On local rings of finite syzygy representation type

TL;DR

This paper characterizes when the completion of a local ring has an isolated singularity, investigates syzygy representation types under completion, and explores Cohen-Macaulay modules and Gorenstein projective modules in this context.

Contribution

It provides new characterizations of isolated singularities, answers Schreyer's conjecture affirmatively, and links finite Cohen-Macaulay representation type to hypersurfaces and Gorenstein projectivity.

Findings

Complete affirmative answer to Schreyer's conjecture.

If finitely many indecomposable MCM modules are locally free, then the ring is a hypersurface or Gorenstein projective modules are projective.

Identifies a new class of virtually Gorenstein rings under finite representation type conditions.

Abstract

Let $R$ be a commutative Noetherian local ring. We characterize when its completion has an isolated singularity, thereby strengthening the Dao-Takahashi refinement of the Auslander-Huneke-Leuschke-Wiegand theorem. We investigate the ascent and descent of finite and countable syzygy representation type along the canonical map from $R$ to its completion. One consequence is a complete affirmative answer to Schreyer's conjecture. We explore analogues of Chen's questions in the context of finite Cohen-Macaulay representation type over Cohen-Macaulay rings. The main result in this direction shows that if $R$ is Cohen-Macaulay and there are only finitely many non-isomorphic indecomposable maximal Cohen-Macaulay modules that are locally free on the punctured spectrum, then either $R$ is a hypersurface or every Gorenstein projective module is projective; moreover, every Gorenstein projective module over the completion of $R$ is a direct sum of finite generated ones. Finally, we study dominant local rings, introduced by Takahashi, under certain finite representation type conditions, and identify a new class of virtually Gorenstein rings.