Homological dimension based on a class of Gorenstein flat modules

Georgios Dalezios; Ioannis Emmanouil

doi:10.5802/crmath.480

Algèbre

Homological dimension based on a class of Gorenstein flat modules

Georgios Dalezios ¹ ; Ioannis Emmanouil ²

¹ Institute of Algebra and Number Theory, University of Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
² Department of Mathematics, University of Athens, Athens 15784, Greece

Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1429-1448.

Résumé

In this paper, we study the relative homological dimension based on the class of projectively coresolved Gorenstein flat modules (PGF-modules), that were introduced by Saroch and Stovicek in [26]. The resulting PGF-dimension of modules has several properties in common with the Gorenstein projective dimension, the relative homological theory based on the class of Gorenstein projective modules. In particular, there is a hereditary Hovey triple in the category of modules of finite PGF-dimension, whose associated homotopy category is triangulated equivalent to the stable category of PGF-modules. Studying the finiteness of the PGF global dimension reveals a connection between classical homological invariants of left and right modules over the ring, that leads to generalizations of certain results by Jensen [24], Gedrich and Gruenberg [17] that were originally proved in the realm of commutative Noetherian rings.

Reçu le : 2022-08-10
Révisé le : 2023-02-19
Accepté le : 2023-02-19
Publié le : 2023-11-10

DOI : 10.5802/crmath.480

Affiliations des auteurs :

Georgios Dalezios ¹ ; Ioannis Emmanouil ²

¹ Institute of Algebra and Number Theory, University of Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
² Department of Mathematics, University of Athens, Athens 15784, Greece

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{CRMATH_2023__361_G9_1429_0,
     author = {Georgios Dalezios and Ioannis Emmanouil},
     title = {Homological dimension based on a class of {Gorenstein} flat modules},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1429--1448},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {361},
     year = {2023},
     doi = {10.5802/crmath.480},
     language = {en},
}

TY  - JOUR
AU  - Georgios Dalezios
AU  - Ioannis Emmanouil
TI  - Homological dimension based on a class of Gorenstein flat modules
JO  - Comptes Rendus. Mathématique
PY  - 2023
SP  - 1429
EP  - 1448
VL  - 361
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.480
LA  - en
ID  - CRMATH_2023__361_G9_1429_0
ER  -

%0 Journal Article
%A Georgios Dalezios
%A Ioannis Emmanouil
%T Homological dimension based on a class of Gorenstein flat modules
%J Comptes Rendus. Mathématique
%D 2023
%P 1429-1448
%V 361
%I Académie des sciences, Paris
%R 10.5802/crmath.480
%G en
%F CRMATH_2023__361_G9_1429_0

Georgios Dalezios; Ioannis Emmanouil. Homological dimension based on a class of Gorenstein flat modules. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1429-1448. doi : 10.5802/crmath.480. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.480/

Bibliographie
Cité par

[1] Maurice Auslander; Mark Bridger Stable Module Theory, Memoirs of the American Mathematical Society, 94, American Mathematical Society, 1969

[2] Maurice Auslander; Idun Reiten; Sverre O. Smalø Representation Theory of Artin Algebras, Cambridge Studies in Advanced Mathematics, 36, Cambridge University Press, 1995 | DOI

[3] Jiu-Hong Bai; Li Liang PGF-modules and strongly semi-Gorenstein-projective modules, J. Shandong Univ., Nat. Sci., Volume 56 (2021) no. 8, pp. 105-110 | DOI

[4] Apostolos Beligiannis Cohen-Macaulay modules, (co)torsion pairs and virtually Gorenstein algebras, J. Algebra, Volume 288 (2005) no. 1, pp. 137-211 | DOI | MR | Zbl

[5] Apostolos Beligiannis; Idun Reiten Homological and homotopical aspects of torsion theories, Memoirs of the American Mathematical Society, 883, American Mathematical Society, 2007

[6] Driss Bennis A note on Gorenstein flat dimension, Algebra Colloq., Volume 18 (2011) no. 1, pp. 155-161 | DOI | MR | Zbl

