The commutative cohomology of finite semigroups
Let \(S\) be a commutative semigroup. \(S\) as a small (Leech) category is denoted by \({\mathcal H} (S)\). An abelian group valued functor \(A=(A, \alpha)\) on \({\mathcal H} (S)\) assigns to each \(x\in S\) an abelian group \(A_x\) and to each pair \((x,t)\in S\times S^1\) a homomorphism \(\alpha_{x,t}: A_x\to A_{xt}\) so that \(\alpha_{x,1}\) is the identity on \(A_x\) and \(\alpha_{xt,u} \alpha_{x,t}=\alpha_{x,tu}\) for all \(x,t,u\in S\). 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 \(u=(u_a )_{a\in S}\) such that \(u_{ab}=\alpha_{a,b} u_a+\alpha_{b,a} u_b\in A_{ab}\) for all \(a,b\in S\). A 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 \(\alpha_{ab,c} s_a+s_{ab,c}=s_{a,bc}+\alpha_{bc,a} s_{b,c}\) for all \(a,b,c\in S\). A 2-coboundary is a 2-cochain \(c\) for which there exists a 1-cochain \(u=(u_a )_{a\in S}\) 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 \(C^1=C^1 (S,A)=\otimes_{a\in S} A_a\), \(Z^1=Z^1 (S,A) \subseteq C^1\), \(C^2=C^2 (S,A) \subseteq \otimes_{a,b\in S} A_{ab}\), \(Z^2=Z^2 (S,A) \subseteq C^2\) and \(B^2=B^2 (S,A)=\delta C^1 \subseteq Z^2\). By definition, \(H^2 (S, A)=Z^2 (S, A)/ B^2 (S,A)\). The author defines minimal 1-cocycles and minimal 1-coboundaries and proves the main result: \(H^2 (S,A) \cong MZ^1 (F,A)/ MB^1 (F,A)\) for any commutative semigroup \(S\), where \(MZ^1 (F,A)\) is a subgroup of all minimal 1-cocycles and \(MB^1 (F,A)\) a subgroup of all minimal 1-coboundaries. Some special cases (for semigroups with one defining relation) are also considered.
scientific article; zbMATH DE number 810222