Homology, Homotopy and Applications, vol.3, No.5, 2001, pp.101–110

METHODS OF CALCULATING COHOMOLOGICAL AND HOCHSCHILD–MITCHELL DIMENSIONS OF FINITE PARTIALLY ORDERED SETS

A. A. HUSAINOV and A. PANCAR

(communicated by Ronald Brown)

Abstract Mitchell characterized all finite partially ordered sets with incidence ring of Hochschild dimension 0, 1, and 2. Cheng characterized all finite partially ordered sets of cohomological dimension one. There are no conjectures in other dimensions. This article contains the algorithms for calculating the dimensions of finite partially ordered sets by elementary operations over rows and columns of matrices with integer entries.

1. Cohomological dimension

Denote by N the set of nonnegative integers, Z the additive group of integers, Ab the category of abelian groups and homomorphisms. For any subset M ⊆ N the sup will be considered in {−1} ∪ N ∪ {∞}. Thus we let sup ∅ = −1.

Let (X, ⩽) be a partially ordered set (shortly poset). We consider X as a small category with objects ObX = X, in which for every x, y ∈ X the set X(x, y) of morphisms x → y consists of one element if x ⩽ y in X, and X(x, y) = ∅, otherwise. If x ⩽ y and x ≠ y, then we write x < y. If neither x ⩽ y nor y ⩽ x, then we say that x and y are incompatible. If every pair of distinct elements in X are incompatible, then we say that X is discrete. If for every x, y ∈ X there exists a sequence x_{0} ⩽ x_{1} ⩾ x_{2} ⩽ ⋯ ⩾ x_{2n} of elements in X such that x_{0} = x and x_{2n} = y, then X is connected. Every subset Y ⊆ X will be considered as the poset in which y_{1} ⩽ y_{2} for y_{1}, y_{2} ∈ Y if and only if y_{1} ⩽ y_{2} in X. The maximal connected subsets of X are called the connected components.

Denote by Ab^{X} the category of functors X → Ab. For every F ∈ Ab^{X} and n ∈ N we have the abelian groups

C^{n}(X, F) = ∏_{x_{0}<···<x_{n}} F (x_{n}).

Considering elements of C^{n}(X, F) as functions with φ(x_{0} < ⋯ < x_{n}) ∈ F (x_{n}), we define homomorphisms d^{n} : C^{n}(X, F) → C^{n+1}(X, F) by the formulas

In the preparation of this paper, we have been assisted by a grant from the NATO and TÜBITAC of Turkey. The first author is supported by Russian Federation Education Department Received October 16, 2000, revised March 15, 2001; published on March 23, 2001. 2000 Mathematics Subject Classification: 18G20 Key words and phrases: cohomological dimension, Hochschild dimension, cohomology of categories \begin{matrix}c \\ ◯ \end{matrix} 2001, A. A. Husainov and A. Pancar. Permission to copy for private use granted.

(d^{n}φ)(x_{0} < ⋯ < x_{n+1}) = ∑_{i=0}^{n}(−1)^{i}φ(x_{0} < ⋯ < x̂_{i} < ⋯ < x_{n+1})+
(−1)^{n+1}F (x_{n} < x_{n+1})(φ(x_{0} < ⋯ < x_{n})).

Here (x_{0} < ⋯ < x̂_{i} < ⋯ < x_{n+1}) = (x_{0} < ⋯ < x_{i−1} < x_{i+1} ⋯ < x_{n+1}), for 0 ⩽ i ⩽ n + 1.

It is well known that d^{n+1} ∘ d^{n} = 0 for all n ∈ N.

Let C^{n}(X, F) = 0 for n < 0. We obtain the complex C^{∗}(X, F) = {C^{n}(X, F), d^{n}}_{n∈Z}. Let lim_{X} : Ab^{X} → Ab be the limit functor, lim_{X}^{n} : Ab^{X} → Ab its n-th right derived functors, n ⩾ 0. Then lim_{X}^{n}F are natural isomorphic to groups H^{n}(C^{∗}(X, F)) = Kerd^{n}/Imd^{n−1}.

