Partial cohomology of groups. (Q2259153)

Let \(G\) and \(A\) be a group and a semigroup, respectively. A \textit{partial action} of \(G\) on \(A\) is a collection of semigroup isomorphisms \(\theta_{x^{-1}}\to A_x\) for each \(x\in G\), where \(A_x\) is an ideal of \(A\) and such that (i) \(A_1=A\) and \(\theta_1=\text{Id}_A\); (ii) \(\theta_x(A_{x^{-1}}\cap A_y)=A_x\cap A_{xy}\); and (iii) \(\theta_x\theta_y=\theta_{xy}\) on \(A_{y^{-1}}\cap A_{y^{-1}x^{-1}}\). Using this concept, the authors develop the notion of morphisms of partial actions and when \(A\) is commutative they call \(A\) a \textit{partial} \(G\)-module. They define the category of partial \(G\)-modules accordingly. Given a partial \(G\)-module \(A\), the authors define for \(n\geq 1\) the module of \(n\)-cochains, \(C^n(G,A)\), as the set of functions \(f\colon G^n\to A\) such that \(f(x_1,\ldots,x_n)\) is an invertible element of \(A_{(x_1,\ldots,x_n)}=A_{x_1}A_{x_1x_2}\cdots A_{x_1\cdots x_n}\). The \(0\)-cochains are just the invertible elements of \(A\). The authors observe that these are abelian groups and define a coboundary homomorphism \(\delta\colon C^n(G,A)\to C^{n+1}(G,A)\) and the corresponding \textit{cohomology} which they call the \textit{partial} cohomology of \(G\) with values in \(A\). The authors give a description of \(H^2(G,A)\) as a suitable \textit{partial} Schur multiplier and of \(H^n(G,A)\) using free resolutions of modules in the appropriate category.