PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 98(112) (2015), 137–145 DOI: 10.2298/PIM150503029P

AN EXAMPLE OF BRUNS–GUBELADZE K-THEORY DEFINED BY THREE DIMENSIONAL POLYTOPE

Th. Yu. Popelensky

Abstract. For the Bruns–Gubeladze polytopal K-theory, we describe a new series of three dimensional balanced Col-divisible polytopes. Also we calculate the corresponding elementary groups and as a corollary obtain an expression of the polytopal K-groups in terms of the Quillen K-groups.

1. Introduction

In the series of papers [1–4] Bruns and Gubeladze have investigated polytopal algebras k[P] where k is a field and P is a lattice polytope. The group of graded Rautomorphisms gr.aut_{R}(k[P]) of the algebra k[P] is an analog of the group GL_{n}(k). The paper [1] introduces elementary automorphisms of k[P] and establishes an important fact that every graded automorphism can be diagonalized by a sequence of elementary automorphism. In [2] it was shown that many graded retractions are conjugates of diagonal idempotents. So the natural question arises: is it possible to find polytopal analogs of the higher algebraic K-groups (for rings). The answer is positive and has been given by Bruns and Gubeladze in [3, 4] for a wide class of balanced polytopes. For a commutative ring R and a balanced polytope P the group 𝔼_{R}(P) generated by elementary graded automorphisms is not perfect in general. Bruns and Gubeladze established a highly nontrivial stabilization procedure which on polytopal level works as “doubling along a facet”. As an outcome of the stabilization procedure one obtains the stable elementary group 𝔼(R, P). It is important that the stable group 𝔼(R, P) in general is not a union of corresponding unstable groups, hence the “polytopal part” of the stabilization is essential. The group 𝔼(R, P) is perfect. In [3, 4] the (stable) Steinberg group 𝕊t(R, P) was defined and it was shown that for a balanced polytope P the natural homomorphism 𝕊t(R, P) → 𝔼(R, P) is a universal central extension. Higher polytopal K-groups are defined by

K_{i}(R, P) = π_{i}(B𝔼(R, P)^{+} ), i ≥ 2.

2010 Mathematics Subject Classification: 19C09, 19D06, 19M05.

Key words and phrases: K-theory, lattice polytope, elementary automorphism, Steinberg group.

Partially supported by RFBR grants 14-01-00007, 13-01-00664, SciSchool 581.2014.1.

Communicated by Zoran Rakić.

If P coincides with the standard simplex Δ^{k} one obtains the Quillen K-theory. More detailed account on history of the question and on the motivation can be found in [1, 2, 5].

Some natural questions arise about the polytopal generalization of K-theory. First of all it is important to know equivalence relations on polytopes which lead to naturally equivalent polytopal K-theory. The projective equivalence of polytopes is obvious, but the E-equivalence (see [3, 4] for definition) is more convenient. Also it is interesting to calculate K_{i}(R, P) for various polytopes P. The case of 2-dimensional polytopes (i.e., polygones) was completely solved by Bruns and Gubeladze [3, 4]. They proved that there are 6 classes of E-equivalence of the polygones and also they calculated the corresponding K-groups. The case of 3dimensional polytopes was investigated in [6]. In that work a classification of the balanced 3-dimensional polytopes up to E-equivalence was proposed and the stable elementary groups of Col-divisible 3-dimensional polytopes were identified. In [8] the case of the pyramid over the unit square (it is balanced, but not Col-divisible) was investigated. This polytope appears in [3, 4] several times as a polytope not satisfying some natural conditions (see also [5]). Some calculations for balanced, but not Col-divisible polytopes, can be found in [9].

Bruns and Gubeladze conjectured [5, Conjecture 8.3] that for a commutative ring R and a Col-divisible (balanced) polytope P of arbitrary dimension one has K_{i}(R, P) = K_{i}(R)⊕⋯ ⊕K_{i}(R) (c(P) summands), where c(P) < dimP is a natural number explicitly defined by P (for some polytopes P a technical condition on the ring R is involved). In all known examples the conjecture holds even for balanced not Col-divisible polytopes.

