Perverse sheaves and finite-dimensional algebras

TL;DR

This paper characterizes when categories of p-perverse sheaves on stratified spaces are equivalent to module categories over finite-dimensional algebras, linking stratification finiteness to algebraic properties.

Contribution

It provides necessary and sufficient conditions for p-perverse sheaf categories to be equivalent to finite-dimensional algebra modules, including a construction of projective covers.

Findings

Perverse sheaf categories are equivalent to finite-dimensional algebra modules under certain finiteness conditions.

Finiteness of strata and local systems is crucial for the algebraic equivalence.

Constructed projective covers for simple perverse sheaves to establish the equivalence.

Abstract

Let $X$ be a topologically stratified space, $p$ be any perversity on $X$, and $k$ be a field. We show that the category of $p$-perverse sheaves on $X$, constructible with respect to the stratification and with coefficients in $k$, is equivalent to the category of finite-dimensional modules over a finite-dimensional algebra if and only if $X$ has finitely many strata and the same holds for the category of local systems on each of these. The main component in the proof is a construction of projective covers for simple perverse sheaves.