Varieties and cohomology of infinitely generated modules.
Let \(G\) be a finite group and let \(H^*(G,k)\) be the cohomology ring of \(G\) with values in a field \(k\). Suppose that the \(p\)-rank of \(G\) is at least \(3\) and that \(k\) is of characteristic \(p\). For any \(kG\)-module \(M\) the ring \(H^*(G,k)\) acts on the Ext-algebra \(\text{Ext}^*_{kG}(M,M)\). Suppose that there is a regular sequence of length \(2\), given by \((x,y)\), say. Using these elements the authors construct an infinite \(kG\)-module \(F(x,y)\) with the following property. The set of primes in \(H^*(G,k)\) containing the annihilator of \(\text{Ext}^*_{kG}(F(x,y),F(x,y))\) in \(H^*(G,k)\) is exactly the set of primes not containing the ideal generated by \(x\) and \(y\). The module \(F(x,y)\) is one of Rickard's idempotent modules. The small support of a module \(N\) is the set of primes occurring in a minimal injective resolution of \(N\). The authors show, heavily using arguments due to Rickard, that the small support of \(F(x,y)\) in \(H^*(G,k)\) consists of all primes of \(H^*(G,k)\). For finitely generated modules the small support coincides with the variety defined as annihilator of the Ext-algebra.
scientific article; zbMATH DE number 5343883