Definition 1.1. Let X be a poset. The cohomological dimension cd X is the sup of n ∈ N for which lim_{X}^{n} ≠ 0.

If X = ∅ then the set of such n is empty, hence cd ∅ = −1.

Cheng [2] characterized all finite posets of cohomological dimension one.

For each abelian group A we denote by Δ_{X}A : X → Ab the functor with Δ_{X}A(x) = A for all x ∈ X, and Δ_{X}A(x ⩽ y) = 1_{A} for all x ⩽ y in X. We denote H^{n}(X, A) = lim_{X}^{n}Δ_{X}A. Let pt be a poset consisting of one element. For every poset X there is precisely one map S : X → pt. This map induces the homomorphisms H^{n}(pt, A) → H^{n}(X, A). Let H̃^{n}(X, A) be cokernels of these homomorphisms. It is not hard to see that H̃^{n}(X, A) ≅ H^{n}(X, A) for all n > 0, and H̃^{0}(X, A) is the cokernel of the homomorphism A → lim_{X}Δ_{X}A which carries each a ∈ A into the thread {a}_{x∈X}.

Let H̃^{−1}(∅, A) = A, and H̃^{n}(∅, A) = 0 for all n ≠ −1. When n < −1 we let H̃^{n}(X, A) = 0 for all posets X.

For any z ∈ X we denote W (z, X) = {x ∈ X : x < z}, X/z = {x ∈ X : x ⩽ z}. For every x, y ∈ X the closed interval is the subset [x, y] = {z ∈ X : x ⩽ z ⩽ y} of X. A subset W ⊆ X is convex if for any x, y ∈ W the closed interval [x, y] is included in W.

Let A be an abelian group, W a convex subset of X. We denote by A[W] : X → Ab the functor such that A[W]|_{W} = Δ_{W} A and A[W](z) = 0 for all z ∉ W. If W = {z} consists of one element z we denote A[{z}] by A[z] : X → Ab.

Lemma 1.2. Let A be an Abelian group, X a poset, z ∈ X an element, n ∈ N a number. Then there is an isomorphism

lim_{X}^{n}A[z] ≅ H̃^{n−1}(W (z, X), A), n ⩾ 0.  (1)

Proof. For n ⩾ 2 the assertion is proved in [8, Lemma 3.1]. We will prove it for n = 0 and n = 1. If z is a minimal element in X then lim_{X}A[z] = A and lim_{X}^{n}A[z] = 0 for all n > 0. If z is not minimal, then lim_{X}A[z] = 0. Therefore, the assertion is true for n = 0. For V = X/z and W = W (z, X) there is an exact sequence

0 → A[z] → Δ_{V} A → A[W] → 0.

The long exact sequence associated with lim_{V}^{k} gives an exact sequence

0 → lim_{V} A[z] → lim_{V} Δ_{V} A → lim_{V} A[W] → lim_{V}^{1} A[z] → 0,

where lim_{V} Δ_{V} A = A. It follows that lim_{V}^{1} A[z] is a cokernel of the homomorphism A → lim_{V} A[W]. Hence lim_{V}^{1} A[z] ≅ H̃^{0}(W, A). It was remarked in the proof of [8, Lemma 3.1] that lim_{V}^{n} A[z] ≅ lim_{X}^{n}A[z]. Therefore lim_{X}^{1}A[z] ≅ H̃^{0}(W, A), and (1) is true for all n ⩾ 0. Q.E.D.

Definition 1.3. The cohomological height c.h.X of a poset X is the sup of n ∈ N, for which H̃^{n}(X, Z) ≠ 0.

If X = ∅, or X is a nonempty totally ordered set then c.h.X = −1.

