Cohomology of the categorical at zero semigroups
In this article we consider the relation between 0−cohomology and extended Eilenberg-MacLane cohomology of categorical at zero semigroups.
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irk-123456789-1573622019-06-21T01:27:40Z Cohomology of the categorical at zero semigroups Kostin, A. In this article we consider the relation between 0−cohomology and extended Eilenberg-MacLane cohomology of categorical at zero semigroups. 2006 Article Cohomology of the categorical at zero semigroups / A. Kostin // Algebra and Discrete Mathematics. — 2006. — Vol. 5, № 1. — С. 51–66. — Бібліогр.: 13 назв. — англ. 1726-3255 2000 Mathematics Subject Classification: 20M50. http://dspace.nbuv.gov.ua/handle/123456789/157362 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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In this article we consider the relation between
0−cohomology and extended Eilenberg-MacLane cohomology of
categorical at zero semigroups. |
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Article |
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Kostin, A. |
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Kostin, A. Cohomology of the categorical at zero semigroups Algebra and Discrete Mathematics |
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Kostin, A. |
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Kostin, A. |
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Cohomology of the categorical at zero semigroups |
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Cohomology of the categorical at zero semigroups |
| title_full |
Cohomology of the categorical at zero semigroups |
| title_fullStr |
Cohomology of the categorical at zero semigroups |
| title_full_unstemmed |
Cohomology of the categorical at zero semigroups |
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cohomology of the categorical at zero semigroups |
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Інститут прикладної математики і механіки НАН України |
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2006 |
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Cohomology of the categorical at zero semigroups / A. Kostin // Algebra and Discrete Mathematics. — 2006. — Vol. 5, № 1. — С. 51–66. — Бібліогр.: 13 назв. — англ. |
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Algebra and Discrete Mathematics |
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AT kostina cohomologyofthecategoricalatzerosemigroups |
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2025-07-14T09:48:16Z |
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1837615281430593536 |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Number 1. (2006). pp. 51 – 66
c© Journal “Algebra and Discrete Mathematics”
Cohomology of the categorical at zero semigroups
Andrey Kostin
Communicated by B. V. Novikov
Abstract. In this article we consider the relation between
0−cohomology and extended Eilenberg-MacLane cohomology of
categorical at zero semigroups.
1. Introduction
The 0−cohomology theory of semigroups with zero element was intro-
duced in the work [7] as a result of investigation devoted to projective
representations of semigroups. This theory was applied also to studying
Brauer monoid [9], matrix algebras [11], calculation of the Eilenberg-
MacLane cohomology (EM-cohomology) of semigroups (see survey [10]
and references there). Unfortunately the 0−cohomology functor is not a
derived functor for all semigroups.
Nevertheless the 0−cohomology functor becomes a derived functor
if we extend the category of coefficients up to the category of covariant
functors from the small category to the category of Abelian groups [5].
On the other hand the 0−cohomology functor is a derived functor for
a certain class of semigroups. B. V. Novikov showed that the categorical
at zero semigroups belong to this class.
More precisely, [7]: let S be a categorical at zero semigroup, then
there is an isomorphism Hn
0 (S,A) ∼= Hn(S̄, A) for all 0−modules under
semigroup S and n ≥ 0. Here H∗
0 denotes the 0−cohomology functor, and
S̄ is the gown of the semigroup S (see sec. 2). This theorem provides a
relation between 0−cohomology of semigroups and EM-cohomology and
takes an important role in the cohomology theory of semigroups.
2000 Mathematics Subject Classification: 20M50.
Key words and phrases: 0−cohomology of semigroups, cohomology of semi-
groups, cohomology of small categories, categorical at zero semigroups.
52 Cohomology of the categorical at zero semigroups
The aim of this article is the generalization of this theorem for the
category of all natural systems.
The paper consists of five sections. The necessary definitions and
theorems are given in second section. The third section is devoted to the
proof of the main result of the article, see theorem 3. The examples of
applications of the main theorem are considered in section 4. The relation
between the theory of Baues’s-Wirshing’s cohomology for small categories
in case of categories without inverse morphisms and 0−cohomology of
semigroups is considered in section 5.
I would like to thank Prof. Boris Novikov for his kind assistance and
encouragement during the work on this paper.
2. Preliminaries
By S1 we will denote a semigroup S with an adjoint identity. A semigroup
S with zero element is called categorical at zero [3] if abc = 0 implies
ab = 0 or bc = 0.
A small category C is called connected if for all objects a and b there
is a sequence of morphisms a −→ c1 ←− c2 −→ . . . −→ cn ←− b of the
category C.
Let F : C −→ D be a covariant functor of small categories and
d ∈ ObD. The comma category (F ↓ d) is the category which objects are
morphisms Fc
α
−→ d, and a morphism from Fk
α
−→ d to Fl
β
−→ d is a
morphism ξ : k −→ l such that β ◦ Fξ = α.
