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 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- nes 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 rst arose from a suggestion of Dennis Gaitsgory that the author study the construction of the unipotent nearby cycles functor R un 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 eliminate the two-sided limit formalism in favor of the straightforward setup indicated briey in (3,x4.2) for D-modules. We also emphasize this construction as a simple demon- stration that R un ( 1) and Verdier duality D commute, and de-emphasize its role in the gluing theorem. Finally, we provide complete proofs; with the exception of the Key Lemma, (1) provides a complete program of proof which is not carried out in detail, making a technical understanding of its contents more dicult given the density of ideas. This paper originated as a learning exercise for the author, so we hope that in its