Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-25T06:56:49.608Z Has data issue: false hasContentIssue false
  • English
  • Français

Cut cotorsion pairs

Part of: Homological algebra Homological methods

Published online by Cambridge University Press:  10 December 2021

Mindy Huerta Open the ORCID record for Mindy Huerta [Opens in a new window] ,
Octavio Mendoza Open the ORCID record for Octavio Mendoza [Opens in a new window]  and
Marco A. Pérez Open the ORCID record for Marco A. Pérez [Opens in a new window]
Mindy Huerta*
Affiliation:
Instituto de Matemáticas. Universidad Nacional Autónoma de México. Circuito Exterior, Ciudad Universitaria. CP04510. Mexico City, MEXICO Instituto de Matemática y EstadÍstica “Prof. Ing. Rafael Laguardia”. Facultad de Ingeniería. Universidad de la República. CP11300. Montevideo, URUGUAY
Octavio Mendoza
Affiliation:
Instituto de Matemáticas. Universidad Nacional Autónoma de México. Circuito Exterior, Ciudad Universitaria. CP04510. Mexico City, MEXICO
Marco A. Pérez
Affiliation:
Instituto de Matemática y EstadÍstica “Prof. Ing. Rafael Laguardia”. Facultad de Ingeniería. Universidad de la República. CP11300. Montevideo, URUGUAY
*
Corresponding author Mindy Huerta. E-mail: mindy@matem.unam.mx
Get access Rights & Permissions [Opens in a new window]

Abstract

We present the concept of cotorsion pairs cut along subcategories of an abelian category. This provides a generalization of complete cotorsion pairs, and represents a general framework to find approximations restricted to certain subcategories. We also exhibit some connections between cut cotorsion pairs and Auslander–Buchweitz approximation theory, by considering relative analogs for Frobenius pairs and Auslander–Buchweitz contexts. Several applications are given in the settings of relative Gorenstein homological algebra, chain complexes, and quasi-coherent sheaves, as well as to characterize some important results on the Finitistic Dimension Conjecture, the existence of right adjoints of quotient functors by Serre subcategories, and the description of cotorsion pairs in triangulated categories as co-t-structures.

