Drinfeld-Lau Descent over Fibered Categories

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 ${\varphi }_k:{\mathcal{X}}_k\to {\mathcal{X}}_k$ 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 ${\alpha }_{\mathcal{M},\mathcal{X}}:\mathcal{M}(\mathcal{X})\to \mathcal{M}({\mathbf{D}}_k(\mathcal{X}))$, where $\mathcal{M}({\mathbf{D}}_k(\mathcal{X}))$ is the category of ${\varphi }_k$-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 étale morphisms or any quasi-separated Deligne-Mumford stack with separated diagonal.

References in corpus (2)

• Approximation of sheaves on algebraic stacks
• Sheafification of linear functors