Notes on Beilinson's "How to glue perverse sheaves"

TL;DR

This paper provides detailed explanations of Beilinson's foundational work on gluing perverse sheaves, including constructions of nearby and vanishing cycles functors, and introduces a new maximal extension functor.

Contribution

It offers a comprehensive, elementary exposition of Beilinson's techniques for gluing perverse sheaves and clarifies the properties of associated functors, including the new maximal extension.

Findings

Nearby and vanishing cycles functors preserve perversity.

These functors respect Verdier duality.

Introduction of a new maximal extension functor.

Abstract

The titular, foundational work of Beilinson not only gives a technique for gluing perverse sheaves but also implicitly contains constructions of the nearby and vanishing cycles functors of perverse sheaves. These constructions are completely elementary and show that these functors preserve perversity and respect Verdier duality on perverse sheaves. The work also defines a new, "maximal extension" functor, which is left mysterious aside from its role in the gluing theorem. In these notes, we present the complete details of all of these constructions and theorems.