Vertex and source determine the block variety of an indecomposable module.
Let \(k\) be a field of finite characteristic \(p\) and let \(G\) be a finite group with order divisible by \(p\). John Carlson defined and studied in the 1980's the cohomology variety \(V_G(M)\) of a module \(M\). This variety depends only on the principal block of \(kG\). Recently Linckelmann defined a variety for any block \(b\) of \(kG\) by some limit construction over the so-called Frobenius category from the usual group cohomology of a fixed defect group \(P\) of \(b\). The main result of the paper is the following. Let \(i\) be a source idempotent and let \(M\) be an indecomposable \(kGb\)-module. Then there is a vertex \(Q\) of \(M\) contained in \(P\) and a source \(U\) of \(M\) such that \(U\) is a direct summand of the restriction of \(iM\) to \(Q\), and such that \(M\) is a direct summand of \(kGi\otimes_{kG}U\). Then, the variety of \(M\) with respect to the block \(b\) equals \(r_Q^*(V_Q(U))\), where \(r_Q\) is given by the composition of inclusion of the block cohomology of \(G\) in the group cohomology of \(P\) and the restriction mapping of usual group cohomology \(H^*(P,k)\to H^*(Q,k)\). This way, the variety of a module in the non principal block is reduced to a variety coming from the usual variety of an appropriate source with respect to an appropriate vertex. Using this result the authors apply results of Carlson on usual cohomology varieties to block cohomology varieties. Namely for indecomposable modules the cohomology variety is connected, and any of its closed homogeneous subvarieties is a module variety for a module of this block.
scientific article; zbMATH DE number 2157516