This note appeared as a result of an attempt to find a counterexample to the Bruns–Gubeladze conjecture. In fact, we did not succeed, but instead we found a series of balanced Col-divisible polytopes which had not been known before and was omitted in the Faramarzi’s classification theorem [6, Theorem 3.2]. We calculate the corresponding elementary groups and as a corollary obtain an expression of the polytopal K-groups in terms of the Quillen K-groups. Despite [6], a classification of balanced (and balanced Col-divisible) 3-dimensional polytopes remains open.

The author is grateful to the referee for his/her valuable suggestions and comments.

2. Basic definitions

The details of the Bruns–Gubeladze construction can be found in their original works [3, 4]. Here we present only an outline of necessary definition and constructions.

Let P be a convex polytope in ℝ^{n} with vertices in the integral lattice ℤ^{n}. We always suppose that n is minimal, that is the minimal affine subspace containing P coincides with ℝ^{n}. Such a polytope is called a lattice polytope. For any facet F there is a unique surjective homomorphism ⟨F, −⟩: ℤ^{n} → ℤ, with the kernel consisting of vectors parallel to the facet F and such that ⟨F, p − q⟩ ≥ 0 for any p ∈ P and q ∈ F.

AN EXAMPLE OF BRUNS–GUBELADZE K-THEORY 139

A vector u ∈ ℤ^{n} is called a column vector for a lattice polytope P if there exists a facet P_{u} ⊂ P such that ⟨P_{u}, u⟩ = −1 and for any other facet F ⊂ P one has ⟨F, u⟩ ≥ 0. In this case the facet P_{u} is called the base facet for the column vector u. For a given column vector the base facet is defined uniquely, but two different column vectors can have the same base facet. A collection of all column vectors with the base facet F is denoted by Col(F). A collection of all column vectors of the lattice polytope P is denoted Col(P).

Assume u ∈ Col(F). Then for any point p ∈ P ∩ ℤ^{n} there exists a unique nonnegative integer k such that p + ku ∈ F. This number is called a height of the point p over the base facet F and is denoted by ht_{F} (p). One has the equality ht_{F} (p) = ⟨F, p − q⟩ where q is an arbitrary point from F.

One can define a natural partial multiplication on the set Col(P). Suppose u, v ∈ Col(P), u + v ∈ Col(P) and P_{u+v} = P_{u}. Then the product uv is defined to be u + v. The product uv is defined not for any pair of column vectors u, v. Obviously if uv exists, then vu is not defined.

As a basic example consider the simplex Δ^{n} in ℝ^{n}, with one vertex (0, …, 0) in the origin and other n vertices of form (0, …, 1, …, 0). The description of column vectors and their partial product is simple. For any two vertices p_{i}, p_{j} ∈ Δ^{n}, there are two column vectors δ_{i}^{j} = p_{j} − p_{i} and δ_{j}^{i} = −δ_{i}^{j}. The base facet of the column vector δ_{j}^{i} is Δ^{n} ∩ {x_{j} = 0} for j ≠ 0 and Δ^{n} ∩ {∑ _{k} x_{k} = 1} for j = 0. The partial product is described by the relation δ_{i}^{j} δ_{j}^{k} = δ_{i}^{k}. There are no other column vectors, and no other product is defined.

Balanced polytopes. A lattice polytope P is called balanced if for any u, v ∈ Col(P) one has ⟨P_{u}, v⟩ ≤ 1. The simplex Δ^{n} is balanced while the triangle conv{(0, 0), (1, 0), (0, 2)} is not.

Note that for a balanced polytope and its base facet F one has inequality |⟨F, u⟩| ≤ 1 for any column vector v. Also ⟨F, u⟩ = −1 iff v ∈ Col(F). Obviously ⟨F, u⟩ = 0 iff u is parallel to F.

