Contravariantly finite resolving subcategories over commutative local rings are trivial ones

Abstract

We show that a contravariantly finite resolving subcategory over a henselian local ring must be the subcategory 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.

arXiv comment8 pages