The decomposition of the trivial module in the complexity quotient category Original Research Article

Suppose that G is a finite group and that k is an algebraically closed field of characteristic p> 0. There exist short exact sequences of kG-modules with the property that one of the terms is a sum of induced modules from the centralizers of maximal elementary abelian p-subgroups of G, a second term is a translate Ωn (k) of the trivial module k, and the variety of the third term is smaller than the full maximal ideal spectrum of the cohomology ring of G. One implication of the result is that any kG-module whose variety intersects the variety of the third term trivially must be an induced module. Also in the quotient category of kG-modules modulo the subcategory of modules with less than maximal complexity, some multiple of the trivial module is a direct sum of induced modules. Thus, by Frobeinus reciprocity, within some complexity factor, the module theory of kG is controlled at the level of the centralizers of the maximal elementary abelian p-subgroups.