Doubling along a facet. Like in the classical Quillen K-theory we need some kind of a stabilization procedure. Let P be a lattice polytope. Choose its facet F. Without loss of generality one can assume that the origin belongs to the facet F and that F is contained in the hyperplane x_{n} = 0. Consider the standard embedding of ℝ^{n} ⊂ ℝ^{n+1} onto the hyperplane x_{n+1} = 0. Let us turn the polytope P by π/2 around the plane x_{n} = x_{n+1} = 0 in ℝ^{n+1} . The image of P under the rotation is denoted by P^{|}.

The polytope P^{⌟F} is defined as the convex hull of P and P^{|} and is called doubling of P along the facet F. If v ∈ Col(F) then one can write P^{⌟v} instead of P^{⌟F}.

After doubling along F the number of facets increases by 1. For any facet G different from F denote by G^{⌟F} the facet of P^{⌟F} which is the convex hull of G and its image G^{|} after rotation of the polytope P by π/2. Let F^{⌟F} = P^{|} for the facet F.

Let us describe the structure of Col(P^{⌟F}). Choose a column vector v ∈ Col(P) with base facet G. Then v is also a column vector for P^{⌟F} with the base facet G^{⌟F} (think of the inclusion P ⊂ P^{⌟F}).

Describe new column vectors which arise after doubling. First of all there are two column vectors δ^{+} and δ^{−} = −δ^{+} with the base facets P^{|} and P^{−} = P correspondingly. Secondly, for a column vector v ∈ Col(P) the vector v^{|} (the image of v after rotation of P by π/2) is a column vector for P^{⌟F}. If the vector v is parallel to the facet F, then v^{|} coincides with v. If v is not parallel to F then v ≠ v^{|} and we have two possibilities. Namely, if v ∈ Col(F), then one has the relations

(1)  δ^{−} v = v^{|}, δ^{+} v^{|} = v,

and if v ∉ Col(F) (therefore ⟨F, v⟩ > 0), then one has another relations

(2)  vδ^{+} = v^{|}, v^{|}δ^{−} = v.

One can show that for a balanced polytope P its doubling P^{⌟F} is also balanced and that the set Col(P^{⌟F}) is a union (not necessarily disjoint)

Col(P^{⌟F}) = Col(P) ∪ Col(P)^{|} ∪ {δ^{+} , δ^{−} }.

A sequence of polytopes 𝔓 = (P = P_{0} ⊂ P_{1} ⊂ P_{2} ⊂ …) is called a doubling spectrum if (1) P_{k+1} is a doubling of P_{k} along a base facet and (2) for any i ∈ ℤ_{+} , v ∈ Col(P_{i}) there exists j ≥ i, such that P_{j+1} = P_{j}^{⌟v}.

For any doubling spectrum there is a natural inclusion Col(P_{i}) ⊂ Col(P_{i+1} ), therefore the direct limit Col(𝔓) = limCol(P_{i}) is defined.

Elementary automorphisms and Steinberg group. Consider a lattice polytope P. Let S(P) be the additive semigroup generated by pairs (p, 1) ∈ ℝ^{n+1} where p ∈ P ∩ℤ^{n}. For a given associative commutative ring R with unit consider the semigroup ring R[P] = R[S(P)]. It has natural grading defined on the generators of the ring by formula deg(p, d) = d.

Denote by gr.aut_{R} (R[P]) the group of graded R-automorphisms of R[P]. An element ϕ ∈ gr.aut_{R} (R[P]) is called an elementary automorphism if there exist a column vector v ∈ Col(P) and an element λ ∈ R such that for every x ∈ S(P), one has

ϕ(x) = (1 + λv)^{ht_{Pv} (x)} x.

Denote this automorphism by e_{v}^{λ}. The subgroup of gr.aut_{R} (R[P]) generated by elementary automorphisms is denoted by 𝔼_{R}(P).

