Group cohomology for Picard groups (Q759829)

The author considers a group G and a category \({\mathcal A}\) endowed with a coherent G-module structure as defined in one of his articles on categories with group structure [ibid. 81, 279-294 (1983; Zbl 0522.18007)]. He defines cohomology groups \(H^ n(G,{\mathcal A})\), \(n\geq 0\), by a sequence \[ 0\to {\mathcal C}^ 0(G,{\mathcal A})\to...\to {\mathcal C}^ n(G,{\mathcal A})\to {\mathcal C}^{n+1}(G,{\mathcal A})\to... \] where \({\mathcal C}^ n(G,{\mathcal A})\) is the category of all maps \(G^ n\to Ob {\mathcal A}\). The prominent example of this situation occurs when G operates on a commutative ring A and \({\mathcal A}\) is the Picard category Pic(A) of projective A-modules of constant rank 1. The author wants to solve a problem posed by Zelinsky, namely to interpret the group \(H^ 3(G,Pic A)\). He does that in his abstract setting by exhibiting \(H^ 3(G,{\mathcal A})\) as a group of equivalence classes of certain extensions of the form \(1\to {\mathcal A}\to {\mathcal D}\to G\to 1\) where \({\mathcal D}\) is a category with a coherent group structure. For a discrete category \({\mathcal A}\), that is when \({\mathcal A}=A\) is a G-module in the usual sense, his result gives the Eilenberg-Mac Lane isomorphism \(H^ 2(G,A)\cong Ext(G,A)\).