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arXiv · 2507.17097

On local rings of finite syzygy representation type

Souvik Dey·Kaito Kimura·Jian Liu·Yuya Otake
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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.

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Souvik Dey, Kaito Kimura, Jian Liu, Yuya Otake. 2025-07-23. On local rings of finite syzygy representation type. https://arxiv.org/abs/2507.17097

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