Global Dimension of Polynomial Rings in Partially Commuting Variables

Let A be any Abelian category. By [1, Chapter XII, §4], extension groups Ext n (A, B) are consisted of congruence classes of exact sequences 0 Here N is the set of nonnegative integers. (We set sup ∅ = -1 and sup N = ∞.) For a ring R with 1, gl dim R is the global dimension of the category of left R-modules.

As it is well known [2,Theorem 4.3.7], for any ring R with 1, gl dim R[x 1 , . . . , x n ] = n + gl dim R.

Moreover, by [3,Theorem 2.1], if A is any Abelian category with exact coproducts and C a bridge category, then gl dim A C = 1 + gl dim A. It follows that gl dim A N n = n + gl dim A for the free commutative monoid N n generated by n elements. We will get one of possible generalizations of this formula. Let M(E, I) be a free partially commutative monoid with a set of variables E, where I ⊆ E × E is an irreflexive symmetric relation assigning the pairs of commuting variables. In this paper, we prove that gl dim A M (E,I) = n + gl dim A for any Abelian category with exact coproducts where n is the sup of numbers of mutually commuting distinct elements of

is the polynomial ring in variables E = {x 1 , x 2 , x 3 , x 4 } with the commuting pairs (x i , x j ) corresponding to adjacent vertices of the graph demonstrated in Figure 1, then for any ring R with 1 we have gl dim R[M(E, I)] = 2+gl dim R.

x 4 x 3

x 1 x 2

The free partially commutative monoids have numerous applications in combinatorics and computer sciences [4]. Our interest in their homology groups is concerned with the studying a topology of mathematical models for concurrency [5].

Throughout this paper let Ab the category of Abelian groups and homomorphisms, Z the additive group of integers, and N the set of nonnegative integers or the free monoid with only one generator. For any category A and a pair A 1 , A 2 ∈ Ob A, denote by A(A 1 , A 2 ) the set of all morphisms A 1 → A 2 . A diagram C → A is a functor from a small category C to a category A. Given a small category C we denote by A C the category of diagrams C → A and natural transformations. For A ∈ Ob A, let ∆ C A (shortly ∆A) denote a diagram C → A with constant values A on objects and 1 A on morphisms.

In this section, we recall some results from the cohomology theory of small categories.

Recall a definition of a nerve of the category and properties of homology groups of simplicial sets. We refer the reader to [1] and [6] for the proofs.

Let C be a small category. Its nerve N * C is the simplicial set in which N n C consists of all sequences of composable morphisms c 0

for n > 0 and N 0 C = Ob C . For n > 0 and 0 i n, boundary operators

Here c 0

The map d n 0 removes α 1 with c 0 and d n n removes α n with c n . Degeneracy operators

Let X be a simplicial set given by boundary operators d n i and degeneracy operators s n i for 0 i n. Consider a chain complex C * (X) of free Abelian groups C n (X) generated by the sets X n for n 0. Differentials

The groups H n (X) = Ker d n / Im d n+1 are called n-th homology groups of the simplicial set X. The groups H n (X) are isomorphic to n-th singular homology groups of the geometric realization of X by the Eilenberg theorem [6, Appl. 2].

For a small category C , let H n (C ) denote the n-th homology group of the nerve N * C . For a simplicial set X and an Abelian group A, cohomology groups H n (X, A) are defined as cohomology groups of the complex Hom(C * (X), A). Let C be a small category. We introduce its cohomology groups

Chapter III, Theorem 4.1] that there is the following exact sequnce (Universal Coefficient Theorem)

Recall the definition and properties of right derived functors lim ← -n C : Ab C → Ab of the limit functor.

Let C be a small category. For every family {A i } i∈I of Abelian groups we consider the direct product i∈I A i as the Abelian group of maps ϕ :

For any functor F : C → Ab, consider the sequnce of Abelian groups

Let C n (C , F ) = 0 for n < 0. The equalities δ n+1 δ n = 0 hold for all integer n. The obtained cochain complex will be denoted by C * (C , F ). Abelian groups

where indices run the sequences c 0 Proof. The first assertion follows from Corollary 1.2. Since the geometric realization of the nerve of Θ n is the n-dimensional torus,

Here n k is the binomial coefficients. Universal Coefficient Theorem for the cohomology groups of the nerve of Θ n gives H n (Θ n , A) ∼ = A.

A small category C is acyclic if H n (C ) = 0 for all n > 0 and H 0 (C ) = Z. Let S : C → D be a functor from a small category to an arbitrary category. For any d ∈ Ob D, a fibre (or comma-category) S/d is the category which objects are given by pairs (c, α) where c ∈ Ob(C ) and α ∈ D(S(c), d).

Let N be the set of nonnegative integer numbers. We will be consi

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