Rings without a Gorenstein analogue of the Govorov-Lazard Theorem

TL;DR

This paper identifies a broad class of commutative noetherian rings where the Gorenstein analogue of the Govorov-Lazard Theorem does not hold, expanding understanding of Gorenstein module theory.

Contribution

It establishes conditions under which the Gorenstein analogue of the Govorov-Lazard Theorem fails for certain rings, using a new approach based on Gorenstein projective envelopes.

Findings

The analogue fails for rings with a dualizing complex, that are henselian, non-Gorenstein, and have a non-free Gorenstein projective module.

Finitely generated Gorenstein projective modules form an enveloping class iff the ring is Gorenstein or all such modules are free.

The work extends previous results by Christensen et al. on Gorenstein projective modules.

Abstract

It was proved by Beligiannis and Krause that over certain Artin algebras, there are Gorenstein flat modules which are not direct limits of finitely generated Gorenstein projective modules. That is, these algebras have no Gorenstein analogue of the Govorov-Lazard Theorem. We show that, in fact, there is a large class of rings without such an analogue. Namely, let R be a commutative local noetherian ring. Then the analogue fails for R if it has a dualizing complex, is henselian, not Gorenstein, and has a finitely generated Gorenstein projective module which is not free. The proof is based on a theory of Gorenstein projective (pre)envelopes. We show, among other things, that the finitely generated Gorenstein projective modules form an enveloping class in mod R if and only if R is Gorenstein or has the property that each finitely generated Gorenstein projective module is free. This is analogous to a recent result on covers by Christensen, Piepmeyer, Striuli, and Takahashi, and their methods are an important input to our work.