Let C_{n}X be the free abelian group generated by the set of all sequences x_{0} < ⋯ < x_{n} in X. For each n ⩾ 0 we define the homomorphism C_{n+1}X ⟶^{∂_{n}} C_{n}X on elements of basis as ∂_{n}(x_{0} < ⋯ < x_{n+1}) = ∑_{k=0}^{n+1}(−1)^{k}(x_{0} < ⋯ < x̂_{k} < ⋯ < x_{n+1}). It is well known that homology groups of C_{∗}X are isomorphic to the integer homology groups of the nerve of X [4]. We call it the integer homology groups of X and denote by H_{n}X. The complex Ab(C_{∗}X, Z) is isomorphic to C^{∗}(X, Δ_{X}Z). Hence we have the exact sequence of Universal Coefficient Theorem [10, Ch.III, Theorem 4.1]

0 ⟶ Ext(H_{n−1}X, A) ⟶ H^{n}(X, A) ⟶ Ab(H_{n}X, A) ⟶ 0  (2)

Proposition 1.4. Let X be a nonempty finite poset. Then

cd X = 1 + sup{c.h.W (z, X) : z ∈ X}.  (3)

Proof. Let n = cd X. The poset X is nonempty, hence n ⩾ 0. There is F ∈ Ab^{X} such that lim_{X}^{n}F ≠ 0. It follows by [8, Corollary 2.2] that there are A ∈ Ab and z ∈ X satisfying lim_{X}^{n}A[z] ≠ 0, and hence H̃^{n−1}(W (z, X), A) ≠ 0. For k > n the groups lim_{X}^{k}A[z] are trivial, therefore H̃^{k−1}(W (z, X), A) = 0. Let n > 1. For W = W (z, X) the following sequence is exact by (2)

0 ⟶ Ext(H_{n−2}W, A) ⟶ H^{n−1}(W, A) ⟶ Ab(H_{n−1}W, A) ⟶ 0

For every Abelian group B we have H^{n}(W, B) ≅ lim_{X}^{n+1}B[z] = 0. Therefore, either H_{n−1}W is a nontrivial free group, or H_{n−2}W has torsion. Each one of these conditions implies H^{n−1}(W, Z) ≠ 0. If n = 1 then H̃^{0}(W, A) ≠ 0, hence W is nonempty and has more than one components, and H̃^{0}(W, Z) ≠ 0 in this case. Therefore, if n = cd X ⩾ 1 then H̃^{n−1}(W (z, X), Z) for some z ∈ X, and consequently n ⩽ 1+c.h.W (z, X). There are isomorphisms lim_{X}^{k}Z[x] ≅ H̃^{k−1}(W (x, X), Z), hence H̃^{k−1}(W (x, X), Z) for all x ∈ X and k > n. Thus, if cd X ⩾ 1 then the inequality cd X ⩾ 1 + c.h.W (x, X) holds for all x ∈ X.

We have proved cd X = 1 + sup{c.h.W (z, X) : z ∈ X} when cd X ⩾ 1. If cd X = 0 then by Laudal [9] each connected component of X contains the smallest element. In this case H̃^{n}(W (z, X), Z) = 0 for all n ⩾ 0, we obtain cd W (z, X) = −1 for all z ∈ X. Therefore, if cd X = 0 then sup{c.h.W (z, X) : z ∈ X} = −1. Thus (3) is true for all nonempty posets X. Q.E.D.

2. Hochschild–Mitchell dimension

Let X be a poset. Let X′ be denoted the set of all closed intervals in X ordered by the inclusion:

[x_{1}, y_{1}] ⩽ [x_{2}, y_{2}] ⇔ [x_{1}, y_{1}] ⊆ [x_{2}, y_{2}].

If it is convenient we consider closed intervals as pairs a ⩽ b ordered by (x_{1} ⩽ y_{1}) ⩽ (x_{2} ⩽ y_{2}) ⇔ x_{2} ⩽ x_{1} ⩽ y_{1} ⩽ y_{2}. Baues and Wirshing [1] call X′ the factorization category.

Using [6] we take the following definition of the Hochschild–Mitchell dimension for posets:

Definition 2.1. Let X be a poset. The Hochschild–Mitchell dimension dim X of X is the cohomological dimension cd X′ of the poset of closed intervals in X.

It has been proved in [6] that Mitchell’s definition of the Hochschild–Mitchell dimension is equivalent to the one above for partially ordered sets.

