Cohomology of block ideals of finite group algebras and stable elements. (Q368653)

Let \(G\) be a finite group, let \(p\) be a prime number dividing \(|G|\), and let \(k\) be an algebraically closed field of characteristic \(p\). In [Algebr. Represent. Theory 2, No. 2, 107-135 (1999; Zbl 0934.16010)], \textit{M. Linckelmann} defined the cohomology ring of a block \(B\) of \(kG\) with defect group \(D\) as the subring of \(H^*(D,k)\) consisting of elements which satisfy a stability condition with respect to the fusion of Brauer pairs associated with \(B\). In this paper, the author shows that an element of \(H^*(D,k)\) belongs to the cohomology ring of \(B\) if and only if its embedding into the Hochschild cohomology ring \(HH^*(kD)\) is stable with respect to the source algebra of \(B\).