Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The derived $\infty$-category of Frobenius modules

Klaus Mattis, Timo Weiß

TL;DR

The paper addresses the problem of identifying the derived $\\infty$-category of generalized Frobenius modules with the Frobenius-module structure in the derived category over a geometric $\\mathbb{F}_p$-scheme. It develops the functorial framework for Frobenius modules, reduces to the affine case via Zariski descent, and applies Schwede–Shipley to realize the categories as $R[F]$-module spectra; it also proves left-completeness and descent properties. The main contributions are a t-exact equivalence $\\mathcal{D}(\\mathrm{Frob}(\\mathrm{QCoh}(X), F_*)) \simeq \\mathrm{Frob}(\\mathcal{D}(\\mathrm{QCoh}(X)), \\mathcal{D}(F_*))$ for geometric $X$, together with Zariski descent and a precise affine-model identification with $R[F]$-modules. These results broaden the applicability of Frobenius and Cartier module techniques in positive characteristic and provide essential descent and completeness properties for derived-category methods.

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.

The derived $\infty$-category of Frobenius modules

TL;DR

The paper addresses the problem of identifying the derived -category of generalized Frobenius modules with the Frobenius-module structure in the derived category over a geometric -scheme. It develops the functorial framework for Frobenius modules, reduces to the affine case via Zariski descent, and applies Schwede–Shipley to realize the categories as -module spectra; it also proves left-completeness and descent properties. The main contributions are a t-exact equivalence for geometric , together with Zariski descent and a precise affine-model identification with -modules. These results broaden the applicability of Frobenius and Cartier module techniques in positive characteristic and provide essential descent and completeness properties for derived-category methods.

Abstract

We prove that for a quasi-compact -scheme with affine diagonal (e.g.\ quasi-compact and separated) there is a t-exact equivalence of stable -categories. Here, denotes the -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 -schemes. As a byproduct we prove that the derived -category of Frobenius (and Cartier) modules satisfies Zariski descent.
Paper Structure (6 sections, 60 theorems, 49 equations, 1 table)

This paper contains 6 sections, 60 theorems, 49 equations, 1 table.

Table of Contents

  1. Introduction
  2. Functoriality of Frobenius modules
  3. Frobenius modules as modules over a ring
  4. Zariski descent of Frobenius modules
  5. Left-completeness on geometric schemes
  6. Compact projective generators

Key Result

Theorem 1

Let $X$ be an $\mathbb{F}_p$-scheme. Then there is a canonical t-exact equivalence of presentable stable $\infty$-categories If $X$ is moreover regular Noetherian, then there is a canonical t-exact equivalence of presentable stable $\infty$-categories

Theorems & Definitions (126)

  • Theorem : cartmod
  • Definition
  • Example
  • Theorem 1: \ref{['lem:ab4:main-thm']}
  • Theorem 2: \ref{['lem:zariski:main-thm']}
  • Remark
  • Theorem 3: \ref{['lem:zariski:left-sep-derived-category-sheaf']}
  • Theorem 4
  • Corollary 5
  • Definition 2.1
  • ...and 116 more