ⊗ N) ⊗ Γ_{an,con}. Pick a basis e_{1}, …, e_{n} of M, and define the map f : M → M ⊗ N by

f (∑ c_{i}e_{i}) = ∑_{i} e_{i} ⊗ dc_{i}.

Then ∇ − f is Γ_{an,con}-linear, so we may view it as an element v of M^{∨} ⊗ M ⊗ N ⊗ Γ_{an,con}. That element satisfies F v − v = w for some w ∈ M^{∨} ⊗ M ⊗ N by the commutation relation between F and ∇. However, M^{∨} ⊗ M is unitroot, so the (generic and special) slopes of M^{∨} ⊗ M ⊗ N are all positive by Remark 7.1.4. By Theorem 4.5.7 and Proposition 3.3.7(b1), it follows that v ∈ M^{∨} ⊗ M ⊗ N, so ∇ acts on M, as desired. (Compare [19, Theorem 6.12] and [3, Lemme V.14].)

Remark 7.1.8. We suspect there is a more conceptual way of saying this in terms of splitting a certain exact sequence of σ-modules.

7.2 The p-adic local monodromy theorem

We recall briefly how the slope filtration theorem, plus a theorem of Tsuzuki, yields the p-adic local monodromy theorem (formerly Crew’s conjecture).

Convention 7.2.1. Throughout this section, retain notation as in Section 2.3, i.e., Γ_{an,con} is the Robba ring. Note that in this case, the integrability condition in Definition 7.1.5 is vacuous, since Ω_{Γ_{an,con}/O}^{1} is free of rank 1.

Definition 7.2.2. We say a ∇-module M over Γ_{an,con} is constant if it is spanned by the kernel of ∇; equivalently, M is isomorphic to a direct sum of trivial ∇-modules. (The “trivial” ∇-module here means Γ_{an,con} itself with the connection given by d.) We say M is quasi-constant if M ⊗ Γ_{an,con}^{L} is constant for some finite separable extension L of K. We say a (σ, ∇)-module is (quasi-)constant if its underlying ∇-module is (quasi-)constant.

The key external result we need here is the following result essentially due to Tsuzuki [35]. A simplified proof of Tsuzuki’s result has been given by Christol [5]; however, see [2] for the corrections of some errors in [5].

Proposition 7.2.3. Let M be a unit-root (σ, ∇)-module over Γ_{con}[π^{−1}]. Then M ⊗ Γ_{an,con} is quasi-constant; in fact, for some finite separable extension L of K, M ⊗ Γ_{con}^{L}[π^{−1}] admits a basis in the kernel of ∇.

Proof. Suppose first that M = M_{0} ⊗ Γ_{con}[π^{−1}] for a unit-root (σ, ∇)-module M_{0} over Γ_{con}. Then our desired statement is Tsuzuki’s theorem [35, Theorem 4.2.6]. To reduce to this case, apply Proposition 5.1.2 to obtain a σ-stable Γ_{con}-lattice M_{0}; by applying Frobenius repeatedly, we see that M_{0} must also be ∇-stable. Thus Tsuzuki’s theorem applies to give the claimed result.

Definition 7.2.4. We say a ∇-module M over Γ_{an,con} is said to be unipotent if it admits an exhaustive filtration by ∇-submodules whose successive quotients are constant. We say M is quasi-unipotent if M ⊗ Γ_{an,con}^{L} is unipotent for some finite separable extension L of K. We say a (σ, ∇)-module is (quasi-)unipotent if its underlying ∇-module is (quasi-)unipotent.

Theorem 7.2.5 (p-adic local monodromy theorem). With notations as in Section 2.3 (i.e., Γ_{an,con} is the Robba ring), let M be a (σ, ∇)-module over Γ_{an,con}. Then M is quasi-unipotent.

Proof. By Proposition 7.1.6, each step of the HN-filtration of M is ∇-stable. It is thus enough to check that each successive quotient is quasi-unipotent; in other words, we may assume that M itself is isoclinic. Note that the definition of unipotence does not depend on the Frobenius lift, so there is no harm in either applying the functor [b]_{∗} or in twisting. We may thus reduce to the case where M is isoclinic of slope zero (i.e., is unit-root). Applying Proposition 7.2.3 then yields the desired result.

