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arXiv · 2608.25626

Contravariantly finite resolving subcategories over commutative local rings are trivial ones

Yuki Mifune·Gen Tanigawa
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Abstract

We show that a contravariantly finite resolving subcategory over a henselian local ring must be the category of free modules or the whole module category or the subcategory of maximal Cohen--Macaulay modules. This removes the Gorenstein assumption from a theorem of Takahashi. We also show that a resolving subcategory of finite type other than the subcategory of free modules must be the subcategory of maximal Cohen--Macaulay modules. As a consequence, every Cohen--Macaulay local ring of finite CM type is uniformly dominant, thereby establishing a stronger form of a conjecture of Takahashi.

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Yuki Mifune, Gen Tanigawa. 2026-08-26. Contravariantly finite resolving subcategories over commutative local rings are trivial ones. https://arxiv.org/abs/2608.25626

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