On the (non)rigidity of the Frobenius Endomorphism over Gorenstein Rings

TL;DR

This paper investigates the properties of the Frobenius endomorphism over Gorenstein rings, revealing conditions under which rigidity fails and establishing new local algebra results related to Picard groups and projective dimensions.

Contribution

It provides new insights into the (non)rigidity of the Frobenius endomorphism over Gorenstein rings and offers a local algebra proof of a known result about the Picard group of punctured spectra.

Findings

Picard group of punctured spectrum has no p-torsion for certain rings

Rigidity of Frobenius endomorphism can fail in Gorenstein rings with isolated singularities

Length criterion for modules of finite length and finite projective dimension

Abstract

It is well-known that for a large class of local rings of positive characteristic, including complete intersection rings, the Frobenius endomorphism can be used as a test for finite projective dimension. In this paper, we exploit this property to study the structure of such rings. One of our results states that the Picard group of the punctured spectrum of such a ring $R$ cannot have $p$-torsion. When $R$ is a local complete intersection, this recovers (with a purely local algebra proof) an analogous statement for complete intersections in projective spaces first given in SGA and also a special case of a conjecture by Gabber. Our method also leads to many simply constructed examples where rigidity for the Frobenius endomorphism does not hold, even when the rings are Gorenstein with isolated singularity. This is in stark contrast to the situation for complete intersection rings. Also, a related length criterion for modules of finite length and finite projective dimension is discussed towards the end.