$F$-finite schemes have a dualizing complex

TL;DR

This paper proves that all Noetherian F-finite schemes possess a canonical dualizing complex compatible with finite type maps, including the Frobenius morphism, using Gabber's result and a new symmetric monoidal structure.

Contribution

It establishes the existence of a canonical dualizing complex for F-finite schemes and introduces a novel symmetric monoidal structure to identify it.

Findings

Existence of a canonical dualizing complex for F-finite schemes.

Compatibility of the dualizing complex with finite type maps.

Introduction of the $!$-tensor product as a new symmetric monoidal structure.

Abstract

In this paper we show that any Noetherian $F$-finite scheme has a dualizing complex $\omega^{\bullet}_{X}$ with the property that for all finite type maps $f \colon X \to Y$ between $F$-finite Noetherian schemes there is a canonical isomorphism $\omega^{\bullet}_{X} \xrightarrow{\cong} f^!\omega^{\bullet}_{Y}$ in $D^b_{coh}(X)$. This, in particular, applies to the Frobenius morphism $F \colon X \to X$ so that we obtain a canonical isomorphism $\omega^{\bullet}_{X} \xrightarrow{\cong} F^!\omega^{\bullet}_{X}$. To prove this, we rely on a result of Gabber that every Noetherian $F$-finite ring is a quotient of a regular ring, from which it follows that every $F$-finite Noetherian scheme has a (potentially non-canonical) dualizing complex. To make this canonical, we identify the dualizing complex of any $F$-finite Noetherian scheme as a unit of an alternate symmetric monoidal structure on $D^b_{coh}(X)$ we call the $!$-tensor product. We also sketch an alternate approach to finding this canonical dualizing complex following the more classical approach to Grothendieck duality.