Mitchell [11] characterized the finite posets of Hochschild–Mitchell dimensions 0, 1 and 2; dim X = 0 if and only if X is discrete, dim X ⩽ 1 if and only if X is isomorphic to a free category. Related questions was studied in term of the incidence algebra in [3], [5], [13].

Let X and Y be posets. The ordinal sum X +Y is the set X ∐ Y ordered in such a way that x ⩽ y for every x ∈ X and y ∈ Y, and orders on X and Y are preserved. We consider the product X^{op} ×Y as the subset of (X +Y)′ consisting of pairs x < y with x ∈ X and y ∈ Y. A nondecreasing map f : X → Y is called strong coinitial if for all y ∈ Y the integer homology groups of the subset {x ∈ X : f(x) ⩽ y} are isomorphic to homology groups of the point.

Lemma 2.2. Let X and Y are posets, T : (X + Y)′ → Ab a functor. If T |_{X′} = 0 and T |_{Y′} = 0 then lim_{(X+Y)′}T = 0, lim_{(X+Y)′}^{1}T ≅ limT |_{X^{op}×Y}, ⋯, lim_{(X+Y)′}^{n}T ≅ lim^{n−1}T |_{X^{op}×Y}, ⋯, ∀n ⩾ 1.

Proof. It is proved in [7] that inclusions X′ ⊆ X′ ∪ (X^{op} × Y) and Y′ ⊆ Y′∪(X^{op}×Y) are strong coinitial. Applying [7, Lemma 5.1] to the cover of (X +Y)′ by U = X′ ∪ (X^{op} × Y) and V = Y′ ∪ (X^{op} × Y) we obtain the exact sequence

0 → lim_{(X+Y)′}T → limT |_{X′} ⊕ limT |_{Y′} → limT |_{X^{op}×Y} ⋯
→ lim_{(X+Y)′}^{n}T → lim^{n}T |_{X′} ⊕ lim^{n}T |_{Y′} → lim^{n}T |_{X^{op}×Y} ⋯

which with the conditions T |_{X′} = 0 and T |_{Y′} = 0 gives the desired isomorphisms. Q.E.D.

The subset ]x, y[= {z ∈ X : x < z < y} is called the open interval. We denote [x, y[= {z ∈ X : x ⩽ z < y} and ]x, y] = {z ∈ X : x < z ⩽ y}.

Lemma 2.3. Let X be a poset, A an abelian group. Then for every a < b in X there exist isomorphisms

lim_{X′}^{n}A[a < b] ≅ H̃^{n−2}(]a, b[, A)

for all n ⩾ 0. If a = b then lim_{X′}A[a ⩽ b] = A and lim_{X′}^{n}A[a ⩽ b] = 0 for n > 0.

Remark 2.4. In particular, if ]a, b[= ∅ then lim^{1}A[a < b] ≅ H̃^{−1}(∅, A) = A.

Proof. For n ⩾ 3 this was proved in [8, Lemma 3.2]. It follows from [8, Lemma 3.1] that lim_{X′}^{n}A[a ⩽ b] ≅ lim_{[a,b]′}^{n}A[a ⩽ b]. Let a < b. Application of Lemma 2.2 to [a, b[= {x ∈ X : a ⩽ x < b} and {b} provides isomorphisms

lim_{[a,b]′}^{n}A[a < b] ≅ lim_{[a,b[^{op}×{b}}^{n−1}A[a < b] ≅ lim_{[a,b[^{op}}^{n−1}A[a] ≅ H̃^{n−2}(]a, b[, A),

for a ≠ b. If a = b then lim_{[a,b]′}A[a ⩽ b] = 0 for n > 0. The assertion is evident for n = 0, since lim_{X′}A[a < b] = 0. Q.E.D.

Theorem 2.5. Let X be a poset which is not discrete. Then

dim X = 2 + sup{c.h.(]a, b[) : a < b}.

