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PreprintarXiv licencearXiv

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 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.

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Cite

BibTeXRIS
BibTeX
@misc{mifune2026contravariantly,
  title = {{Contravariantly finite resolving subcategories over commutative local rings are trivial ones}},
  author = {Yuki Mifune and Gen Tanigawa},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.25626},
  eprint = {2608.25626},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.25626},
}
RIS
TY  - UNPB
TI  - Contravariantly finite resolving subcategories over commutative local rings are trivial ones
AU  - Yuki Mifune
AU  - Gen Tanigawa
PY  - 2026
DA  - 2026/10/10
PB  - arXiv
DO  - 10.48550/arxiv.2608.25626
UR  - https://arxiv.org/abs/2608.25626
AB  - 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.
ER  -