Example 7.2.6. In Theorem 7.2.5, it can certainly happen that M fails to be quasi-constant. For instance, if u^{σ} = qu, and M has rank two with a basis e_{1}, e_{2} such that

F e_{1} = e_{1}, F e_{2} = qe_{2}

∇e_{1} = 0, ∇e_{2} = e_{1} ⊗ \frac{du}{u},

then M is a (σ, ∇)-module which is unipotent but not quasi-constant.

Remark 7.2.7. The fact that quasi-unipotence follows from the existence of a slope filtration as in Theorem 6.4.1 was originally pointed out by Tsuzuki [36, Theorem 5.2.1]. Indeed, that observation was the principal motivation for the construction of slope filtrations of σ-modules in [19].

Remark 7.2.8. We remind the reader that Theorem 7.2.5 has also been proved (independently) by André [1] and by Mebkhout [30], using the index theory for p-adic differential equations developed in a series of papers by Christol and Mebkhout [6], [7], [8], [9]. This represents a completely orthogonal approach to ours, as it primarily involves the structure of the connection rather than the Frobenius. The different approaches seem to have different strengths. For example, on one hand, the Christol-Mebkhout approach seems to say more about p-adic differential equations on annuli over p-adic fields which are not discretely valued. On the other hand, our approach has a certain flexibility that the Christol-Mebkhout approach lacks; for instance, it carries over directly to the q-difference situation considered by André and di Vizio in [1], whereas the analogue of the Christol-Mebkhout theory seems much more difficult to develop. It also carries over to the setting of “fake annuli” arising in the problem of semistable reduction for overconvergent F -isocrystals: in this setting, one replaces k((t)) by the completion of k[t_{1}, t_{1}^{−1}, …, t_{n}, t_{n}^{−1}] for a valuation which totally orders monomials (i.e., the valuations of t_{1}, …, t_{n} are linearly independent over ℚ). See [25] for further details.

7.3 Generic versus special revisited

The adjectives “generic” and “special” were introduced in Chapter 5 to describe the two paradigms for attaching slopes to σ-modules over Γ_{con}[π^{−1}]. Here is a bit of clarification as to why this was done. Throughout this section, retain notation as in Section 2.3. Let X → Spec k⟦t⟧ be a smooth proper morphism, for k a field of characteristic p > 0. Then the crystalline cohomology of X, equipped with the action of the absolute Frobenius, gives a (σ, ∇)-module M over the ring O⟦u⟧; the crystalline cohomology of the generic fibre corresponds to M ⊗ Γ, whereas the crystalline cohomology of the special fibre corresponds to M ⊗ (O⟦u⟧/uO⟦u⟧). However, the latter is isomorphic to M ⊗ (Γ_{an,con} ∩ O[π^{−1}]⟦u⟧) by “Dwork’s trick” [13, Lemma 6.3]. Thus the generic and special HN-polygons correspond precisely to the Newton polygons of the generic and special fibres; the fact that the special HN-polygon lies above the generic HN-polygon (i.e., Proposition 5.5.1) in this case follows from Grothendieck’s specialization theorem. This gives a theoretical explanation for why Proposition 5.5.1 holds, but a more computationally explicit example may also be useful. (Thanks to Frans Oort for suggesting this presentation.) Suppose that σ is chosen so that u^{σ} = u^{p}. Let M be the rank 2 σ-module over Γ_{con} defined by

F v_{1} = v_{2}, F v_{2} = pv_{1} + uv_{2}.

Then v_{1} is a cyclic vector and F^{2}v_{1} − uF v_{1} − pv_{1} = 0, so by Lemma 5.2.4, the generic HN-polygon of M has the same slopes as the Newton polygon of the polynomial x^{2} − ux − p, namely 0 and 1. On the other hand, Dwork’s trick implies that the special HN-polygon of M has slopes 1/2 and 1/2.

7.4 Splitting exact sequences (again)

For reference, we collect here some more results about computing H^{1} of σmodules.

Proposition 7.4.1. For any σ-modules M_{1}, M_{2} over Γ_{con}[π^{−1}], the map Ext(M_{1}, M_{2}) → Ext(M_{1} ⊗ Γ_{an,con}, M_{2} ⊗ Γ_{an,con}) is surjective.