Proof. The poset X is not discrete, hence dim X ⩾ 1 [11].By Lemma 2.3 there are isomorphisms lim_{X′}^{n}A[a < b] ≅ H̃^{n−2}(]a, b[, A) for all n ∈ N, a < b in X, and A ∈ Ab. The application of [8, Corollary 2.2] to X′ gives that the inequality n ⩾ dim X is equivalent to the existence of A ∈ Ab and a < b for which H̃^{n−2}(]a, b[, A) ≠ 0. Hence, if n = dim X then there exists a < b and A, for which H̃^{n−2}(]a, b[, A) ≠ 0. Remark that if k > n then the groups H̃^{k−2}(]a, b[, B) are trivial for all Abelian groups B. If n = dim X = 1 then H̃^{i}(]a, b[, A) = 0 for i ⩾ 0 and so c.h.(]a, b[) = −1. If n = dim X = 2 then we obtain H̃^{0}(]a, b[, A) ≠ 0, it follows that ]a, b[ is nonconnected and consequently H̃^{0}(]a, b[, Z) ≠ 0. If n = dim X > 2 then H̃^{n−1}(]a, b[, B) = 0 for all Abelian groups B, and it follows from the exact sequence of Universal Coefficient Theorem

0 → Ext(H_{n−3}(]a, b[), A) ⟶ H^{n−2}((]a, b[), A) ⟶ Ab(H_{n−2}(]a, b[), A) → 0

that either H̃_{n−2}(]a, b[) is non trivial free, or H̃_{n−3}(]a, b[) has a torsion. Every of the conditions implies that H̃^{n−2}((]a, b[), Z) ≠ 0. Therefore dim X ⩽ 2+sup{c.h.(]a, b[) : a < b}. On the other hand if H̃^{n−2}((]a, b[), Z) ≠ 0, for n ⩾ 2, then cd X′ ⩾ n. Hence dim X ⩾ 2 + sup{c.h.(]a, b[) : a < b}. Thus the equality holds.

3. Method of calculating the cohomological height

Let P be the set of all prime numbers p > 1, Z_{p} the additive group of integers mod p. We introduce a p-cohomological height for the description of calculating the cohomological height.

Definition 3.1. For p ∈ P the p-cohomological height of a poset X is the sup of n ∈ N for which H̃^{n}(X, Z_{p}) ≠ 0.

Remark 3.2. The equality c.h._{p}X = −1 holds for every p ∈ P if and only if the groups H^{i}(X, Z_{p}) are zero for all i > 0 and X is connected or empty. If H^{i}(X, Z_{p}) = 0 for all i > 0, then the rank of the vector space H^{0}(X, Z_{p}) over the field Z_{p} is equal to ∑_{i⩾0}(−1)^{i}|X^{(i)}| where X^{(k)} is the set of all sequences x_{0} < x_{1} < ⋯ < x_{k}, and |X^{(k)}| the number of these sequences. Hence in this case c.h._{p}X = −1 if and only if ∑_{i⩾0}(−1)^{i}|X^{(i)}| ⩽ 1, and the equality ∑_{i⩾0}(−1)^{i}|X^{(i)}| = 0 holds if and only if X = ∅.

Lemma 3.3. Let X be an arbitrary finite poset. Then

c.h.X = sup{c.h._{p}X : p ∈ P}.

Proof. If X = ∅ then the equality is clear. Let X ≠ ∅ and c.h.X = −1. Then H^{i}(X, Z) = 0 for all i > 0 and H^{0}(X, Z) = Z. Using the exact sequences

0 ⟶ Ext(H_{n−1}X, Z) ⟶ H^{n}(X, Z) ⟶ Ab(H_{n}X, Z) ⟶ 0  (4)

we establish that H_{i}X = 0 for all i > 0 and H_{0}X = Z. Applying Universal Coefficient Theorem (2) we obtain c.h._{p}X = −1. If c.h.X = 0, then H^{i}(X, Z) = 0 for all i > 0, it follows from Universal Coefficient Theorem that H_{i}X = 0 for all i > 0, and consequently H^{i}(X, Z_{p}) = 0 for all i > 0. In this case we obtain c.h._{p}X = 0. Considering (4) in the case of c.h.X > 0 we obtain that either H_{n}X is free, or H_{n−1}X includes Z_{p} for some p ∈ P. It follows that H^{n}(X, Z_{p}) ≠ 0 for some p ∈ P. Hence sup{c.h._{p}X : p ∈ P} ⩾ c.h.X. On the other hand, the following sequences are exact for all k ⩾ n

