The derived $\infty$-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 $\mathcal{D}(\mathrm{Frob}(\mathrm{QCoh}(X),F_{\ast }))\to \mathrm{Frob}(\mathcal{D}(\mathrm{QCoh}(X)),\mathcal{D}(F_{\ast }))$ 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.

24 pages, final published version