For any v ∈ Col(𝔓) there exists i ∈ ℕ such that v ∈ Col(P_{j}) for all j ≥ i. Hence elementary automorphisms e_{v}^{λ} ∈ 𝔼_{R}(P_{j}), j ≥ i, form a compatible system. Therefore they define a graded automorphism of R[𝔓], which we call “elementary” and denote by e_{v}^{λ}. The group 𝔼(R, 𝔓) is the subgroup of gr.aut_{R} (R[𝔓]) generated by elementary automorphisms.

In [3, 4] it was shown that the group 𝔼(R, 𝔓) does not depend on a choice of a doubling spectrum of the polytope P, hence one uses notation 𝔼(R, P) instead of 𝔼(R, 𝔓). The group 𝔼(R, P) is perfect. For a balanced lattice polytope P elementary automorphisms satisfy relations which are similar to the relations between elementary matrices:

(i) e_{v}^{λ}e_{v}^{μ} = e_{v}^{λ+μ} for all v ∈ Col(𝔓) and λ, μ ∈ R;

(ii) for all u, v ∈ Col(𝔓) and λ, μ ∈ R

[e_{u}^{λ}, e_{v}^{μ}] = \{ \begin{matrix}e_{uv}^{−λμ} , & if uv is defined, \\ 1, & if u + v ∉ Col(P) ∪ \{0\}. \end{matrix}

Fix a doubling spectrum 𝔓 of P. Define the Steinberg group 𝕊t(R, P) to be the group generated by symbols x_{v}^{λ}, v ∈ Col(𝔓), λ ∈ R, and relations

(3) x_{v}^{λ}x_{v}^{μ} = x_{v}^{λ+μ} for all v ∈ Col(𝔓), λ, μ ∈ R;

(4) [x_{u}^{λ}, x_{v}^{μ}] = \{ \begin{matrix}x_{uv}^{−λμ} , & if product uv is defined, \\ 1, & if u + v ∉ Col(𝔓) ∪ \{0\} \end{matrix}

for all u, v ∈ Col(𝔓) and λ, μ ∈ R.

The group 𝕊t(R, P) does not depend on a choice of a doubling spectrum 𝔓 of P. For a balanced polytope P the natural epimorphism 𝕊t(R, P) → 𝔼(R, P) is a universal central extension.

Example: For P = Δ^{n} the groups 𝔼(R, P) and 𝕊t(R, P) are isomorphic to usual E(R) and St(R).

3. Polytopes F (t, s)

For integers t ≥ s ≥ 1 define the polytope F (t, s) to be the convex hull of the points (0, 0, 0), (2t, 0, 0), (1, 2, 0), (t + 1, 1, 0), (0, 0, 1), (1, 0, 2), (2t − s + 1, 0, 1) ∈ ℝ^{3}.

The Col-structure of F (t, s) can be described as follows. There are two base facets: F_{w} = F (t, s) ∩ {y = 0} with the column vectors v_{τ} = (τ − 1, −1, 0), τ = 1, …, t, and w = (0, −1, 1) and F_{u} = F (t, s) ∩ {z = 0} with the column vectorsv_{σ} = (σ − 1, 0, −1), σ = 1, …, s. The only relations between the column vectors are wu_{σ} = v_{σ}, σ = 0, …, s. If t > s, then the column vectors v_{s+1} , …, v_{t} cannot be decomposed as a product of other column vectors.

From this description it follows that the polytopes F (t, s) are balanced and Col-divisibe. Note that

(1) for t = s > 1 the polytope F (t, t) is E-equivalent to the polytope P_{e_{4}} (t),

(2) for t > 1 the polytope F (t, 1) is E-equivalent to the polytope P_{e}(t, 1) (for definition of P_{e_{4}} (t) and P_{e}(t, 1) see [6]),

(3) the polytope F (1, 1) (though it is three dimensional) is E-equivalent to the polygon of type (c) (for definition see [4]).

It appears that the condition t ≥ s is essential.