Let B be a category, A is a subcategory of B. A functor R : B −→ A
is said to be a reflector [2], if for all objects b ∈ B there is a morphism
ηb : b −→ Rb such that each arrow g : b −→ a ∈ A can be represented as
g = fηb for unique morphism f ∈ MorA(Rb, a).
Let us consider the category Sem0 whose objects are semigroups with
zero and morphisms are mappings f : S \ 0 −→ T such that f(xy) =
f(x)f(y) for xy 6= 0. As it was shown in [8] there is a reflector R :
Sem0 −→ Sem where Sem is the category of all semigroups.
The reflector’s value RS is called the gown of a semigroup S and is
denoted by S̄ [7].
The gown S̄ consists of tuples < s1, . . . , sn > with si 6= 0, sisi+1 = 0.
In case if S is a categorical at zero semigroup, the product of elements
s =< s1, . . . , sn > and t =< t1, . . . , tm > is defined by the formula:
st =
{
< s1, . . . , snt1, . . . , tm >, if snt1 6= 0,
< s1, . . . , sn, t1, . . . , tm >, if snt1 = 0.
For simplicity we sometime will omit brackets in products like α <
u > or < u > α, where α ∈ S̄ and u ∈ S.
A. Kostin 53
By the symbol |s| we will denote the number of elements in a tuple
s =< s1, . . . , sn >.
By a nerve of a category C we will call the set Nn(C) of all tuples
(α1, . . . , αn) which components are morphisms of the category C such
that the composition αi+1αi exists for all 1 ≤ i ≤ n− 1 and n ≥ 1. The
nerve N0(C) consists of all objects of the category C.
Let Cn(C) denote a free Abelian group with the set of generators
Nn(C). We define a coboundary homomorphism dn : Cn+1(C) −→
Cn(C) on the generating set by the formula:
dn(α1, . . . , αn+1) = (α2, . . . , αn+1) +
n
∑
i=1
(−1)i(α1, . . . , αiαi+1, . . . , αn+1) +
(−1)n+1(α1, . . . , αn).
The homology of the complex {Cn(C), dn}
∞
n=0 is called an integral ho-
mology of the nerve of the category C and is denoted by Hn(C) (see, for
example, [12]).
By the symbol ∆ : Ab −→ Ab
C we denote the diagonal functor,
which maps an Abelian group A to the constant functor ∆A : C −→ Ab.
For an object c ∈ ObC the value of the functor ∆A is the Abelian group
A and (∆A)(f) = idA for all morphisms f of the category C.
The right adjoint functor to the functor ∆ is called an inverse limit
lim←−C : Ab
C −→ Ab.
Theorem 1. [6] Let τ : C −→ E be a functor of small categories. Then
the following conditions are equivalent:
a) The category (τ ↓ e) is connected and Hn(τ ↓ e) = 0 for all n > 0, e ∈
E;
b) For every functor F : E −→ Ab the canonical morphism:
lim←−
n
E
F −→ lim←−
n
C
Fτ
is an isomorphism for all n > 0.
Let C be a small category. The category of factorization FC [4] is the
category whose objects are all morphisms of C and the set MorC(f, g)
consists of three-tuples (α, f, β) such that αfβ = g. A covariant functor
D : FC −→ Ab is called a natural system on category C.
If we consider semigroup S1 as a category with a single object, we
obtain the correspondent definitions for the category of factorizations FS1
in semigroup S and natural system FS1 −→ Ab on S.
54 Cohomology of the categorical at zero semigroups
The cohomology of a category C with coefficients in the natural system
D [4] is the Abelian groups Hn(C, D) = Extn(Z,D) for all n ≥ 0 where
symbol Ext denotes the derived functor of Hom-functor and Z : FC −→
Ab is the constant functor: Z(c) ∼= Z.
Analogously to [13], [4] we define the category of 0-factorizations in
the semigroup S with zero F0S
1, whose objects are elements from S \ 0,
and morphism’s set Mor(a, b) consists of three-tuples (α, a, β) where α, β
are elements from S, such that αaβ = b [5].
A 0-natural system on semigroup S with zero is a covariant functor
D : F0S
1 −→ Ab. For simplicity we will denote the value of a functor
D on an object a ∈ ObF0S
1 by Da. Let us denote α∗ = D(α, a, 1) and
β∗ = D(1, a, β), then D(α, a, β) = α∗β
∗ for each morphism (α, a, β).
For given natural number n ≥ 1 let us denote by NnS
1 the set of
all n-tuples (a1, . . . , an) of elements from S1 such that a1 · · · an 6= 0 (the
nerve of semigroup S1). In case n = 0 let N0S
1 = {1}. The map with
the domain on the nerve of S1 that sends each a = (a1, . . . , an) to an
element from Da1···an
is called a n-cochain. The set of all n-cochains is an
Abelian group Cn
0 (S1, D) with respect to pointwise addition. For n = 0
let C0
0 (S1, D) = D1.
