Three forms, under a specific condition
An arXiv preprint reports that a contravariantly finite resolving subcategory of finitely generated modules over a henselian local ring can take only one of three forms: add R, the free-module category; mod R, the whole category of finitely generated modules; or CM(R), the category of maximal Cohen-Macaulay modules. The last option requires R itself to be Cohen-Macaulay.
In ordinary language, contravariant finiteness is an approximation condition involving right approximations from the chosen subcategory. The result concerns selected classes of finitely generated modules and says that, once this condition is imposed under the stated ring hypothesis, only the three listed forms can occur.
A proof built from module structure
This is a theoretical study of commutative noetherian local rings, finitely generated modules and the maximal Cohen-Macaulay subcategory CM(R). It uses no empirical participants or dataset. The proof strategy uses a minimal free resolution and the associated Hom complexes, so the conclusions are algebraic implications under explicit assumptions.
A key lemma in the argument gives a conditional two-branch conclusion. Under its own stated hypotheses, one module, denoted A, has finite projective dimension, or another module, denoted B, has finite injective dimension. The alternatives depend on those hypotheses and are not presented as an unconditional statement.
A separate result for finite type
The paper treats finite type as a separate finiteness condition for resolving subcategories. Under that condition, there are two outcomes: the subcategory is add R, the free-module category; or R is Cohen-Macaulay of finite CM type and the subcategory is CM(R), the maximal Cohen-Macaulay category.
The finite-type analysis is reduced to the complete case. That reduction belongs to the separate finite-type classification, while the main three-way result remains tied to henselian local rings.
A conditional result on injective dimension
Another theorem addresses homological dimensions. If a resolving subcategory contains a module of infinite projective dimension, a minimal right approximation of the module denoted k exists, and eventual Ext vanishing holds for a module denoted M, the theorem concludes that M has finite injective dimension. The conclusion is conditional on all of these requirements.
Restrictions from right approximations
The framework is also applied to right approximations of k. If k has a right approximation from a chosen subcategory X, the paper gives three possibilities: k belongs to X; X equals add R, the free-module category; or R is Cohen-Macaulay and X is contained in CM(R), the maximal Cohen-Macaulay category.
Cohen-Macaulay and semidualizing cases
For a henselian Cohen-Macaulay local ring with a canonical module, the paper states an exact equivalence: a resolving subcategory is contravariantly finite precisely when it is add R, CM(R), or mod R. In that setting, the same three categories are exactly the possible contravariantly finite resolving subcategories.
A further application examines a semidualizing module C and its associated category G_C(R). If the module k has a minimal right G_C(R)-approximation, the category is either add R, or R is Cohen-Macaulay and C is canonical. The corresponding finite-type classification says that G_C(R) is of finite type exactly when it is add R, or when R is Cohen-Macaulay of finite CM type and C is canonical.
A consequence for dominant rings
The paper also reports a consequence for dominant local rings: every Cohen-Macaulay local ring of finite CM type is uniformly dominant. It further states that the two cited Takahashi conjectures hold true.
The conditions define the boundary
The results have a defined scope. The main classification is conditional on the ring being henselian and on contravariant finiteness. The finite-type conclusion carries its own Cohen-Macaulay and finite-CM-type conditions, while the homological conclusion depends on the stated approximation and Ext-vanishing assumptions.
The paper therefore does not classify arbitrary resolving subcategories that lack the stated contravariant-finiteness, finite-type, approximation or Ext-vanishing conditions. Its conclusions are mathematical implications under explicit hypotheses, rather than empirical observations or quantitative estimates.
Paper data and sources
Original title: Contravariantly finite resolving subcategories over commutative local rings are trivial ones
Authors: Yuki Mifune, Gen Tanigawa
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
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