Local rings with quasi-decomposable maximal ideal

Abstract

Let $(R,\mathfrak{m})$ be a commutative noetherian local ring. In this paper, we prove that if $\mathfrak{m}$ is decomposable, then for any finitely generated $R$-module $M$ of infinite projective dimension $\mathfrak{m}$ is a direct summand of (a direct sum of) syzygies of $M$. Applying this result to the case where $\mathfrak{m}$ is quasi-decomposable, we obtain several classfications of subcategories, including a complete classification of the thick subcategories of the singularity category of $R$.

18 pages. Some minor changes throughout the paper; statement of Corollary 6.5 improved; Remark 6.7, Corollary 6.8, and reference [2] are added. Final version to appear in the Mathematical Proceedings of the Cambridge Philosophical Society

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