Comptes Rendus
Algèbre, Théorie des représentations
Resolving subcategories whose finitely presented module categories are abelian
Ryo Takahashi1
1 Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya 464-8602, Japan
Comptes Rendus. Mathématique, Volume 359 (2021) no. 5, pp. 577-592.

Let 𝒳 be an additive full subcategory of an abelian category. It is a classical fact that if 𝒳 is contravariantly finite, then the category mod𝒳 of finitely presented right 𝒳-modules is abelian. In this paper, we consider the question asking when the converse holds true for a resolving subcategory of the category of finitely generated modules over a commutative noetherian henselian local ring. We give both affirmative answers and negative answers to this question.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/crmath.197
Classification : 13C60, 18A25, 18E10
Ryo Takahashi 1

1 Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya 464-8602, Japan
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{CRMATH_2021__359_5_577_0,
     author = {Ryo Takahashi},
     title = {Resolving subcategories whose finitely presented module categories are abelian},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {577--592},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {359},
     number = {5},
     year = {2021},
     doi = {10.5802/crmath.197},
     language = {en},
}
TY  - JOUR
AU  - Ryo Takahashi
TI  - Resolving subcategories whose finitely presented module categories are abelian
JO  - Comptes Rendus. Mathématique
PY  - 2021
SP  - 577
EP  - 592
VL  - 359
IS  - 5
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.197
LA  - en
ID  - CRMATH_2021__359_5_577_0
ER  - 
%0 Journal Article
%A Ryo Takahashi
%T Resolving subcategories whose finitely presented module categories are abelian
%J Comptes Rendus. Mathématique
%D 2021
%P 577-592
%V 359
%N 5
%I Académie des sciences, Paris
%R 10.5802/crmath.197
%G en
%F CRMATH_2021__359_5_577_0
Ryo Takahashi. Resolving subcategories whose finitely presented module categories are abelian. Comptes Rendus. Mathématique, Volume 359 (2021) no. 5, pp. 577-592. doi : 10.5802/crmath.197. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.197/

[1] Takuma Aihara; Tokuji Araya; Osamu Iyama; Ryo Takahashi; Michio Yoshiwaki Dimensions of triangulated categories with respect to subcategories, J. Algebra, Volume 399 (2014), pp. 205-219 | DOI | MR | Zbl

[2] Tokuji Araya A homological dimension related to AB rings, Beitr. Algebra Geom., Volume 60 (2019) no. 2, pp. 225-231 | DOI | MR | Zbl

[3] Tokuji Araya; Ryo Takahashi; Yuji Yoshino Homological invariants associated to semi-dualizing bimodules, J. Math. Kyoto Univ., Volume 45 (2005) no. 2, pp. 287-306 | MR | Zbl

[4] Maurice Auslander Coherent functors, Proceedings of the Conference on Categorical Algebra (La Jolla, Calif., 1965) (1966), pp. 189-231 | DOI | Zbl

[5] Maurice Auslander; Mark Bridger Stable module theory, Memoirs of the American Mathematical Society, 94, American Mathematical Society, 1969 | MR | Zbl

[6] Maurice Auslander; Ragnar-Olaf Buchweitz The homological theory of maximal Cohen–Macaulay approximations, Mém. Soc. Math. Fr., Nouv. Sér., Volume 38 (1989), pp. 5-37 Colloque en l’honneur de Pierre Samuel (Orsay, 1987) | Numdam | Zbl

[7] Maurice Auslander; Idun Reiten Applications of contravariantly finite subcategories, Adv. Math., Volume 86 (1991) no. 1, pp. 111-152 | DOI | MR | Zbl

[8] Maurice Auslander; Sverre O. Smalø Almost split sequences in subcategories, J. Algebra, Volume 69 (1981) no. 2, pp. 426-454 | DOI | MR | Zbl

[9] Luchezar L. Avramov; Alex Martsinkovsky Absolute, relative, and Tate cohomology of modules of finite Gorenstein dimension, Proc. Lond. Math. Soc., Volume 85 (2002) no. 2, pp. 393-440 | DOI | MR | Zbl

[10] Winfried Bruns; Jürgen Herzog Cohen–Macaulay rings, Cambridge Studies in Advanced Mathematics, 39, Cambridge University Press, 1998 | Zbl

[11] Lars W. Christensen Gorenstein dimensions, Lecture Notes in Mathematics, 1747, Springer, 2000 | MR | Zbl

[12] Lars W. Christensen Semi-dualizing complexes and their Auslander categories, Trans. Am. Math. Soc., Volume 353 (2001) no. 5, pp. 1839-1883 | DOI | MR | Zbl

[13] Lars W. Christensen; Greg Piepmeyer; Janet Striuli; Ryo Takahashi Finite Gorenstein representation type implies simple singularity, Adv. Math., Volume 218 (2008) no. 4, pp. 1012-1026 | DOI | MR | Zbl

[14] Hailong Dao; Ryo Takahashi Classification of resolving subcategories and grade consistent functions, Int. Math. Res. Not., Volume 2015 (2015) no. 1, pp. 119-149 | MR | Zbl

[15] Edgar E. Enochs; Overtoun M. G. Jenda Gorenstein injective and projective modules, Math. Z., Volume 220 (1995) no. 4, pp. 611-633 | DOI | MR | Zbl

[16] E. Graham Evans; Phillip Griffith Syzygies, London Mathematical Society Lecture Note Series, 106, Cambridge University Press, 1985 | MR | Zbl

[17] Evgeniĭ S. Golod G-dimension and generalized perfect ideals, Proc. Steklov Inst. Math., Volume 165 (1984) no. 3, pp. 67-71 translation of Tr. Mat. Inst. Steklova 165 (1984), p. 62-66 | MR | Zbl

[18] Craig Huneke; David A. Jorgensen Symmetry in the vanishing of Ext over Gorenstein rings, Math. Scand., Volume 93 (2003) no. 2, pp. 161-184 | DOI | MR | Zbl

[19] Arash Sadeghi; Ryo Takahashi Resolving subcategories closed under certain operations and a conjecture of Dao and Takahashi, Mich. Math. J., Volume 70 (2021) no. 2, pp. 341-367 | DOI

[20] Ryo Takahashi On G-regular local rings, Commun. Algebra, Volume 36 (2008) no. 12, pp. 4472-4491 | DOI | MR | Zbl

[21] Ryo Takahashi Contravariantly finite resolving subcategories over commutative rings, Am. J. Math., Volume 133 (2011) no. 2, pp. 417-436 | DOI | MR | Zbl

[22] Ryo Takahashi Classification of dominant resolving subcategories by moderate functions (2020) (to appear in Ill. J. Math., available at https://www.math.nagoya-u.ac.jp/~takahashi/papers.html) | DOI | MR | Zbl

[23] Yuji Yoshino Cohen–Macaulay modules over Cohen–Macaulay rings, London Mathematical Society Lecture Note Series, 146, London Mathematical Society, 1990

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

Homological dimension based on a class of Gorenstein flat modules

Georgios Dalezios; Ioannis Emmanouil

C. R. Math (2023)

Relative global dimensions and stable homotopy categories

Li Liang; Junpeng Wang

C. R. Math (2020)

Cohomologie des G-faisceaux en caractéristique positive

Niels Borne

C. R. Math (2003)