Transfer in Hochschild cohomology of blocks of finite groups (Q1303796)

Let \(\mathcal O\) be a complete discrete valuation ring with residue field of characteristic \(p>0\), let \(G\) be a finite group, and let \(b\) be a block (idempotent) of the group algebra \({\mathcal O}G\) with maximal Brauer \(b\) subpair \((P,e_P)\). The author defines the cohomology ring \(H^*(G,b)\) as the subring of \(H^*(P,{\mathcal O})\) consisting of elements whose restriction to \(Q\) is \(N_G(Q,e_Q)\)-stable for every subpair \((Q,e_Q)\) of \((P,e_P)\). This definition is independent of the choice of \((P,e_P)\) (up to isomorphism) and yields the familiar cohomology ring \(H^*(G,{\mathcal O})\) in the case of the principal block. In the general case, the author constructs an embedding of \(H^*(G,b)\) into the Hochschild cohomology ring \(HH^*({\mathcal O}Gb)\). This makes use of transfer maps, between Hochschild cohomology rings, and the author develops the necessary machinery for arbitrary symmetric \(\mathcal O\)-algebras \(A,B\) and chain complexes of \(A\)-\(B\)-bimodules.