The derived $\infty$-category of Frobenius modules

TL;DR

This paper establishes an equivalence of stable $d$-categories for Frobenius modules over certain schemes, extending previous results and proving Zariski descent for derived categories of Frobenius modules.

Contribution

It generalizes prior work by proving a t-exact equivalence for Frobenius modules over broader classes of schemes and shows that their derived categories satisfy Zariski descent.

Findings

Proves t-exact equivalence of Frobenius module categories for quasi-compact schemes with affine diagonal.

Extends previous results from regular Noetherian schemes to more general schemes.

Demonstrates Zariski descent for the derived $d$-category of Frobenius modules.

Abstract

We prove that for $X$ a quasi-compact $\mathbb{F}_p$-scheme with affine diagonal (e.g.\ $X$ quasi-compact and separated) there is a t-exact equivalence $\mathcal D(\mathrm{Frob}(\mathrm{QCoh}(X),F_*)) \to \mathrm{Frob}(\mathcal D(\mathrm{QCoh}(X)),\mathcal D(F_*))$ of stable $\infty$-categories. Here, $\mathrm{Frob}(-,-)$ denotes the $\infty$-category of generalized Frobenius modules as introduced in arXiv:2410.17102. This generalizes our result from arXiv:2410.17102, where we proved the above for regular Noetherian $\mathbb{F}_p$-schemes. As a byproduct we prove that the derived $\infty$-category of Frobenius (and Cartier) modules satisfies Zariski descent.