Generalizations of intersection homology and perverse sheaves with duality over the integers

TL;DR

This paper generalizes intersection homology and perverse sheaves to arbitrary PID coefficients, establishing a broad duality framework that encompasses classical theorems and introduces new classes of spaces with self-duality.

Contribution

It extends the Deligne sheaf construction and Poincaré duality to more general settings, including arbitrary PID coefficients and spaces beyond classical IP spaces.

Findings

Duality holds over arbitrary PIDs without local cohomology conditions.

Introduces a new class of spaces with self-duality beyond classical IP spaces.

Contains torsion-sensitive Deligne sheaves within categories of perverse sheaves.

Abstract

We provide a generalization of the Deligne sheaf construction of intersection homology theory, and a corresponding generalization of Poincar\'e duality on pseudomanifolds, such that the Goresky-MacPherson, Goresky-Siegel, and Cappell-Shaneson duality theorems all arise as special cases. Unlike classical intersection homology theory, our duality theorem holds with ground coefficients in an arbitrary PID and with no local cohomology conditions on the underlying space. Self-duality does require local conditions, but our perspective leads to a new class of spaces more general than the Goresky-Siegel IP spaces on which upper-middle perversity intersection homology is self dual. We also examine categories of perverse sheaves that contain our "torsion-sensitive" Deligne sheaves as intermediate extensions.