Proof. Let

0 → M_{2} ⊗ Γ_{an,con} → M → M_{1} ⊗ Γ_{an,con} → 0

be a short exact sequence of σ-modules. Choose a basis of M_{2}, then lift to M a basis of M_{1}; the result is a basis of M. Let A be the matrix via which F acts on this basis; after rescaling the basis of M_{2} suitably, we can put ourselves into the situation of Lemma 6.1.1. We can now perform the iteration of Lemma 6.1.1 in such a way as to respect the short exact sequence (i.e., take u_{ijlm} = 0 whenever the pair (i, j) falls in the lower left block); as in the proof of Proposition 6.2.2, we end up with a model M_{b} of M over Γ_{con}[π^{−1}], which by construction sits in an exact sequence 0 → M_{2} → M_{b} → M_{1} → 0. This yields the desired surjectivity.

We next give some generalizations of parts of Proposition 3.3.7.

Proposition 7.4.2. Let M be a σ-module over Γ_{con}[π^{−1}] whose generic HNslopes are all nonnegative. Then the map H^{1}(M) → H^{1}(M ⊗ Γ_{an,con}) is injective.

Proof. By Proposition 5.1.2, we can choose an F -stable Γ_{con}-lattice M_{0} of M. Let e_{1}, …, e_{n} be a basis of M_{0}, and define the matrix A over Γ_{con} by F e_{j} = ∑_{i} A_{ij}e_{i}. Choose r > 0 such that A has entries in Γ_{r} and w_{r}(A) ≥ 0. Suppose v ∈ M and w ∈ M ⊗ Γ_{an,con} satisfy v = w − F w, and write v = ∑_{i} x_{i}e_{i} and w = ∑_{i} y_{i}e_{i} with x_{i} ∈ Γ_{con}[π^{−1}] and y_{i} ∈ Γ_{an,con}; then x_{i} = y_{i} − ∑_{j} A_{ij}y_{j}^{σ}. If w ∉ M, we can choose m < 0 such that v_{m}(x_{i}) = ∞ and 0 < min_{i} min_{l≤m}{v_{l,r}(y_{i})} < ∞. Then

min_{l≤m}{v_{l,r}(y_{i})} > q^{−1} min_{l≤m}{v_{l,r}(y_{i})}

≥ q^{−1} min_{l≤m} min_{j}{v_{l,r}(A_{ij}y_{j}^{σ})}

≥ q^{−1} min_{l≤m} min_{j}{v_{l,r}(y_{j}^{σ})}

= q^{−1} min_{l≤m} min_{j}{rv_{l}(y_{j}^{σ}) + l}

≥ min_{l≤m} min_{j}{rv_{l}(y_{j}) + l}

= min_{l≤m} min_{j}{v_{l,r}(y_{j})};

taking the minimum over all i yields a contradiction. Hence w ∈ M, yielding the injectivity of the map H^{1}(M) → H^{1}(M ⊗ Γ_{an,con}).

Proposition 7.4.3. Let M be a σ-module over Γ[π^{−1}] whose HN-slopes are all positive. Then F − 1 is a bijection on M, i.e., H^{0}(M) = H^{1}(M) = 0.

Proof. By Proposition 5.1.2, we can choose an F -stable Γ-lattice M_{0} of M such that F (M_{0}) ⊆ πM_{0}. Then F − 1 is a bijection on M_{0}/πM_{0}, hence also on M_{0} and M.

Lemma 7.4.4. For L/K a finite Galois extension, and M a σ-module over Γ_{an,con}, the map H^{1}(M) → H^{1}(M ⊗ Γ_{an,con}^{L}) is injective.

Proof. Put G = Gal(L/K). Given w ∈ M, suppose that there exists v ∈ M ⊗ Γ_{an,con}^{L} such that w = v − F v. Then we also have w = v′ − F v′ for

v′ = \frac{1}{#G} ∑_{g∈G} v^{g},

which is G-invariant and hence belongs to M.

Theorem 7.4.5. Let M be a σ-module over Γ_{an,con} whose HN-slopes are all positive. Then the exact sequence

0 → M → N → Γ_{an,con} → 0  (7.4.6)

splits if and only if N has smallest HN-slope zero.