Let us define the coboundary homomorphism δn : Cn
0 (S1, D) −→
Cn+1
0 (S1, D) by the formula (n ≥ 1)
(δf)(a1, . . . , an+1) = a1∗f(a2, . . . , an+1)
+
n
∑
i=1
(−1)if(a1, . . . , aiai+1, . . . , an+1) + (−1)n+1a∗n+1f(a1, . . . , an).
For the case n = 0 we set (δf)(x) = x∗f −x
∗f for f ∈ D1, x ∈ S
1 \0.
It is simple to prove that δnδn−1 = 0. The group of cocycles Zn
0 (S1, D)
is the kernel of the coboundary δn. Let Bn
0 (S1, D) denote the group
of coboundaries which is the image of δn−1. The 0−cohomology of a
semigroup S with coefficients in 0−natural system D is the Abelian groups
Hn
0 (S1, D) = Zn
0 (S1, D)/Bn
0 (S1, D).
Theorem 2. [5] Let S be a semigroup with zero, D : F0S
1 −→ Ab is a
0-natural system. Then there is an isomorphism
Hn
0 (S1, D) ∼= Extn(Z,D),
which is natural in D, where Z is the constant 0−natural system
Z(s) ∼= Z.
A. Kostin 55
3. Main theorem
Let S be a categorical at zero semigroup and S̄ be the grown of S. Con-
sider the embedding functor i : F0S
1 −→ FS̄1 which is defined by the
formula:
i(l) = < l >, (1)
i(s, l, p) = (< s >,< l >,< p >),
for all objects l ∈ ObF0S
1 and morphisms (s, l, p) ∈ MorF0S
1.
Lemma 1. The embedding i is full.
Proof. Let us consider the arrow < l >
(σ,τ)
−→< t > for some σ, τ ∈ S̄1 and
l, t ∈ ObF0S
1. Then < t >= σ < l > τ , what implies the following
equality |σ < l > τ | = 1. This is possible only if σ, τ ∈ S1 and σlτ 6= 0,
i.e. (σ, l, τ) ∈ MorF0S
1.
Let us formulate the main result of the article.
Theorem 3. Let S be a categorical at zero semigroup, i : F0S
1 −→ FS̄1
is the embedding functor which was defined in (1), D : FS̄1 −→ Ab is
a natural system on S̄1. Then the functor i induces an isomorphism of
cohomology
Hn(S̄1, D) ∼= Hn
0 (S1, Di)
for all n ≥ 0.
Let us sketch the main steps of the proof. In the first step we consider
cohomology functor of semigroup as a derived functor of the limit functor
under some factorization category. Further, we explore the integral com-
plex of the nerve of comma category (i ↓ s) for s ∈ S̄1. For completion of
the proof, due to the theorem 1, it is sufficient to show that the comma
category (i ↓ s) is connected for all s ∈ S̄1 and that the integral homology
of nerve (i ↓ s) is acyclic. The lemmas 2 and 3 are devoted to this aim.
Proof. In [4] it was shown that the cohomology of a small category C
with coefficients in natural systems is the derived functor of the inverse
limit functor:
Hn(C, E) = lim←−
n
FCE, n ≥ 0,
whereE : FC −→ Ab is a natural system. If we consider the monoid S̄1 as
a category with a single object, then we have the following isomorphism:
Hn(S̄1, D) ∼= lim←−
n
FS̄1
D for all n ≥ 0.
56 Cohomology of the categorical at zero semigroups
Since the 0−cohomology functor is a derived functor (see theorem 2),
it is easy to show that Hn
0 (S1, D) ∼= Extn(Z,D) ∼= lim←−
n
F0S1D for all n ≥ 0.
The embedding functor i : F0S
1 −→ FS̄1 induces the natural homor-
phism:
Ψ : lim←−
n
FS̄1D −→ lim←−
n
F0S1Di, n ≥ 0.
By the theorem 1 let us show that under conditions of the theorem 3 the
homorphism Ψ is an isomorphism of cohomologies.
Let us adduce auxiliary notations which will be useful in future. Let
s =< s1, . . . , sm > be an element of the gown S̄. Then define
ek = (< s1, . . . , sk >, 1, < sk+1, . . . , sm >) ∈ (i ↓ s), 0 ≤ k ≤ m,
s̄k = (< s1, . . . , sk−1 >,< sk >,< sk+1, . . . , sm >) ∈ (i ↓ s), 1 ≤ k ≤ m.
Lemma 2. Let S be a categorical at zero semigroup, S̄ is the gown of
S, i : F0S
1 −→ FS̄1 is the embedding functor. Then the comma category
(i ↓ s) is connected for each s ∈ S̄1.