Proposition 1. Suppose P is a polytope such that there are at least two base facets F_{u} and F_{w}. Assume that for the base facet F_{u} there are s column vectors u_{σ}, 1 ≤ σ ≤ s, which are parallel to F_{w} and that for the base facet F_{w} there is a column vector w which is not parallel to the F_{u}. Then the vectors v_{σ} = w + u_{σ} are column vectors for the base facet F_{w}.

In other words, in assumptions of the proposition there are at least s column vectors for the base facet F_{w} such that wu_{σ} = v_{σ}.

Proof. Fix σ = 1, …, s. As w and u_{σ} are column vectors, one has ⟨G, w + u_{σ}⟩ = ⟨G, w⟩ + ⟨G, u_{σ}⟩ ≥ 0 for any facet G of the polytope P different from F_{u} and F_{w}. For the facet F_{u} one has ⟨F_{u}, w + u_{σ}⟩ = ⟨F_{u}, w⟩ − 1 ≥ 0 and for the facet F_{w}: ⟨F_{w}, w + u_{σ}⟩ = −1 + 0 = −1. Hence w + u_{σ} is a column vector for the base facet F_{w}. □

Fix integers t ≥ s ≥ 1 and consider a doubling spectrum P_{0}, P_{1}, P_{2}, … of the polytope P_{0} = F (t, s). Let us describe the column structure of the polytopes P_{j} (by induction on j). From this we obtain the description of the elementary group and the Steinberg group for the polytope P_{0} and identify the corresponding K-theory.

First of all we describe the structure of the base facets of the polytopes P_{j}. For every polytope P_{j} we divide its base facets into two families A^{1}(0), A^{2}(0) and enumerate the facets in the families. For the polytope P_{0} define A^{1}(0) = {F_{w}}, A^{2}(0) = {F_{u}}.

Suppose P_{j+1} is a doubling of P_{j} along a facet F, which belongs to A^{1}(j). Assume A^{1}(j) = {A_{1}, A_{2}, …, A_{n}}. Define A^{1}(j + 1) to be {A′_{1}, A′_{2}, …, A′_{n}, A′_{n+1} }, where A′_{n+1} = P_{j} (recall that P_{j} is a facet of P_{j+1} ) and A′_{j} = A_{j}^{⌟F} for j ≤ n. Assume A^{2}(j) = {B_{1}, B_{2}, …, B_{m}}. Define A^{2}(j + 1) to be {B′_{1}, B′_{2}, …, B′_{m}}, B′_{j} = B_{j}^{⌟F}. Definition of A^{r}(j + 1) for the case F ∈ A^{2}(j) is analogous.

Now describe the structure of Col-vectors. We shall do it in two steps.

First of all we describe what happens to the vectors u_{σ}, v_{τ}, w under consecutive doublings. Let a_{r}(j) = #A^{r}(j)

Lemma 1. For the polytope P_{j} there are the column vectors w_{i}^{k}, u_{σ,k} , v_{τ,i}, where 1 ≤ i ≤ a_{1}(j), 1 ≤ k ≤ a_{2}(j) such that:

(1) w_{i}^{k}u_{σ,k} = v_{σ,i} , for all k and 1 ≤ σ ≤ s,

(2) v_{τ,i} is a column vector for the i-th base facet from A^{1}(j), and it is parallel to all other base facets,

(3) u_{σ,k} is a column vector for the k-th base facet from A^{2}(j), and it is parallel to all other base facets,

(4) w_{i}^{k} is a column vector for the i-th base facet from A^{1}(j), it has height 1 over the k-th base facet from A^{2}(j) and it is parallel to all other base facets.

Proof. The case of P_{0} is obvious.

By induction assume that doubling of P_{j} was made along the facet A_{i} ∈ A^{1}(j). Then besides the vectors δ^{±} we obtain new vectors (w_{i}^{k})^{|} and v_{τ,i}^{|} for all k and τ. Denote them by w_{a_{1} (j)+1}^{k} and v_{τ,a_{1} (j)+1} correspondingly. All these vectors are column vectors for the new base facet A_{a_{1} (j)+1} .