Proof. If the sequence splits, then P (N) = P (M) + P (Γ_{an,con}) by Proposition 4.7.2, so N has smallest HN-slope zero. We verify the converse first in the case where K is algebraically closed, so that Γ_{an,con} = Γ_{an,con}^{alg}. We proceed by induction on rank(M). In case M is isoclinic, then the inequality P (N) ≥ P (M) + P (Γ_{an,con}) from Proposition 4.7.2 and the hypothesis that N has smallest HN-slope zero together force P (N) = P (M) + P (Γ_{an,con}); by Proposition 4.7.2 again, (7.4.6) splits. In case M is not isoclinic, let M_{1} be the first step in the HN-filtration of M. By Proposition 4.7.2, we have

P (N) ≥ P (M_{1}) + P (N/M_{1}) ≥ P (M_{1}) + P (M/M_{1}) + P (Γ_{an,con});

since P (N) has smallest slope 0, so does P (M_{1}) + P (N/M_{1}). Since P (M_{1}) has positive slope, P (N/M_{1}) must have smallest slope zero. Hence by the induction hypothesis, the exact sequence 0 → M/M_{1} → N/M_{1} → Γ_{an,con} → 0 splits, say as N/M_{1} ≅ M/M_{1} ⊕ M′ with M′ ≅ Γ_{an,con}. Let N′ be the inverse image of M′ under the surjection N → N/M_{1}; it follows now that (7.4.6) splits if and only if 0 → M_{1} → N′ → M′ → 0 splits. By Proposition 4.7.2 again, P (N) ≥ P (N′)+P (M/M_{1}) ≥ P (M_{1})+P (M/M_{1})+P (Γ_{an,con}), and P (M/M_{1}) has all slopes positive, so P (N′) has smallest slope zero. Again by the induction hypothesis, 0 → M_{1} → N′ → M′ → 0 splits, yielding the splitting of (7.4.6). To summarize, we have proved that if N has smallest HN-slope zero, then (7.4.6) splits after tensoring with Γ_{an,con}^{alg}; it remains to descend this splitting back to Γ_{an,con}. To do this, apply Proposition 6.2.2 to produce a σmodule M_{0} over Γ_{con}^{L}[π^{−1}], for some finite Galois extension L of K, with M_{0} ⊗ Γ_{an,con}^{L} ≅ M ⊗ Γ_{an,con}^{L}. Then apply Proposition 7.4.1 to descend the given exact sequence, after tensoring up to Γ_{an,con}^{L}, to an exact sequence 0 → M_{0} → N_{0} → Γ_{con}^{L}[π^{−1}] → 0. We have shown that the exact sequence 0 → M_{0} ⊗ Γ_{an,con}^{alg} → N_{0} ⊗ Γ_{an,con}^{alg} → Γ_{an,con}^{alg} → 0 splits; by Proposition 7.4.2, the exact sequence 0 → M_{0} ⊗ Γ_{con}^{alg}[π^{−1}] → N_{0} ⊗ Γ_{con}^{alg}[π^{−1}] → Γ_{con}^{alg}[π^{−1}] → 0 splits. Choose w ∈ M_{0} whose image in H^{1}(M_{0}) corresponds to the exact sequence 0 → M_{0} → N_{0} → Γ_{con}^{L}[π^{−1}] → 0; we have now shown that the class of w in H^{1}(M_{0} ⊗ Γ_{con}^{alg}[π^{−1}]) vanishes. By Proposition 7.4.3, there exists a unique v ∈ M_{0}⊗Γ^{L}[π^{−1}] with w = v−F v. Since the class of w in H^{1}(M_{0}⊗Γ_{con}^{alg}[π^{−1}]) vanishes, we have v ∈ M_{0} ⊗ Γ_{con}^{alg}[π^{−1}]; since M_{0} is free over Γ_{con}^{L}[π^{−1}] and Γ_{con}^{alg}[π^{−1}] ∩ Γ^{L}[π^{−1}] = Γ_{con}^{L}[π^{−1}], we have v ∈ M_{0}. Hence the sequence (7.4.6) splits after tensoring with Γ_{an,con}^{L}. By Lemma 7.4.4, (7.4.6) also splits, as desired.

7.5 Full faithfulness

Here is de Jong’s original application of the reverse filtration [13, Proposition 8.2].

