Degenerations for modules over blocks of group algebras (Q1271524)

Let \(K\) be an algebraically closed field of arbitrary characteristic \(p\), and \(G\) be a finite group. Then the group algebra \(KG\) is a product of connected algebras \(B_i\), \(i=1,\ldots,m\), called the blocks of \(KG=B_1\times\cdots\times B_m\). Let \(B\) be a block of \(KG\). The authors prove that every degeneration of finite dimensional \(B\)-modules is given by short exact sequences if and only if the block \(B\) is of finite representation type. As a result, for such block \(B\) the partial orders \(\leq_{\text{deg}}\) and \(\leq_{\text{ext}}\) on the set of isomorphism classes of finite dimensional \(B\)-modules [see \textit{G. Zwara}, J. Algebra 198, No. 2, 563-581 (1997; Zbl 0902.16015)] coincide for all \(B\)-modules.