Assume doubling of P_{j} was made along the facet B_{k} ∈ A^{2}(j). Then besides the vectors δ^{±} we obtain new vectors (w_{i}^{k})^{|} and u_{σ,k}^{|} for all i and τ. Denote them

by w_{i}^{a_{2} (j)+1} and v_{τ,a_{2} (j)+1} correspondingly.

Statement (1) is a straightforward consequence of the relation wu_{σ} = v_{σ}. Also note that there is no such relation for the vectors v_{σ+1,i} , …, v_{τ,i}. □

Secondly, we describe column vectors which appear as the δ^{±} -vectors or vectors they produce under doublings.

Lemma 2. For any P_{j} and for any two different facets A_{k}, A_{l} ∈ A^{r}(j) there is a column vector δ_{k}^{l}(r) for the base facet A_{k}, which has height 1 over the facet A_{l} and is parallel to all other base facets. The vectors δ_{k}^{l}(r) satisfy the relations δ_{k}^{l}(r)δ_{l}^{p}(r) = δ_{k}^{p}(r) for all r, k, l, p.

Proof. For j = 0 the statement is trivial.

Assume by induction that doubling of P_{j} was made along the facet A_{i} ∈ A^{1}(j). Then we have new column vectors δ^{±} and δ_{i}^{k}(1)^{|}, δ_{k}^{i}(1)^{|} for all k ≠ i. Denote δ_{i}^{k}(1)^{|} by δ_{a_{1} (j)+1}^{k} (1), δ_{k}^{i}(1)^{|} by δ_{k}^{a_{1} (j)+1} (1), δ^{+} by δ_{i}^{a_{1} (j)+1} , and δ^{−} by δ_{a_{1} (j)+1}^{i} . If one of the indices k, l, p coincides with a_{1}(j) + 1, then the relation δ_{k}^{l}(r)δ_{l}^{p}(r) = δ_{k}^{p}(r) follows from one of relations (1), (2). For example relation (2) (δ_{k}^{i})^{|}δ^{−} = δ_{k}^{i} is the same as δ_{k}^{a_{1} +1} δ_{a_{1} +1}^{i} = δ_{k}^{i} and relation (1) δ^{+} (δ_{i}^{k})^{|} = δ_{i}^{k} is the same as δ_{i}^{a_{1} +1} δ_{a_{1} +1}^{k} = δ_{i}^{k}.

The collection of the vectors δ_{k}^{l}(2) is unchanged as all of them are parallel to the facet A_{i}.

The case of doubling along the facet from A^{2}(j) is analogous. □

Lemma 3. The vectors δ_{i}^{k}(r), u_{σ,p} , v_{τ,i} , w_{i}^{p} satisfy the relations

δ_{k}^{i}(1)v_{τ,i} = v_{τ,k} (5)

δ_{q}^{p}(2)u_{σ,p} = u_{σ,q} (6)

δ_{k}^{i}(1)w_{i}^{p} = w_{k}^{p} (7)

w_{i}^{p}δ_{p}^{q}(2) = w_{i}^{q} (8)

for all i, k, p, q, r.

The relations (5)-(7) follow from (1), and the relation (8) follows from (1).

4. Representation of 𝕊t(R, F (t, s))

The purpose of this section is to construct a kind of a matrix representation of 𝕊t(R, F (t, s)) and to deduce from it the description of 𝔼(R, F (t, s)). From now on fix t ≥ s ≥ 1 and choose a doubling spectrum 𝔓 of P_{0} = F (t, s).

Denote by St(n), n ≥ 0, a group generated by the symbols x_{v}^{λ} and the Steinberg relations (3), (4) where v ∈ ColP_{n} and λ ∈ R. There is a canonical homomorphism ϕ_{n} : St(n) → St(n + 1). It can be shown that 𝕊t(R, P_{0}) = limSt(n).

