High Frobenius pushforwards generate the bounded derived category

TL;DR

The paper develops explicit generator results for the bounded derived category $\mathsf{D}^{\mathrm{b}}(\operatorname{coh}X)$ of noetherian schemes in characteristic $p$ by using Frobenius pushforwards. A key mechanism is a local-global principle combined with a nilpotence-type property of Frobenius on local rings, which ties the global generation to a bound in terms of the codepth $\operatorname{codepth} X$. It shows that for $F$-finite $X$ and $e>\log_p(\operatorname{codepth} X)$, high Frobenius pushforwards $F^e_*$ of a generator $G$ of $\mathsf{Perf}\,X$ generate $\mathsf{D}^{\mathrm{b}}(\operatorname{coh}X)$, and if $X$ is locally complete intersection, even $F_* G$ suffices. In the affine case, this yields strong generators like $F^e_*\mathcal{O}_X$, and the results extend to Veronese subrings and relate to Kuznetsov components for complete intersections, providing new structural insight into how Frobenius geometry controls derived-category generation and singularity structure.

Abstract

This work concerns generators for the bounded derived category of coherent sheaves over a noetherian scheme $X$ of prime characteristic. The main result is that when the Frobenius map on $X$ is finite, for any compact generator $G$ of $\mathsf{D}(X)$ the Frobenius pushforward $F ^e_*G$ generates the bounded derived category whenever $p^e$ is larger than the codepth of $X$, an invariant that is a measure of the singularity of $X$. The conclusion holds for all positive integers $e$ when $X$ is locally complete intersection. The question of when one can take $G=\mathcal{O}_X$ is also investigated. For smooth projective complete intersections it reduces to a question of generation of the Kuznetsov component.

Theorems & Definitions (72)

• Theorem A
• Corollary B
• Corollary C
• Theorem D
• Corollary E
• Theorem 1.7
• proof : Proof of \ref{['th:local-global']}
• Corollary 1.10
• Lemma 1.12
• proof