0 ⟶ Ext(H_{k}X, Z_{p}) ⟶ H^{k+1}(X, Z_{p}) ⟶ Ab(H_{k+1}X, Z_{p}) ⟶ 0,

therefore H^{k+1}(X, Z_{p}) = 0. It leads to c.h._{p}X ⩽ c.h.X for all p ∈ P. Thus, the equality is true. Q.E.D.

For arbitrary matrix with integer entries A = (a_{ij}) the p-rank is the rank of the matrix A mod p = (a_{ij}modp) over the field Z_{p}. The absolute rank is the number rk A = inf{rk_{p}A : p ∈ P}. It is well known (see [14, §I.6(D)], for example) that for every matrix A there are invertible matrices S and T, with the determinants ±1, for which the matrix Ã = S ∘ A ∘ T has entries ã_{ij} = 0 for i ≠ j, where ã_{ii} > 0 divide ã_{i+1i+1} for all 1 ⩽ i ⩽ n, where n is a largest number for which ã_{nn} ≠ 0.

Remark that rk A is the number of ones in Ã.

It is true for all n ∈ N that rk d^{n−1} ⩽ ∑_{i⩾n}(−1)^{i−n}|X^{(i)}|. If X is a nonempty poset then there exists n ∈ N for which this inequality is strong, since otherwise the equalities rk H^{0}(X, Z) = rk d^{−1} = 0 lead to X = ∅.

Lemma 3.4. Let X be a nonempty finite poset. Denote

K_{n} = ∑_{i⩾n}(−1)^{i−n}|X^{(i)}|.

Let n be the largest number for which the strong inequality rk d^{n−1} < K_{n} holds. If n > 0 then c.h.X = n. If n = 0 then

c.h.X = \{ \begin{matrix}−1, if K_{0} = 1, \\ 0, otherwise. \end{matrix}

Proof. Let N be largest for which the set of sequences x_{0} < x_{1} < ⋯ < x_{N} is nonempty. Consider complex of abelian groups C^{n} = C^{n}(X, Δ_{X}Z_{p}) and homomorphisms

0 ⟶^{d^{−1}} C^{0} ⟶^{d^{0}} C^{1} ⟶^{d^{1}} ⋯ ⟶^{d^{N−1}} C^{N} ⟶^{d^{N}} 0

We have Kerd^{N} = C^{N}. Hence rk_{p}Kerd^{N} = |X^{(N)}| for all p ∈ P. If there is p ∈ P such that rk_{p}d^{N−1} < |X^{(N)}| then the equality H^{N}(X, Z_{p}) = C^{N}/Im d^{N−1} implies that c.h._{p}X = N and consequently c.h.X = N. Otherwise for every p ∈ P the exact sequence

0 ⟶ Kerd^{N−1} ⟶ C^{N−1} ⟶ Imd^{N−1} ⟶ 0

gives the equality rk_{p}Kerd^{N−1} = |X^{(N−1)}|−|X^{(N)}|. If n > c.h.X, then H^{n}(X, Z_{p}) = 0 for all p ∈ P and hence the following sequences are exact

0 ⟶ Kerd^{n} ⟶ C^{n} ⟶^{d^{n}} ⋯ ⟶^{d^{N−1}} C^{N} ⟶ 0,

consequently rk_{p}Kerd^{n} = |X^{(n)}|−⋯+(−1)^{N−n}|X^{(N)}|. For n = c.h.X there exists p ∈ P, such that rk_{p}Kerd^{n} > rk_{p}Imd^{n}. Therefore, for n = c.h.X there is the inequality

rk_{p}Imd^{n−1} < |X^{(n)}| − ⋯ + (−1)^{N−n}|X(N)|.