Let a_{r}(n) = #A^{r}(n) (for simplicity we write a_{r}). Denote by M_{kl} (or M_{kl}(R)) a set of all matrices (with k rows and l columns) with entries in R. Let M(n) be a set of block matrices of the form

( \begin{matrix}M_{a_{1} a_{1}} & M_{a_{1} a_{2}} & ⨁ _{1≤τ≤t} M_{a_{1} 1} \\ 0 & M_{a_{2} a_{2}} & ⨁ _{1≤σ≤s} M_{a_{2} 1} \\ 0 & 0 & 1 \end{matrix} )

(here M_{a_{1} a_{1}} acts on ⨁ _{τ} M_{a_{1} 1} diagonally, etc.).

We define the map ψ_{n} : St(n) → M(n) as follows. Consider the “scheme”

( \begin{matrix}δ(1) & w & v_{τ} \\ 0 & δ(2) & u_{σ} \\ 0 & 0 & 1 \end{matrix} ) .

Let v be one of the column vectors δ_{i}^{j} (r), w_{i}^{j}, v_{τ,i} or u_{σ,i} (r = 1, 2) of the polytope P_{n}. Define ψ_{n} on the generator x_{v}^{λ} to be the matrix from M(n) with zero entries except 1 on the diagonal and λ placed in the block with the same “name” as the vector v at the intersection of i-th row and j-th column if v is δ_{i}^{j} (r) or w_{i}^{j}, or just in i-th row if v = u_{σ,i} or v = v_{τ,i}.

Proposition 2. The map ψ_{n} is an epimorphism of St(n) onto the group

E(n) = ( \begin{matrix}E_{a_{1}} (R) & M_{a_{1} a_{2}} (R) & ⨁ _{τ} M_{a_{1} 1}(R) \\ 0 & E_{a_{2}} (R) & ⨁ _{σ} M_{a_{2} 1}(R) \\ 0 & 0 & 1 \end{matrix}) .

The proof is straightforward.

While passing from P_{n} to P_{n+1} one of the numbers a_{1}, a_{2} increases by 1. So we have obvious stabilization maps η_{n} : M(n) → M(n + 1) and their restrictions η_{n} : E(n) → E(n + 1). The diagram

St(n) −−−−→^{ϕ_{n}} St(n + 1)
↓^{ψ_{n}}  ↓^{ψ_{n+1}}
E(n) −−−−→^{η_{n}} E(n + 1)

commutes, hence we have a homomorphism of the stable groups ψ : 𝕊t(R, P_{0}) → E(∞), where E(∞) is the group

( \begin{matrix}E(R) & M(R) & ⨁ _{τ} V (R) \\ 0 & E(R) & ⨁ _{σ} V (R) \\ 0 & 0 & 1 \end{matrix}) ,

E(R) = lim_{n} E_{n}(R), M(R) = lim_{m,n} M_{mn} (R), V (R) = lim_{n} M_{n1} (R).

5. Elementary group 𝔼(R, F (t, s)) and K-theory

The representation ψ of 𝕊t(R, F (t, s)) is not exact.

Proposition 3. kerψ = Z(St(R, P_{0})).

Proof. The inclusion kerψ ⊂ Z(St(R, P_{0})) is obvious since Z(E(∞)) = 0.

For the inverse inclusion we need the following statement, which generalizes Milnor’s arguments from [7, Theorem 5.1].

