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

中文翻译：

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具有准可分解极大理想的局部环

让 （R, 𝔪) 是一个可交换的诺特局部环。在本文中，我们证明如果𝔪是可分解的，那么对于任何有限生成R-模块米无限射影维度𝔪是米. 将此结果应用于 𝔪 是准可分解的情况，我们获得了几个子类别的分类，包括奇点类别的厚子类别的完整分类R.