Drinfeld-Lau Descent over Fibered Categories

TL;DR

This paper extends Drinfeld's lemma to fibered categories over finite fields, establishing equivalences of categories for various algebraic stacks and gerbes, broadening its applicability in algebraic geometry.

Contribution

The authors generalize Drinfeld's lemma to a wider class of algebraic stacks and fibered categories, providing new equivalences in the context of algebraic geometry over finite fields.

Findings

Drinfeld's lemma holds for proper algebraic stacks and affine gerbes.

The functor alpha is an equivalence for very general algebraic stacks.

Equivalence established for stacks of immersions and certain Deligne-Mumford stacks.

Abstract

Let ${\mathcal X}$ be a category fibered in groupoids over a finite field $\mathbb{F}_q$, and let $k$ be an algebraically closed field containing $\mathbb{F}_q$. Denote by $\phi_k\colon {\mathcal X}_k\to {\mathcal X}_k$ the arithmetic Frobenius of ${\mathcal X}_k/k$ and suppose that ${\mathcal M}$ is a stack over $\mathbb{F}_q$ (not necessarily in groupoids). Then there is a natural functor $\alpha_{{\mathcal M},{\mathcal X}}\colon{\mathcal M}({\mathcal X})\to{\mathcal M}({\mathbf D_k}({\mathcal X}))$, where ${\mathcal M}({\mathbf D_k}({\mathcal X}))$ is the category of $\phi_k$-invariant maps ${\mathcal X}_k\to {\mathcal M}$. A version of Drinfeld's lemma states that if ${\mathcal X}$ is a projective scheme and ${\mathcal M}$ is the stack of quasi-coherent sheaves of finite presentation, then $\alpha_{{\mathcal M},{\mathcal X}}$ is an equivalence. We extend this result in several directions. For proper algebraic stacks or affine gerbes ${\mathcal X}$, we prove Drinfeld's lemma and deduce that $\alpha_{{\mathcal M},{\mathcal X}}$ is an equivalence for very general algebraic stacks ${\mathcal M}$. For arbitrary ${\mathcal X}$, we show that $\alpha_{{\mathcal M},{\mathcal X}}$ is an equivalence when ${\mathcal M}$ is the stack of immersions, the stack of quasi-compact separated \'etale morphisms or any quasi-separated Deligne-Mumford stack with separated diagonal.