Quillen stratification for block varieties.
Let \(G\) be a finite group and let \(k\) be an algebraically closed field. By a classical result of Quillen the maximal ideal spectrum of \(H^*(G, k)\) -- which is an affine variety -- has a stratification labeled by the conjugacy classes of nontrivial elementary Abelian \(p\)-subgroups of \(G\). Later Carlson defined the cohomology variety \(V_G(M)\) of a finitely generated \(kG\)-module \(M\). Avrunin and Scott proved that \(V_G(M)\) has a stratification similar to the one in Quillen's theorem. Now suppose that \(M\) belongs to a \(p\)-block \(bkG\) where \(b\) is a primitive central idempotent in \(kG\). If \(b\) does not define the principal block then \(V_G(M)\) is, in general, not an invariant of \(M\) regarded as a \(bkG\)-module. Motivated by this lack the author defined [in J. Algebra 215, No. 2, 460-480 (1999; Zbl 0960.20008)] a variety \(V_G(M)\) related to the block cohomology \(H^*(G,b)\). This variety allows a surjective morphism onto \(V_G(M)\) which in case of the principal block turns out to be an isomorphism. The paper under review contains essentially two main theorems. In the first one the author describes a technique to compute \(V_{G,b}(M)\) via the truncation \(kP\)-module \(iM\) where \(i\) is a source idempotent and \(P\) a defect group of \(b\). In the other theorem he constructs a stratification of \(V_{G,b}(M)\) which in case of the principal block coincides with the Quillen stratification due to Avrunin and Scott. An illuminating example at the end of the paper clarifies the fact that one really has to work with the truncation \(iM\) and not only with the restriction \(M|_P\) of \(M\) to the defect group \(P\).
scientific article; zbMATH DE number 1787474
scientific article