Proposition 7.5.1. Suppose that K admits a valuation p-basis. Let M be a σ-module over Γ_{con}[π^{−1}] admitting an injective F -equivariant Γ_{con}[π^{−1}]-linear morphism ϕ : M → Γ[π^{−1}](m) for some integer m. Then ϕ^{−1}(Γ_{con}[π^{−1}]) is a σ-submodule of M of rank 1, and its extension to Γ_{con}^{alg}[π^{−1}] is equal to the first step of the reverse filtration of M. In particular, M has highest generic slope m with multiplicity 1.

Proof. We first suppose K is algebraically closed (this case being [13, Corollary 5.7]). Let M_{1} be the first step of the reverse filtration of M. Then M_{1} ⊗ Γ^{alg}[π^{−1}] is isomorphic to a direct sum of standard σ-modules of some slope s_{1} = c/d; by Lemma 5.4.1 (applied to ([d]_{∗}(M_{1} ⊗ Γ_{con}^{alg}[π^{−1}]))(−c)), M_{1} itself is isomorphic to a direct sum of standard σ-modules of slope s_{1}. The map ϕ induces a nonzero F -equivariant map M_{1} ⊗ Γ^{alg}[π^{−1}] → Γ^{alg}[π^{−1}](m), and hence a nonzero element of H^{0}(M_{1}^{∨} ⊗ Γ^{alg}[π^{−1}](m)). By Corollary 4.1.4, we must have m = s_{1}, and so M_{1} ≅ Γ_{con}^{alg}[π^{−1}](m)^{⊕n} for some n. By Proposition 3.3.4, we have H^{0}(M_{1}^{∨}(m)) = H^{0}(M_{1}^{∨} ⊗ Γ^{alg}[π^{−1}](m)), and so ϕ actually induces an injective F -equivariant map M_{1} → Γ_{con}^{alg}[π^{−1}](m). To summarize, M has highest generic slope m, and the first step of the reverse filtration is contained in ϕ^{−1}(Γ_{con}^{alg}[π^{−1}]). Since the latter is a σ-submodule of M of rank no more than 1, we have the desired result. We now suppose K is general. Put M′ = M ⊗_{Γ_{con}[π^{−1}]} Γ_{con}^{alg}[π^{−1}]; by Proposition 2.2.21(c), the composite map

ψ : M′ →^{ϕ⊗1} Γ[π^{−1}] ⊗_{Γ_{con}[π^{−1}]} Γ_{con}^{alg}[π^{−1}] →^{μ} Γ^{alg}[π^{−1}]

is also injective. By the above, ψ^{−1}(Γ_{con}^{alg}[π^{−1}]) is a σ-submodule of M′ of rank 1, and coincides with the first step of the reverse filtration of M′. Let e_{1}, …, e_{n} be a basis of M, put v = ψ^{−1}(1), and write v = ∑ x_{i}e_{i} with x_{i} ∈ Γ_{con}^{alg}[π^{−1}]. Then 1 = ψ(v) = ∑ x_{i}ϕ(e_{i}); by Proposition 2.2.21(b), we have x_{i} ∈ Γ_{con}[π^{−1}] for i = 1, …, n. Hence v ∈ ϕ^{−1}(Γ_{con}[π^{−1}]), so the latter is a σ-submodule of M of rank 1. This yields the desired result.

Remark 7.5.2. Proposition 7.5.1 can be used to reduce instances of showing H^{0}(M) = H^{0}(M ⊗ Γ[π^{−1}]), for M an F -module over Γ_{con}[π^{−1}], to showing that a certain class in H^{1}(N), where N is a related F -module over Γ_{con}[π^{−1}] with positive HN-slopes, vanishes. Thanks to Proposition 7.4.2, this in turn reduces to checking vanishing of the class in H^{1}(N ⊗ Γ_{an,con}), where either Dwork’s trick, in the case of [13], or the p-adic local monodromy theorem, in the case of [20], can be brought to bear. Note that by Theorem 7.4.5, it is enough to check vanishing of a class in H^{1}(N ⊗ Γ_{an,con}) after replacing K by a finite separable extension.

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Kiran S. Kedlaya Dept. of Mathematics Massachusetts Institute of Technology 77 Massachusetts Avenue Cambridge, MA 02139 USA kedlaya@mit.edu 526

Documenta Mathematica 10 (2005)