Blocks and support varieties. (Q492988)

Let \(k\) be a field of characteristic \(p>0\), and let \(G\) be a finite group. The block decomposition of the category \(\mathbf{mod}(kG)\) of finitely generated modules over the group algebra \(kG\) is a classical tool in representation theory. The authors define and investigate block decompositions of related categories, e.g. of the stable category \(\mathbf{stmod}(kG)\) and of thick subcategories \(\mathcal C\) of \(\mathbf{stmod}(kG)\). These are refinements of the classical block decomposition of \(\mathcal C\). In order to describe one of the main results of the paper, let \(\mathcal C_V\) be the thick subcategory of \(\mathbf{stmod}(kG)\) defined by a closed homogeneous subvariety \(V\) of the maximal ideal spectrum \(V_G(k)\) of the cohomology ring \(H^*(G,k)\). Moreover, let \(B\) be a block of \(kG\), let \(V_{G,B}\) be the block variety defined by Linckelmann, and let \(\varrho_B\colon V_{G,B}\to V_G(k)\) be the canonical map. The authors prove that the blocks of \(\mathcal C_V\cap B\) (called ext-blocks in the paper) are parametrized by the connected components of the projective variety associated to \(\varrho^{-1}_B(V)\).