纤维化范畴上的Drinfeld-Lau下降
Drinfeld-Lau Descent over Fibered Categories
AI总结：
本文将Drinfeld引理推广至真代数栈、仿射芽及任意纤维化范畴，证明了在多种广义栈条件下自然函子均为等价。
摘要：
设 ${\mathcal X}$ 为有限域 $\mathbb{F}_q$ 上的群胚纤维化范畴，$k$ 为包含 $\mathbb{F}_q$ 的代数闭域。记 $ϕ_k\colon {\mathcal X}_k\to {\mathcal X}_k$ 为 ${\mathcal X}_k/k$ 的算术 Frobenius，并设 ${\mathcal M}$ 为 $\mathbb{F}_q$ 上的栈（不必为群胚）。则存在自然函子 $α_{{\mathcal M},{\mathcal X}}\colon{\mathcal M}({\mathcal X})\to{\mathcal M}({\mathbf D_k}({\mathcal X}))$，其中 ${\mathcal M}({\mathbf D_k}({\mathcal X}))$ 为 $ϕ_k$-不变映射 ${\mathcal X}_k\to {\mathcal M}$ 的范畴。Drinfeld 引理的一个版本指出，若 ${\mathcal X}$ 为射影概形且 ${\mathcal M}$ 为有限表示拟凝聚层栈，则 $α_{{\mathcal M},{\mathcal X}}$ 为等价。我们在多个方向推广了此结果。对于真代数栈或仿射芽 ${\mathcal X}$，我们证明了 Drinfeld 引理并推出 $α_{{\mathcal M},{\mathcal X}}$ 对非常一般的代数栈 ${\mathcal M}$ 为等价。对任意 ${\mathcal X}$，我们证明当 ${\mathcal M}$ 为浸入栈、拟紧分离平展态射栈或具对角分离的任意拟分离 Deligne-Mumford 栈时，$α_{{\mathcal M},{\mathcal X}}$ 为等价。
英文摘要：
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 $ϕ_k\colon {\mathcal X}_k\to {\mathcal X}_k$ 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 $α_{{\mathcal M},{\mathcal X}}\colon{\mathcal M}({\mathcal X})\to{\mathcal M}({\mathbf D_k}({\mathcal X}))$, where ${\mathcal M}({\mathbf D_k}({\mathcal X}))$ is the category of $ϕ_k$-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 $α_{{\mathcal M},{\mathcal X}}$ 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 $α_{{\mathcal M},{\mathcal X}}$ is an equivalence for very general algebraic stacks ${\mathcal M}$. For arbitrary ${\mathcal X}$, we show that $α_{{\mathcal M},{\mathcal X}}$ 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.
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