Proposition 4. [4, proof of Proposition 8.2] Assume that 𝔔 = (Q = Q_{0} ⊂ Q_{1} ⊂ Q_{2} ⊂ …) is a doubling spectrum of a polytope Q. For every i ∈ ℕ∪{0} define two sets of column vectors U^{i+1} = {u ∈ Col(Q_{i+1} ) | ⟨Q_{i}, u⟩ = 1} and V^{i+1} = {v ∈ Col(Q_{i+1} ) | ⟨Q_{i}, v⟩ = −1}. Consider subgroups 𝔘^{i+1} , 𝔙^{i+1} ⊂ St_{i+1} (R, Q) generated by all x_{u}^{λ} and x_{v}^{μ} correspondingly (here u ∈ U^{i+1} and v ∈ V^{i+1} , λ, μ ∈ R). Suppose for a group G, there is given an epimorphism π: St(R, Q) → G which is injective on 𝔘^{i+1} and 𝔙^{i+1} . Then kerπ ⊂ Z(St(R, Q)).

Apply this proposition to the doubling spectrum 𝔓 of P_{0}, G = E(∞) and π = ψ. From Lemmas 1 and 2 we can identify the sets U^{n} and V^{n}.

First of all suppose that doubling P_{n−1} ⊂ P_{n} was done along a facet A_{i} ∈ A^{1}. Then the set U^{n} consists of the vectors δ_{j}^{i}(1), j = 1, …, a_{1}(n), j ≠ i. The set V^{n} consists of the vectors v_{τ,i}, δ_{i}^{j} (A^{1}), j = 1, …, a_{1}(n), j ≠ i, and the vectors w_{i}^{k}, k = 1, …, a_{2}(n).

Image of 𝔙^{n} consists of the matrices in E(n) ⊂ E(∞) with 1 on the diagonal and other nonzero entries in the i-th rows of the blocks δ(1), w and ⊕v_{τ}. Image of 𝔘^{n} consists of matrices in E(n) ⊂ E(∞) with 1 on the diagonal and other nonzero entries in the i-th column of the block δ(A^{1}).

To prove the injectivity one should note that from the Steinberg relations, it follows that 𝔘^{n} are 𝔙^{n} are abelian groups. Moreover, using arguments from Lemma 5.2 of [7] or from Proposition 8.2 of [4] one can show that these abelian groups are isomorphic to R^{N} for suitable numbers N ∈ ℕ. Then simple counting of dimensions shows that ψ is injective on 𝔘^{n} and 𝔙^{n}.

The case of doubling P_{n−1} ⊂ P_{n} along a facet from A^{2} is analogous. □

Theorem 1. The groups E(∞) and 𝔼(R, P) are naturally isomorphic.

Proof. The kernel of the natural homomorphism 𝕊t(R, P_{0}) → 𝔼(R, P_{0}) coincides with Z(𝕊t(R, P_{0})) as 𝕊t(R, P_{0}) is the universal central extension of 𝔼(R, P_{0}). It Proposition 3 it was shown that ker(ψ : 𝕊t(R, P_{0}) → 𝔼(∞)) also coincides with Z(𝕊t(R, P_{0})). Therefore there exists a natural isomorphism of E(∞) and 𝔼(R, P). □

Recall that a ring R is called an S(n)-ring if there exist elements x_{1}, …, x_{n} ∈ R^{∗} such that sum of any subset of them is a unit. The ring R has many units if R is an S(n)-ring for any n ∈ ℕ.

Corollary 1. There is a natural isomorphism

K_{i}(R, F (t, s)) = K_{i}(R) ⊕ K_{i}(R), i ≥ 2,

provided R has many units.

The proof can be done in the same way as the proof of Theorem 9.2 from [3] with suitable minor changes.

References

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7. J. Milnor, Introduction to Algebraic K-theory, Princeton, 1971.

8. Th. Yu. Popelensky, M. V. Prikhodko, Bruns–Gubeladze K-groups for quadrangular pyramid, Sovrem. Mat., Fundam. Napravl. 51 (2013), 141–150.

9. Th. Yu. Popelenskiĭ, Bruns–Gubeladze K-theory defined by some three-dimensional polytopes, Russ. J. Math. Phys. 22(3) (2015), 400–409.

Moscow State Lomonosow University (Received 20 12 2012) Moscow, Russia (Revised 29 04 2015)

    popelens@mech.math.msu.su