Thus the lemma is proved for n > 0. Let n = 0. Then H^{i}(X, Z) = 0 for all i > 0, consequently rk H^{0}(X, Z) = K_{0}. The poset X is not empty, hence K_{0} > 0. If K_{0} = 1, then H̃^{0}(X, Z) = 0 and c.h.X = −1. If K_{0} > 1 then H̃^{0}(X, Z) ≠ 0 and c.h.X = 0. Q. E. D.

4. Algorithms

Calculating the absolute rank rk A of integer matrix. We transform the matrix A to the diagonal form Ã by the well-known algorithm (see [14], for example). Then the absolute rank rk A is equal to the number of ones in Ã. Hence the process is finished if we see an element on the diagonal of Ã which is not equal to 1.

An elementary column (row) operation is a composition of the following transformations:

(1) interchange any two columns (rows),

(2) multiply any column (row) by −1,

(3) add of one column (row) to another.

If S and T are square matrices with detS = ±1, detT = ±1, for which is defined S ∘ A ∘ T, then the matrix Ã = S ∘ A ∘ T may be obtained from A by elementary column and row operations.

The transformation to the diagonal form consists of the following actions:

(a) If a_{11} divides all elements of the first column and of the first row, then we subtract from the i-th row the first row multiplied by a_{i1}/a_{11} for each i ≠ 1. Then we obtain a matrix with a_{i1} = 0 for all i ≠ 1. Analogously we make a_{1j} = 0 for all j ≠ 1.

(b) Let ||A|| be smallest of the |a_{ij}| among a_{ij} ≠ 0. We interchange two columns and two rows in such a way that we obtain |a_{11}| = ||A||. If a_{11} < 0 then we multiply the first row by −1. We consider the case when (a) does not hold. Let i be a number for which a_{11} does not divide a_{i1}. If a_{i1} < 0 then we multiply the i-th row by −1. There are q > 0 and 0 < r < a_{11} such that a_{i1} = a_{11}q + r. We subtract the first row multiplied by q from the i-th row and have obtained a matrix A′ with ||A′|| < ||A||. We do analogous actions over columns if a_{11} does not divide a_{1j} for some j.

(c) Thus we can suppose that a_{i1} = 0 for i > 1 and a_{1j} = 0 for j > 1.

If a_{ij} is not divided by a_{11} for some (i, j) then we add the j-th column to the first column and subtract from i-th row the first row multiplied by q where a_{i1} = a_{11}q+r, 0 < r < a_{11}. We obtain A′ with ||A′|| < ||A|| and then apply (b).

We obtain that all entries are divided by a_{11}. Now, if a_{11} = 1, then we apply the process to the submatrix without the first column and row, and if a_{11} ≠ 1, then we stop. Calculating the cohomological height. Let X be a finite poset. If X = ∅ then c.h.X = −1. We will work with X ≠ ∅. Let N be the largest number for which the set of sequences x_{0} < x_{1} < ⋯ < x_{N} is not empty.

Step 0. Let n = N, K_{n} = |X^{(N)}|.

Step 1. If rk d^{n−1} = K_{n} and n ⩾ 0, then we let K_{n−1} = |X^{(n−1)}| − K_{n}, n = n − 1 and go to step 1.

Step 2. If n > 0, then we let c.h.X = n. If n = 0, then c.h.X = −1 in the case of K_{0} = 1, and c.h.X = 0 in the case of K_{0} > 1.

Example 4.1. Let X be the poset depicted in the following Hasse diagram (directed downward)

[figure: 0; 1; 2; 3; 4]

The set X^{(k)} consists of all sequences x_{0} < x_{1} < ⋯ < x_{k}, for every k ⩾ 0. So X^{(0)} = {0, 1, 2, 3, 4}; X^{(1)} = {0 < 2, 0 < 3, 0 < 4, 1 < 2, 1 < 3, 1 < 4, 2 < 4}; X^{(2)} = {0 < 2 < 4, 1 < 2 < 4}; X^{(3)} = ∅. The homomorphism d^{k} : C^{(k)} → C^{(k+1)} is given by |C^{(k)}| × |C^{(k+1)}| matrix of integers, which is transpose of the matrix defined by the action

∂_{k}(x_{0} < ⋯ < x_{k+1}) = ∑_{i=0}^{k+1}(−1)^{i}(x_{0} < ⋯ < x̂_{i} < ⋯ < x_{i+1}).

