Drinfeld-Lau Descent over Fibered Categories
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866914889927229440 |
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| author | Di Proietto, Valentina Tonini, Fabio Zhang, Lei |
| author_facet | Di Proietto, Valentina Tonini, Fabio Zhang, Lei |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2012_14075 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Drinfeld-Lau Descent over Fibered Categories Di Proietto, Valentina Tonini, Fabio Zhang, Lei Algebraic Geometry Algebraic Topology Category Theory Number Theory 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. |
| title | Drinfeld-Lau Descent over Fibered Categories |
| topic | Algebraic Geometry Algebraic Topology Category Theory Number Theory |
| url | https://arxiv.org/abs/2012.14075 |