The decomposition of the trivial module in the complexity quotient category (Q1910746)

Let \(G\) be a finite group and \(k\) an algebraically closed field of characteristic \(p>0\). The main aim of this paper is to show that there are short exact sequences of \(kG\)-modules in which one term is a sum of induced modules from the centralisers of maximal elementary abelian \(p\)-subgroups of \(G\), a second is a translate, \(\Omega^n(k)\), of the trivial \(G\)-module \(k\) and the variety of the third term is less than the full maximal ideal spectrum of the cohomology ring of \(G\). The complexity of a finitely generated \(kG\)-module, \(M\), is the dimension of the variety of \(M\) and the main tools used in the paper include the quotient of the stable category of \(G\)-modules with less than maximal complexity. The implications include that in this complexity quotient category some multiple of the trivial module is a direct sum of induced modules.