Proof. Consider the elements (α, l, β), (γ, t, δ) ∈ Ob(i ↓ s). We define
the path which connects these objects. Let s =< s1, . . . , sn >.
a) If n = 1, then (1, s, 1) ∈ Ob(i ↓ s) and the commutative diagram
s
l
(α
, β
) -
t
� (γ, δ)
s
?� (γ
, δ
)(α, β) -
gives the required path.
b) Let n ≥ 2. It is obvious that the diagram
s̄p
�
(sp, 1)
ep
(1, sp+1)
- s̄p+1
shows us that the set {ek}
⋃
{s̄p} ⊂ Ob(i ↓ s) is connected.
Suppose (α, l, β) = (< s1, . . . , sk−1, αk >, l,< β1, sk+1, . . . , sn >) ∈
(i ↓ s) does not belong to the set {ek}
⋃
{s̄p}. It is easy to check that the
morphism (u, l, v) : (α, l, β) −→ s̄k of the comma category (i ↓ s) which
is defined by the equation
(u, l, v) =
(αk, l, β1), if αklβ1 6= 0,
(αk, l, 1), if αkl 6= 0 and lβ1 = 0,
(1, l, β1), if αkl = 0 and lβ1 6= 0,
for some 1 ≤ k ≤ n, defines the path between objects (α, l, β) and s̄k.
A. Kostin 57
We start the proof of the fact that the complex {Cn(i ↓ s), dn}
∞
n=0,
s ∈ ObFS̄1 is acyclic.
By the definition Cn(i ↓ s), n ≥ 1, is a free Abelian group which
is generated by the set of all tuples (f1, . . . , fn) of morphisms from the
category (i ↓ s), such that the composition fi+1fi exists and C0(i ↓ s) =
ZOb(i ↓ s).
Let Ω = ((u1, v1), . . . , (un, vn)) be an element of the nerve Nn(i ↓
s), n ≥ 1, s ∈ ObFS̄1. This means that the diagram
l0
(u1, v1)
- l1
(u2, v2)
- l2 - . . . - ln−1
(un, vn)
- ln
. . .
s
�
(αn
, βn
)
�
-
-
(α0 , β0)
-
is commutative where li ∈ S
1 and αi, βi ∈ S̄
1. In other words the follow-
ing equalities
li+1 = ui+1livi+1,
αi = αi+1ui+1,
βi = vi+1βi+1,
exist for all 0 ≤ i ≤ n−1. It implies that the nerve’s element Ω is defined
by the cortege
[αn, un, . . . , u1, l0, v1, . . . , vn, βn].
In such a way the basis of the group Cn(i ↓ s) is the set Nn(i ↓ s) of el-
ements [α, un, . . . , u1, l, v1, . . . , vn, β], such that unun−1 · · · l · · · vn−1vn ∈
S1, αunun−1 . . . l · · · vn−1vnβ = s for some α, β ∈ S̄1, n ≥ 1.
On basis elements of Cn(i ↓ s) the coboundary dn−1 : Cn(i ↓ s) −→
Cn−1(i ↓ s), n ≥ 1 acts by the formula:
dn−1[α, un, . . . , u1, l, v1, . . . , vn, β] = [α, un, . . . , u1lv1, . . . , vn, β] +
n−1
∑
i=1
(−1)i[α, un, . . . , ui+1ui, . . . , u1, l, v1, . . . , vivi+1, . . . , vn, β] +
(−1)n[αun, . . . , u1, l, v1, . . . , vnβ].
It is simple to prove that the groups Cn(i ↓ s) with their coboundary
maps form the simplicial object in the category of Abelian groups. By the
theorem of normalization [1] it is sufficient to prove that the normalized
complex {Ĉn(i ↓ s), dn}
∞
n=0 is acyclic. Here Ĉn(i ↓ s), n ≥ 1 is a free
Abelian group with the set of generators:
N̂n(i ↓ s) = {[α, un, . . . , l, . . . , vn, β] ∈ Nn(i ↓ s)|
(uk, vk) 6= (1, 1), α, β ∈ S̄1, ∀k}.
58 Cohomology of the categorical at zero semigroups
The group Ĉ0 coincides with C0 by the definition.
Lemma 3. The complex {Ĉn(i ↓ s), dn}
∞
n=0 is acyclic.
