Equivariant sheaves for profinite groups

TL;DR

This paper develops a comprehensive theory of equivariant sheaves over profinite spaces and groups, introducing new constructions and structural results that underpin algebraic models for rational G-spectra.

Contribution

It constructs a robust notion of equivariant presheaves and sheafification, and establishes the abelian category structure of Weyl-G-sheaves, advancing the algebraic modeling of rational G-spectra.

Findings

Weyl-G-sheaves of R-modules form an abelian category with enough injectives.

Equivariant sheaves can be built as colimits of sheaves over finite discrete spaces.

The paper provides explicit constructions of infinite products and equivariant skyscraper sheaves.

Abstract

We study equivariant sheaves over profinite spaces, where the group is also taken to be profinite. We resolve a serious deficit in the existing theory by constructing a good notion of equivariant presheaves, with a suitable equivariant sheafification functor. Using equivariant sheafification, we develop the general theory of equivariant sheaves of modules over a ring, give explicit constructions of infinite products and introduce an equivariant analogue of skyscraper sheaves. These results underlie recent work by the authors which proves that there is an algebraic model for rational G-spectra in terms of equivariant sheaves over profinite spaces. That model is constructed in terms of Weyl-G-sheaves over the space of closed subgroups of G, where the term Weyl indicates that the stalk over H is H-fixed. In this paper, we prove that Weyl-G-sheaves of R-modules form an abelian category with enough injectives and is a coreflective subcategory of equivariant sheaves of R-modules. We end the paper with a structural result that provides another way to conveniently build equivariant sheaves from simpler data. We prove that a G-equivariant sheaf over a profinite base space X is a colimit of equivariant sheaves over finite discrete spaces X_i with actions of finite groups G_i, where X is the limit of the X_i and G is the limit of the G_i.