A remark on Deligne's finiteness theorem

TL;DR

This paper discusses a finiteness theorem for irreducible lisse sheaves over schemes over finite fields, extending Deligne's results by relaxing conditions and analyzing different notions of ramification boundedness.

Contribution

It extends Deligne's finiteness theorem to normal schemes and clarifies the necessary conditions and notions of bounded ramification.

Findings

Finiteness of irreducible lisse sheaves with bounded rank and ramification.

Normal schemes suffice for Deligne's theorem, not just smooth.

Discussion of various boundedness notions of ramification.

Abstract

Over a connected geometrically unibranch scheme $X$ of finite type over a finite field, we show finiteness of the number of irreducible $\bar \Q_\ell$-lisse sheaves, with bounded rank and bounded ramification in the sense of Drinfeld, up to twist by a character of the finite field. On $X$ smooth, with bounded ramification in the sense of bounding the Swan conductors on curves, this is Deligne's theorem. Version 2: We prove also that for Deligne's finiteness theorem, it is enough to assume $X$ normal. However, the proof uses the smooth case, unlike the proof in the case of bounded ramification in the sense of Drinfeld. Finally we discuss the various notions of boundedness of ramification used.