The entries of d^{0} and d^{1} are given in the tables

\begin{aligned}& 0<2 & 0<3 & 0<4 & 1<2 & 1<3 & 1<4 & 2<4 \\ 0 & -1 & -1 & -1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & -1 & -1 & -1 & 0 \\ 2 & 1 & 0 & 0 & 1 & 0 & 0 & -1 \\ 3 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\ 4 & 0 & 0 & 1 & 0 & 0 & 1 & 1 \end{aligned} \begin{aligned}& 0<2<4 & 1<2<4 \\ 0<2 & 1 & 0 \\ 0<3 & 0 & 0 \\ 0<4 & -1 & 0 \\ 1<2 & 0 & 1 \\ 1<3 & 0 & 0 \\ 1<4 & 0 & -1 \\ 2<4 & 1 & 1 \end{aligned}

We remark that d^{0} and d^{1} will be reduced to the following diagonal matrices

d̃^{1} = ( \begin{matrix}1 & 0 \\ 0 & 1 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{matrix} ) ,  d̃^{0} = ( \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} )

Step 0. n = 2, K_{2} = |X^{(2)}| = 2.

Step 1. We reduce d^{1} by elementary row and column operators to a diagonal matrix d̃^{1} and obtain rk d^{1} = 2. We have rk d^{1} = K_{2}, let n = 2 − 1, K_{1} = |X^{(1)}| − K_{2} = 5, and go to step 1.

Step 1. We reduce d^{0} to a diagonal matrix d̃^{0} and obtain rk d^{0} = 4. We have rk d^{0} = 4 < K_{1}, and take the next step.

Step 2. n = 1 > 0. Therefore, c.h.X = 1.

Calculating the cohomological dimension. For every x ∈ X we compute c.h.W (x, X) where W (x, X) = {y ∈ X : x < y}. If X = ∅ then we let cd X = −1, otherwise cd X is equal to 1 + m, where m is the largest among c.h.W (x, X). Calculating the Hochschild–Mitchell dimension. X = ∅ if and only if dim X = −1. If X is discrete then dim X = 0. Otherwise we compute c.h.]x, y[ for all pairs x < y. If m is the largest of c.h.(]x, y[) then we let dim X = m + 2. Calculating the cohomological dimension of a finite topological space. Let X_{top} = (X, τ_{X}) be a topological space. The cohomological dimension cd X_{top} of a topological space is the sup of n ∈ N, for which H^{n}(X_{top}, −) : Sh(X_{top}) → Ab are not zeros. Here Sh(X_{top}) is denoted the category of sheaves over X_{top}, and H^{n}(X_{top}, F) cohomology groups of X_{top} with coefficients in the sheaf.

Let O(x) be the smallest open subset containing x ∈ X. We let x ⩽ y ⇔ O(x) ⊆ O(y). Then we obtain the preordered set X. It is known that cd X_{top} = cd (X, ⩽). Cohomological dimensions of equivalent sets are equal, hence cd (X, ⩽) is equal to the cohomological dimension of the corresponding poset. Calculating the Hochschild dimension of incidence ring. Let X be a small category. The incidence ring Z(X) consists of sums ∑_{α∈MorX} n_{α}α where n_{α} ∈ Z are integers almost all of which are zeros. If X is a category with finite set of objects, then by [11] the dimension dim X is equal to Hochschild dimension of the ring Z(X). Therefore, we can calculate the Hochschild dimension of the incidence ring of a finite poset as the Hochschild–Mitchell dimension.

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A. A. Husainov husainov@knastu.ru

Department of Computer Technologies, Komsomolsk-on-Amur State Technical University, prosp. Lenina, 27, Komsomolsk-on-Amur, Russia, 681013

A. Pancar apancar@samsun.omu.edu.tr

Department of Mathematics, Fen Edebiyat Faculty, Ondokuz Mayis University, Kurupelit, Samsun, Turkey, 55139