The commutative cohomology of nilsemigroups
Let \(S\) be a finite commutative semigroup. The category \({\mathcal H}(S)\) is defined as follows: the objects of \({\mathcal H}(S)\) are the elements of \(S\); a morphism from \(a\) to \(b\) is an ordered pair \((a,t)\) with \(t \in S^ 1\) and \(b=at\); composition is given by \((b,u)(a,t)=(a,tu)\). An abelian group valued functor \(A=(A,\alpha)\) on \({\mathcal H}(S)\) consists of an abelian group \(A_ a\) for each \(a \in S\), and a homomorphism \(\alpha_{a,t}: A_ \alpha \to A_{at}\) for every \(a \in S\), \(t \in S^ 1\). A 1- cochain on \(S\) with coefficients in \(A\) is a family \(u=(u_ a)_{a\in S}\) such that \(u_ a \in A_ a\) for all \(a\in S\). A 1-cocycle is a 1- cochain \((u_ a)\) such that \(u_{ab}=\alpha_{a,b}u_ a + \alpha_{b,a}u_ b \in A_{ab}\) for all \(a,b \in S\). A (symmetric) 2- cochain is a family \(c=(c_{a,b})_{a,b\in S}\) such that \(c_{a,b} \in A_{ab}\) and \(c_{a,b}=c_{b,a}\) for all \(a,b \in S\). A 2-cocycle is a 2-cochain \(s\) such that \(\alpha_{ab,c}s_{a,b} + s_{ab,c}=s_{a,bc} + \alpha_{bc,a}s_{b,c}\) for all \(a,b,c \in S\). A 2-coboundary is a 2- cochain \(c\) for which there exists a 1-cochain \(u\) such that \(c=\delta u\), where \((\delta u)_{a,b}=\alpha_{a,b}u_ a + \alpha_{b,a}u_ b - u_{ab}\) for all \(a,b\in S\). Under componentwise addition these form abelian groups \(C^ 1=C^ 1(S,A)=\oplus_{a \in S}A_ a\), \(Z^ 1=Z^ 1(S,A)\), \(C^ 2=C^ 2(S,A) \subseteq \oplus_{a,b\in S}A_{ab}\), \(Z^ 2=Z^ 2(S,A)\) and \(B^ 2=B^ 2(S,A)=\delta C^ 1\). By definition, \(H^ 2(S,A)=Z^ 2(S,A)/B^ 2(S,A)\). Now let \(N^ 1\) be a finite nontrivial commutative nilsemigroup with identity adjoined and let \(F\) be a finitely generated free commutative semigroup such that \(N^ 1=F^ 1/{\mathcal C}\) for some congruence \(\mathcal C\). The main result of the article proves that \(H^ 2(N^ 1,A) \cong MZ^ 1(F^ 1,A)/MB^ 1(F^ 1,A)\), where \(MZ^ 1(F^ 1,A)\) and \(MB^ 1(F^ 1,A)\) are groups of minimal 1-cocycles and minimal 1- coboundaries, respectively. As an application, it is proved that for a finite commutative nilsemigroup \(N\), if \(H^ 2(N^ 1,A)=0\) for every abelian group valued functor \(A\) on \({\mathcal H}(S)\) such that \(A_ 0=0\), then \(N\) is 0-free.
finitely generated free commutative semigroup
scientific article; zbMATH DE number 166145