[7] Driss Bennis; Najib Mahdou Global Gorenstein dimensions, Proc. Am. Math. Soc., Volume 138 (2010) no. 2, pp. 461-465 | DOI | MR | Zbl

[8] Theo Bühler Exact categories, Expo. Math., Volume 28 (2010) no. 1, pp. 1-69 | DOI | MR | Zbl

[9] Henri Cartan; Samuel Eilenberg Homological Algebra, Princeton Mathematical Series, 19, Princeton University Press, 1956

[10] Lars Winther Christensen; Sergio Estrada; Peder Thompson Gorenstein weak global dimension is symmetric, Math. Nachr., Volume 294 (2021) no. 11, pp. 2121-2128 | DOI | MR | Zbl

[11] Lars Winther Christensen; Anders Frankild; Henrik Holm On Gorenstein projective, injective and flat dimensions—a functorial description with applications, J. Algebra, Volume 302 (2006) no. 1, pp. 231-279 | DOI | MR | Zbl

[12] Jonathan Cornick; Peter H. Kropholler On complete resolutions, Topology Appl., Volume 78 (1997) no. 3, pp. 235-250 | DOI | MR | Zbl

[13] Georgios Dalezios; Sergio Estrada; Henrik Holm Quillen equivalences for stable categories, J. Algebra, Volume 501 (2018), pp. 130-149 | DOI | MR | Zbl

[14] Ioannis Emmanouil On certain cohomological invariants of groups, Adv. Math., Volume 225 (2010) no. 6, pp. 3446-3462 | DOI | MR | Zbl

[15] Ioannis Emmanouil On the finiteness of Gorenstein homological dimensions, J. Algebra, Volume 372 (2012), pp. 376-396 | DOI | MR | Zbl

[16] Edgar E. Enochs; Overtoun M. G. Jenda Relative Homological Algebra. Vol. 1, De Gruyter Expositions in Mathematics, 30, Walter de Gruyter, 2011

[17] T. V. Gedrich; Karl W. Gruenberg Complete cohomological functors on groups, Topology Appl., Volume 25 (1987), pp. 203-223 | DOI | MR | Zbl

[18] James Gillespie Model structures on exact categories, J. Pure Appl. Algebra, Volume 215 (2011) no. 12, pp. 2892-2902 | DOI | MR | Zbl

[19] James Gillespie; Alina Iacob Duality pairs, generalized Gorenstein modules, and Ding injective envelopes, C. R. Math. Acad. Sci. Paris, Volume 360 (2022), pp. 381-398 | MR | Zbl

[20] Rüdiger Göbel; Jan Trlifaj Approximations and endomorphism algebras of modules, De Gruyter Expositions in Mathematics, 41, Walter de Gruyter, 2006 | DOI

[21] Dieter Happel Triangulated categories in the representation theory of finite-dimensional algebras, London Mathematical Society Lecture Note Series, 119, London Mathematical Society, 1989

[22] Henrik Holm Gorenstein homological dimensions, J. Pure Appl. Algebra, Volume 189 (2004) no. 1-3, pp. 167-193 | DOI | MR | Zbl

[23] Mark Hovey Cotorsion pairs, model category structures, and representation theory, Math. Z., Volume 241 (2002) no. 3, pp. 553-592 | DOI | MR | Zbl

[24] Christian U. Jensen Les foncteurs dérivés de $lim$ et leurs applications en théorie des modules, Lecture Notes in Mathematics, 254, Springer, 1972

[25] Pooyan Moradifar; Jan Šaroch Finitistic dimension conjectures via Gorenstein projective dimension, J. Algebra, Volume 591 (2022), pp. 15-35 | DOI | MR | Zbl

[26] Jan Šaroch; Jan Štʼovíček Singular compactness and definability for $Σ$ -cotorsion and Gorenstein modules, Sel. Math., New Ser., Volume 26 (2020) no. 2, 23, 40 pages | MR | Zbl

Cité par Sources :

Commentaires - Politique