Functors for a block of a group algebra (Q1322069)

Let \(G\) be a finite group and \(k\) be an algebraically closed field of characteristic \(p\). The main result of the paper is the following Theorem: For each block \(B\) of \(kG\) there is an exact sequence of \(k\)- algebras, indexed by the cells of the Brown complex of chains of \(p\)- subgroups of \(G\): \[ 0 \to \overline{ZB} \to \bigoplus_{{\sigma \in G\setminus \Delta} \atop \dim \sigma = 0} \overline{ZB}_ \sigma \to \bigoplus_{{\sigma \in G\setminus \Delta}\atop \dim \sigma = 1} \overline {ZB}_ \sigma \to \dots \bigoplus_{{\sigma \in G\setminus \Delta}\atop \dim \sigma = n} \overline{ZB}_ \sigma \to 0. \] Here, if \(G_ \sigma\) is the stabilizer in \(G\) of some simplex of \(\Delta\) lying above \(\sigma\), then \(B_ \sigma\) is the sum of the blocks of \(kG_ \sigma\) which are Brauer correspondents of \(B\). Also \(\overline{ZB}\) is the center of \(B\) modulo the ideal of elements that can expressed as \(k\)- linear combination of elements of defect zero, and \(ZB_ \sigma\) is the center of \(B_ \sigma\) modulo the ideal of elements that can be expressed as \(k\)-linear combination of elements of defect group \(\Delta\) in \(G_ \sigma\), where either \(C_ G(\Delta) \nleq G_ \sigma\) or \(\Delta\) is trivial. The morphisms are defined in terms of the Brauer map.