Injective dimension of sheaves of rational vector spaces

TL;DR

This paper explores how the complexity of a profinite space, measured by the Cantor-Bendixson rank, determines the injective dimension of sheaves of rational vector spaces over that space.

Contribution

It establishes a direct link between the Cantor-Bendixson rank of a profinite space and the injective dimension of sheaves of rational vector spaces on it.

Findings

Injective dimension equals the Cantor-Bendixson rank for profinite spaces.

Provides a characterization of sheaves of rational vector spaces in terms of topological complexity.

Connects topological properties with algebraic invariants in sheaf theory.

Abstract

The Cantor-Bendixson rank of a topological space X is a measure of the complexity of the topology of X. The Cantor-Bendixson rank is most interesting when the space is profinite: Hausdorff, compact and totally disconnected. We will see that the injective dimension of the Abelian category of sheaves of rational vector spaces over a profinite space is determined by the Cantor-Bendixson rank of the space.