Base Change Along the Frobenius Endomorphism and the Gorenstein Property

TL;DR

This paper proves that for a local ring in positive characteristic, if the derived base change of a complex via Frobenius has finite injective dimension, then the ring is Gorenstein, linking base change properties to ring structure.

Contribution

It establishes a new criterion connecting the Gorenstein property of a ring with the behavior of complexes under Frobenius base change.

Findings

Derived base change via Frobenius with finite injective dimension implies the ring is Gorenstein.

The result generalizes to contracting endomorphisms beyond Frobenius.

Provides a new characterization of Gorenstein rings in positive characteristic.

Abstract

Let $R$ be a local ring of positive characteristic and $X$ a complex with nonzero finitely generated homology and finite injective dimension. We prove that if derived base change of $X$ via the Frobenius (or more generally, via a contracting) endomorphism has finite injective dimension then $R$ is Gorenstein.