Proof. Let us construct a contracting homotopy εn+1 : Ĉn −→ Ĉn+1,
n ≥ 1. Let Ω = [α, un, . . . , u1, l, v1, . . . , vn, β] ∈ N̂n(i ↓ s) for el-
ements α =< α1, . . . , αk >, β =< β1, . . . , βp >. For all elements
s =< s1, . . . , sm >∈ S̄ let us denote l(s) =< s2, . . . , sm > and r(s) =<
s1, . . . , sm−1 >. Denote by ω the product un · · ·u1lv1 · · · vn. Define the
homotopy by the formula:
εn+1Ω=
0, if αkω = ωβ1 = 0,
(−1)n+1[r(α), αk, un, . . . , vn, β1, l(β)], if αkωβ1 6= 0,
(−1)n+1[r(α), αk, un, . . . , vn, 1, β], if αkω 6= 0, ωβ1 = 0,
(−1)n+1[α, 1, un, . . . , vn, β1, l(β)], if αkω = 0, ωβ1 6= 0,
for all n ≥ 1.
Let us prove that in the case n ≥ 2 the diagram
Ĉn+1
dn
-
�
εn+1
Ĉn
dn−1
-
�
εn
Ĉn−1
yields the following equation:
εndn−1 + dnεn+1 = id
Ĉn
. (2)
Consider three possible cases which correspond to the definition of
εn+1.
Case 1. αkω = ωβ1 = 0. Then dnεn+1Ω = 0 and
εndn−1Ω = εn[α, un, . . . , u1lv1, . . . , vn, β] + (3)
n−1
∑
i=1
(−1)iεn[α, un, . . . , ui+1ui, . . . , u1, l, v1, . . . , vivi+1, . . . , vn, β] +
(−1)nεn[αun, . . . , u1, l, v1, . . . , vnβ].
By associativity of S1 all the summands of equality (3) except the last
one are equal to zero.
Let un = 1 then vn 6= 1. Since S is categorical at zero we conclude
that vnβ1 = 0. Thus we have
εn[αun, un−1, . . . , vn−1, vnβ] = εn[α, un−1, . . . , vn−1, < vn, β1 >] =
(−1)n[α, 1, un−1, . . . , vn−1, vn, β] = (−1)nΩ
A. Kostin 59
The proof of the equation (2) in the case when un 6= 1 and vn = 1 is dual
with the given one. Now let un 6= 1 and vn 6= 1. Since S is categorical at
zero we have αkun = vnβ1 = 0. Hence
εn[αun, un−1, . . . , vn−1, vnβ] =
εn[< . . . , αk, un >, un−1, . . . , vn−1, < vn, β1, . . . >] = (−1)nΩ.
So the proof of the equation (2) is finished for the first case.
Case 2. Let Ω ∈ N̂n(i ↓ s) such that αkωβ1 6= 0. Then
dnεn+1Ω = (−1)n+1dn[r(α), αk, un, . . . , vn, β1, l(β)] =
(−1)n+1([r(α), αk, un, . . . , u1lv1, . . . , vn, β1, l(β)] +
n−1
∑
i=1
(−1)i[r(α), αk, . . . , ui+1ui, . . . , vivi+1, . . . , β1, l(β)] +
(−1)n[r(α), αkun, un−1, . . . , vn−1, vnβ1, l(β)] + (−1)n+1Ω) (4)
Evidently the sum of the first n summands of (3) equals to the correspon-
dent sum of (4) with the opposite sign. By the definition of the product
in S̄1 the last summand of the equality (3) has the form:
(−1)nεn[αun, un−1, . . . , vn−1, vnβ] =
(−1)nεn[< . . . , αk−1, αkun >, un−1, . . . , vn−1, < vnβ1, β2, . . . >] =
(−1)n[r(α), αkun, . . . , vnβ1, l(β)].
This statement ends the proof of the second case.
Case 3. In case when αkω = 0 and ωβ1 6= 0 (the case when αkω 6= 0
and ωβ1 = 0 is dual) the proof of (2) is similar with the second case.
Now we start the proof of equation (2) for n = 1. Let s =< s1, . . . ,
sm >, m ≥ 2 (the case when m = 1 is evident).
Define the homorphism ε1 : C0(i ↓ s) −→ Ĉ1(i ↓ s), m ≥ 2. Introduce
the following auxiliary notations:
sl
k =[< s1, . . . , sk−1 >, sk, 1, 1, < sk+1, . . . , sm >]∈ Ĉ1(i ↓ s), 1 ≤ k ≤ m,
sr
k =[< s1, . . . , sk−1 >, 1, 1, sk, < sk+1, . . . , sm >]∈ Ĉ1(i ↓ s), 1 ≤ k ≤ m.
We give ε1 by the formulas:
ε1ek =
k
∑
i=1
sr
i +
m
∑
i=k+1
sl
i, 0 ≤ k ≤ m,
ε1s̄k =
k
∑
i=1
sr
i +
m
∑
i=k
sl
i, 1 ≤ k ≤ m.
60 Cohomology of the categorical at zero semigroups
Now consider the case when the element γ = [α, u, β] ∈ C0(i ↓ s) does
not belong to the set {ek, s̄l}k,l. Then define
ε1(γ) =
−[r(α), αk, u, 1, 1, β] + ε1s̄k, if u 6= 1, αku 6= 0, uβ1 = 0,
−[α, 1, u, β1, l(β)] + ε1s̄k+1, if u 6= 1, αku = 0, uβ1 6= 0,
−[r(α), αk, u, β1, l(β)] + ε1s̄k, if αkuβ1 6= 0.
