Monomial resolutions of trivial source modules
Let \(G\) be a finite group and \(F\) a field of characteristic \(p>0\). For an \(FG\)-module \(V\), a subgroup \(H\) of \(G\) and a homomorphism \(\phi\colon H\to F^\times\) a generalized Brauer construction \(\overline V(H,\phi)\) was defined by the authors in a previous paper, and it turned out that it is related to the canonical induction formula for trivial source modules. In the present paper, for fixed \(V\) and varying \((H,\phi)\), conjugation, restriction, corestriction and transitivity maps between the spaces \(\overline V(H,\phi)\) are defined and properties of these maps are investigated. In Section 2 of the paper the notion of a `Brauer sheaf' is introduced. Its objects are families of \(F\)-vector spaces indexed by pairs \((H,\phi)\) together with conjugation, restriction, corestriction and transitivity maps satisfying certain natural compatibilities. The generalized Brauer construction gives rise to Brauer sheaves, and another source of examples comes from the category \(_{FG}\mathtt{mon}\) of finite \(G\)-equivariant line bundles. In Section 3 a monomial resolution of a trivial source module \(V\) is defined as a chain complex \(M_*\) in \(_{FG}\mathtt{mon}\) using the embeddings of \(V\) and \(M_*\) into the category of Brauer sheaves in a way compatible with the canonical induction formula for \(V\). A necessary and sufficient condition for the existence of a monomial resolution of \(V\) is the existence of a so called `Brauer filtration' of \(V\). If such a filtration exists, \(V\) can be regarded as an object in another category of sheaves \(\text{Sh}_F(G)\) previously introduced by the first author. This leads in Section 4 to the study of monomial resolutions of sheaves in \(\text{Sh}_F(G)\). The construction of the monomial resolution is not functorial. In the last section conditions are given under which \(FG\)-homomorphisms can be extended to chain maps between monomial resolutions, and a list of open questions is stated.
scientific article; zbMATH DE number 1763579