Notes on Beilinson’s “How to glue perverse sheaves

Abstract. The titular, foundational work of Beilinson not only gives a tech-nique for gluing perverse sheaves but also implicitly contains constructions of the nearby and vanishing cycles functors of perverse sheaves. These con-structions are completely elementary and show that these functors preserve perversity and respect Verdier duality on perverse sheaves. The work also de-fines 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. In this paper we discuss Alexander Beilinson’s “How to glue perverse sheaves” [1] with three goals. The first arose from a suggestion of Dennis Gaitsgory that the author study the construction of the unipotent nearby cycles functor Rψun which, as Beilinson observes in his concluding remarks, is implicit in the proof of his Key Lemma 2.1. Here, we make this construction explicit, since it is invaluable in many contexts not necessarily involving gluing. The second goal is to restructure the pre-sentation around this new perspective; in particular, we have chosen to eliminat