Let Ω = [α, u1, l, v1, β] ∈ N̂1(i ↓ s). Denote ω = u1lv1. Consider three
cases.
Case 1: αkωβ1 6= 0. Then we have:
ε1d0Ω = ε1[α, u1lv1, β]− ε1[αu1, l, v1β] = −[r(α), αk, u1lv1, β1, l(β)] +
ε1s̄k + [r(α), αku1, l, v1β1, l(β)]− ε1s̄k.
In the same time
d1ε2Ω = d1[r(α), αk, u1, l, v1, β1, l(β)] = [r(α), αk, u1lv1, β1, l(β)]−
[r(α), αku1, l, v1β1, l(β)] + Ω.
This ends the exploration of the first case.
Case 2: αkω = 0 and ωβ1 6= 0.
a) l 6= 1. Then
ε1d0Ω = −[α, 1, u1lv1, β1, l(β)] + ε1s̄k+1 + [α, u1, l, v1β1, l(β)]− ε1s̄k+1,
d1ε2Ω = d1[α, 1, u1, l, v1, β1, l(β)] = [α, 1, u1lv1, β1, l(β)]−
[α, u1, l, v1β1, l(β)] + Ω.
b) l = 1, u1 = 1, v1 6= 1. In this case we have:
ε1d0Ω = ε1[α, v1, β]− ε1[α, 1, < v1β1, . . . >] = −[α, 1, v1, β1, l(β)] +
ε1s̄k+1 − ε1ek = −[α, 1, v1, β1, l(β)] + sr
k+1,
d1ε2Ω = d1[α, 1, 1, 1, v1, β1, l(β)] = [α, 1, v1, β1, l(β)]− sr
k+1 + Ω.
c) l = 1, u1 6= 1, v1 = 1. Then we obtain
ε1d0Ω = ε1[α, u1, β]− ε1[< . . . , αk, u1 >, 1, β] = −[α, 1, u1, β1, l(β)] +
ε1s̄k+1 + [α, u1, 1, β1, l(β)]− ε1s̄k+1,
d1ε2Ω = [α, 1, u1, β1, l(β)]− [α, u1, 1, β1, l(β)] + Ω.
This ends our consideration of the second case. It is clear that the proof
in case when αkω 6= 0 and ωβ1 = 0 is similar to the considered one.
Case 3: αkω = 0 and ωβ1 = 0. Then (α, u1lv1, β) = s̄k+1 ∈ C0(i ↓
s), ε2Ω = 0. We have
ε1d0Ω = ε1s̄k+1 − ε1(αu1, l, v1β).
A. Kostin 61
a) l 6= 1 or l = 1 when u1 6= 1 and v1 6= 1. Then ε1d0Ω = ε1s̄k+1 +
Ω− ε1s̄k+1 = Ω.
b) l = 1 and u1 = 1. Then ε1d0Ω = ε1s̄k+1 − ε1ek = sr
k+1 = Ω.
c) l = 1 and v1 = 1. In this case ε1d0Ω = ε1s̄k+1−ε1ek+1 = sl
k+1 = Ω.
In such a way the equation (2) for the case n = 1 is proven. The proof
of the lemma as well as the main theorem is completed.
Remark. There is another form of the described isomorphism. Let i∗ :
Ab
FS̄1
−→ Ab
F0S1
be the restriction functor which is induced by i. Let
us denote by Lani left Kan extension along the functor i (Lani ⊣ i
∗).
Since i is full, we get from [2] (Ch. X)
i∗(LaniG) = G,
for each functor G : F0S
1 −→ Ab. Replacing D by LaniG in the theo-
rem 3, we obtain the other form of the isomorphism:
Hn
0 (S,G) ∼= Hn(S̄, LaniG).
Corollary. [7]. Let S be a categorical at zero semigroup, S̄ its grown.
Then
Hn(S̄, A) ∼= Hn
0 (S,A), n ≥ 0,
for each 0−module A over S.
4. Examples
Example 1. Let us compute the cohomology of the semigroup W with
the representation: 〈ai, bj , w |aibi = w, 1 ≤ i, j ≤ n〉. Let the ideal
I = W \ {ai, bj , w| 1 ≤ i, j ≤ n}. It is simple to check that Rees factor
semigroup S = W/I is categorical at zero and the gown S̄ = W .
Using theorem 3 for computing the cohomology of W , it is sufficient
to calculate 0−cohomology of the semigroup S.
Let F0S
1 be the category of 0−factorizations of S1 and D : F0S
1 −→
Ab is a natural system. Consider the normalized complex:
Ĉ0
0 (S1, D) −→ Ĉ1
0 (S1, D) −→ Ĉ2
0 (S1, D) −→ . . . −→ Ĉn
0 (S1, D) −→ . . .
It is obvious that Ĉp
0 (S1, D) = 0 if p ≥ 3. Let f ∈ Ẑ2
0 (S1, D) =
Ĉ2
0 (S1, D). Then f is defined by its values on the elements of the set
{(ak, bk)}
n
k=1 ⊂ S × S. In that way we have: Ẑ2
0 (S1, D) ∼=
⊕n
i=1Dw.
62 Cohomology of the categorical at zero semigroups
Now let f ∈ B̂2
0(S1, D). It means that there is a function h ∈
Ĉ1
0 (S1, D) such that
f(ai, bi) = D(ai, 1)h(bi)− h(w) +D(1, bi)h(ai), 1 ≤ i ≤ n.
Denote by Mi an Abelian group which consists of elements D(ai, 1)x+
D(1, bi)y for all x ∈ Dbi
, y ∈ Dai
, 1 ≤ i ≤ n.
Let K be a subset of
⊕n
i=1Dw of the tuples (m1, . . . ,mn) where mi ∈
t +Mi for some t ∈ Dw, 1 ≤ i ≤ n. Then H2
0 (S1, D) ∼= (
⊕n
i=1Dw)/K.
Let us compute this factor group.
Consider the map
η : Dw/M1 ⊕Dw/M2 ⊕ · · · ⊕Dw/Mn −→ Dn
w/K, (5)
which is given by the rule:
η(x1 +M1, x2 +M2, . . . , xn +Mn) = (x1, x2, · · · , xn) +K.
We now verify that η is a correctly defined map. Let x′i ∈ xi +
Mi, 1 ≤ i ≤ n. Then η(x1 +M1, x2 +M2, . . . , xn +Mn)−η(x′1 +M1, x
′
2 +
M2, . . . , x
′
n +Mn) = (x1 − x
′
1, . . . , xn − x
′
n) +K = 0 since xi − x
′
i ∈ Mi,
and the map η is correctly defined.
Obviously η is a surjective map. Let us calculate the kernel L of η.
Let η(x1 +M1, x2 +M2, . . . , xn +Mn) = (t +m1, . . . , t +mn) for some
t ∈M, mi ∈Mi, 1 ≤ i ≤ n. It follows that
xi = t+mi, 1 ≤ i ≤ n.
Thus L consists of tuples of cosets (t+M1, . . . , t+Mn) generated by an
element t ∈ Dw. By the Noeter Theorem we conclude that
H2(S1, D) ∼= (
n
⊕
i=1
Dw/Mi)/L, L = {(t+M1, . . . , t+Mn)| t ∈ Dw} (6)
Example 2. Let us compute the second cohomology group of the semi-
group S from the example 1 for more specific case. We will need this
example in section 5.
Let S = W/I be the semigroup from example for n = 3, A be an
Abelian finite group of odd order, D : F0S
1 −→ Ab be a 0−natural
system. For each nonzero element s ∈ S the value of the functor D is the
Abelian group Ds = A
⊕
A. If (s, u, t) be a morphism from F0S
1, a =
(a1, a2) ∈ Du then:
D(s, u, t)a = s∗t
∗a =
a, if s = t = 1,
(a1 + a2, a1 + a2), if s = a1 and t = 1,
(a1 − a2, a2 − a1), if s ∈ {a2, a3} and t = 1,
0, otherwise.
A. Kostin 63
It is easy to check that D is a covariant functor.
Now compute the second 0−cohomology group H2
0 (S,D).
Using notation from example 1 we have:
M1 = a1∗Db1 = {(x, x)| x ∈ A},
Mi = ai∗Dbi
= {(x,−x)| x ∈ A}, i = 2, 3,
K = {(v +m1, v +m2, v +m3)| v ∈ Dw,mi ∈Mi}.
Define the map ϕ :
⊕3
i=1Dw/Mi −→ A3 by the formula
ϕ(l +M1, p+M2, k +M3) = (l1 − l2, p1 + p2, k1 + k2),
where l = (l1, l2), p = (p1, p2) and k = (k1, k2).
Lemma 4. The map ϕ is an isomorphism of Abelian groups. The inverse
map ϕ−1 is defined by the formula ϕ−1(a1, a2, a3) = ((a1, 0) +M1,
(a2, 0) +M2, (a3, 0) +M3).
Consider the epimorphism ψ = η ◦ ϕ−1 : A3 −→ D3
w/K where
η : Dw/M1
⊕
Dw/M2
⊕
Dw/M3 −→ D3
w/K
is the homomorphism which was defined in (5). Using the formula (6) we
obtain H2
0 (S1, D) = A3/Kerψ.
Let us compute the kernel ψ. Let (a1, a2, a3) ∈ Kerψ. Then
((a1, 0), (a2, 0), (a3, 0)) ∈ K ⇔
((a1, 0), (a2, 0), (a3, 0)) = (t+ (l1, l2), t+ (l2,−l2), t+ (l3,−l3))
for some ti, li ∈ A. It follows that a1 = T − P and a2 = a3 = T + P
where T, P ∈ A. Since A has odd order, Kerψ = {(a, b, b), a, b ∈ A}.
Thus we obtain
H2
0 (S1, D) ∼= A. (7)
5. Cohomology of categories without inverse morphisms
A small category C is called a category without inverse morphisms if for
all morphisms f, g from f ◦ g = idx it follows f = g = idx, x ∈ ObC.
Let C be a category without inverse morphisms. Define the semi-
group SNC. The elements of SNC are all nonidentical morphisms of C
and a zero element. For all f, g ∈ SNC \ {0} define the multiplication by
the formula:
fg =
{
f̄ ◦ ḡ, if the composition f̄ ◦ ḡ exists,
0, otherwise.
64 Cohomology of the categorical at zero semigroups
Here and further on for the element f ∈ SNC we denote by f̄ the
correspondent morphism of the category C.
Let us define the map
∗ : MorC −→ MorC
⋃
{1}
by the rule
f∗ =
{
1, if f = idx, x ∈ ObC,
f, otherwise.
Lemma 5. Let C be a small category without inverse morphisms. The
map ∗ can be extended up to the equivalence of categories ∗ : FC
∼
−→
F0SNC
1.
Proof. Let the value of the functor ∗ for object f ∈ ObFC be f∗ ∈
ObF0SNC
1 and (α∗, f∗, β∗) be the image of the morphism (α, f, β) ∈
MorFC. It is simple to check that ∗ is well-defined covariant functor.
Let us prove that the functor ∗ is an equivalence of categories. Denote
by ψ the map MorFC(f, g) −→ MorF0SNC1(f∗, g∗) which is induced by
the functor ∗.
Ensure that ψ is injective. Let (α∗, f∗, β∗) = (α∗, k∗, β∗) for some
morphisms (α, f, β), (α, k, β) ∈ MorFC(f, g). Let g be a nonidentical mor-
phism. Consider two cases. If f∗ = 1 then f = k = iddomα. In case if
f∗ 6= 1 the morphism f = k = f̄∗. If g is the identical morphism, it is
obvious that ψ is injective.
Let us check surjectivity. If (s1, f
∗, s2) be a morphism F0SNC
1 then
s1f
∗s2 6= 0 and the composition s̄1 ◦ f̄∗ ◦ s̄2 exists. It is obvious that
(s̄1, f̄∗, s̄2) is the necessary preimage of (s1, f
∗, s2).
Since the equivalence of the categories implies the isomorphism of
cohomologies, we have proved the following
Theorem 4. Let C be a small category without inverse morphisms, D :
F0SNC
1 −→ Ab is a 0-natural system. Then there is the isomorphism of
cohomologies:
Hn(C, D∗) ∼= Hn
0 (SNC
1, D), n ≥ 0.
A. Kostin 65
Example 3. Let E be a small category which is defined by the commu-
tative diagram
•
•
�
b 1
•
�
b 2
. . . . . . •
b
n
-
•
�
a n
a
2
-
a
1
-
with w = aibi. Obviously E is the category without inverse morphisms
and SNE coincides with the semigroup S from example 1.
Using the result from this example and theorem 4 we get
Proposition. Let P : FE −→ Ab be a natural system on E , τ : F0SNC
1 ∼
−→
FC is an equivalence of categories. Then
H2(E, P )=(
n
⊕
i=1
(Pτ)w/Ki)/L, L = {(m+K1, . . . ,m+Kn)|m ∈ (Pτ)w},
where Ki is a subgroup of (Pτ)w which consists of the elements
(Pτ)(ai, 1)x+ (Pτ)(1, bi)y.
Remark. The cohomology of the category E from example 3 was ex-
plored in [4]. In that work the following result was introduced without
proof
H2(E, P ) = Pw/I2
⊕
Pw/I3, (8)
with I2 = a1∗Pb1 + a2∗Pb2 and I3 = a1∗Pb1 + a3∗Pb3 . It is erroneous.
Indeed, let us consider the functor P = D∗ where D is the 0−natural
system which was defined in (7). Then I2 = M1 + M2 = Dw, I3 =
M1 +M3 = Dw and formula (8) becomes the form H2(E, D∗) = 0.
From the other hand, using the result that was received in (7) we
obtain the formula H2(E, D∗) ∼= A which contradicts with (8).
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Contact information
Andrey Kostin ul. Luk’yanovskaya 21, kv. 99, 04071,
Kiev, Ukraine
E-Mail: kostin_a@inbox.ru
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