Keywords

cut cotorsion pairscut frobenius pairscut Auslander-Buchweitz contexts

MSC classification

Primary: 18G25: Relative homological algebra, projective classes
Secondary: 18G10: Resolutions; derived functors 18G20: Homological dimension 18G35: Chain complexes 16E65: Homological conditions on rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.)
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 64 , Issue 3 , September 2022 , pp. 548 - 585
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Auslander, M. and Buchweitz, R.-O., The homological theory of maximal Cohen-Macaulay approximations, Mém. Soc. Math. France (N.S.), 38 (1989), 5–37. Colloque en l’honneur de Pierre Samuel (Orsay, 1987).CrossRefGoogle Scholar
Bass, H., Finitistic dimension and a homological generalization of semi-primary rings, Trans. Amer. Math. Soc. 95 (1960), 466–488.CrossRefGoogle Scholar
Becerril, V., Mendoza, O., Pérez, M. A. and Santiago, V., Frobenius pairs in abelian categories: Correspondences with cotorsion pairs, exact model categories, and Auslander-Buchweitz contexts, J. Homotopy Relat. Struct. 14 (2019), 1–50.CrossRefGoogle Scholar
Becerril, V., Mendoza, O. and Santiago, V., Relative Gorenstein objects in abelian categories, Comm. Algebra 49(1) (2021), 352–402.CrossRefGoogle Scholar
Bravo, D., Gillespie, J. and Hovey, M., The stable module category of a general ring, Preprint. arXiv:1405.5768, 2014.Google Scholar
Bravo, D., Gillespie, J. and Pérez, M. A., Locally type $\text{FP}_n$ and n-coherent categories, Preprint. arXiv:1908.10987, 2019.Google Scholar
Bravo, D. and Pérez, M. A., Finiteness conditions and cotorsion pairs, J. Pure Appl. Algebra, 221(6) (2017), 1249–1267.CrossRefGoogle Scholar
Christensen, L. W., Estrada, S. and Iacob, A.. A Zariski-local notion of F-total acyclicity for complexes of sheaves, Quaest. Math. 40(2) (2017), 197–214.CrossRefGoogle Scholar
Christensen, L. W., Estrada, S. and Thompson, P., The stable category of Gorenstein flat sheaves on a Noetherian scheme, Proc. Am. Math. Soc. 149(2) (2021), 525–538.CrossRefGoogle Scholar
Cortés Izurdiaga, M., Estrada, S. and Guil Asensio, P. A., A model structure approach to the finitistic dimension conjectures, Math. Nachr. 285(7) (2012), 821–833.CrossRefGoogle Scholar
Efimov, A. I and Positselski, L., Coherent analogues of matrix factorizations and relative singularity categories, Algebra Number Theory 9(5) (2015), 1159–1292.CrossRefGoogle Scholar
Enochs, E. E. and Estrada, S., Relative homological algebra in the category of quasi-coherent sheaves. Adv. Math. 194(2) (2005), 284–295.CrossRefGoogle Scholar
Enochs, E. E., Estrada, S. and Odabaşi, S., Pure injective and absolutely pure sheaves, Proc. Edinb. Math. Soc. (2) 59(3) (2016), 623–640.CrossRefGoogle Scholar
Enochs, E. E. and Jenda, O. M. G., Relative Homological Algebra. Vol. 1. 2nd revised and extended ed., vol. 30 (Walter de Gruyter, Berlin, 2nd revised and extended ed. edition, 2011).Google Scholar
Enochs, E. E. and Jenda, O. M. G., Relative Homological Algebra. Vol. 2. 2nd revised ed., vol. 54, (Walter de Gruyter, Berlin, 2nd revised ed. edition, 2011).Google Scholar
Garca Rozas, J. R., Covers and Envelopes in the Category of Complexes of Modules, vol. 407. Chapman & Hall/CRC Research Notes in Mathematics, (Chapman & Hall/CRC, Boca Raton, FL, 1999).Google Scholar
Gillespie, J., The flat model structure on ${\text{Ch}}(R)$ , Trans. Amer. Math. Soc. 356(8) (2004), 3369–3390.Google Scholar
Gillespie, J., Kaplansky classes and derived categories, Math. Z. 257(4) (2007), 811–843.CrossRefGoogle Scholar
Gillespie, J., Model structures on modules over Ding-Chen rings. Homology Homotopy Appl. 12(1) (2010), 61–73.CrossRefGoogle Scholar
Gillespie, J., AC-Gorenstein rings and their stable module categories, J. Aust. Math. Soc. 107(2) (2019), 181–198.CrossRefGoogle Scholar
Göbel, R. and Trlifaj, J., Approximations and Endomorphism Algebras of Modules, vol. De Gruyter Expositions in Mathematics (Walter de Gruyter GmbH & Co. KG, Berlin, 2006).CrossRefGoogle Scholar
Görtz, U. and Wedhorn, T.. Algebraic Geometry I, (Advanced Lectures in Mathematics. Vieweg + Teubner, Wiesbaden, 2010). Schemes with examples and exercises.CrossRefGoogle Scholar
Green, E. L., Kirkman, E. and Kuzmanovich, J., Finitistic dimensions of finite-dimensional monomial algebras, J. Algebra 136(1) (1991), 37–50.CrossRefGoogle Scholar
Green, E. L. and Zimmermann-Huisgen, B.. Finitistic dimension of Artinian rings with vanishing radical cube. Math. Z. 206(4) (1991), 505–526.CrossRefGoogle Scholar
Hartshorne, R., Algebraic Geometry (Springer-Verlag, New York-Heidelberg, 1977). Graduate Texts in Mathematics, No. 52.Google Scholar
Hu, J. S., Zhang, D. D. and Zhou, P. Y., Proper resolutions and Gorensteinness in extriangulated categories, Front. Math. China 16 (2021).Google Scholar
Huerta, M., Mendoza, O. and Pérez, M. A., n-cotorsion pairs. J. Pure Appl. Algebra 225(5) (2021), 35. Id/No 106556.CrossRefGoogle Scholar
Igusa, K. and Todorov, G., On the finitistic global dimension conjecture for Artin algebras, in Representations of Algebras and Related Topics, vol. 45. Fields Inst. Commun., (Amer. Math. Soc., Providence, RI, 2005), 201–204.CrossRefGoogle Scholar
Iitaka, S., Algebraic Geometry, vol. 76. Graduate Texts in Mathematics, (Springer-Verlag, New York-Berlin, 1982). An introduction to birational geometry of algebraic varieties, North-Holland Mathematical Library, 24.CrossRefGoogle Scholar
Kong, F., Song, K. and Zhang, P., Decomposition of torsion pairs on module categories, J. Algebra 388 (2013), 248–267.CrossRefGoogle Scholar
Krause, H., The spectrum of a locally coherent category, J. Pure Appl. Algebra 114(3) (1997), 259–271.CrossRefGoogle Scholar
Mendoza, O. and Sáenz, C., Tilting categories with applications to stratifying systems, J. Algebra 302(1) (2006), 419–449.Google Scholar
Mendoza Hernández, O., Sáenz Valadez, E. C., Santiago Vargas, V. and Souto Salorio, M. J., Auslander-Buchweitz approximation theory for triangulated categories, Appl. Categ. Struct. 21(2) (2013), 119–139.CrossRefGoogle Scholar
Mendoza Hernández, O., Sáenz Valadez, E. C., Santiago Vargas, V. and Souto Salorio, M. J., Auslander-Buchweitz context and co-t-structures, Appl. Categ. Struct. 21(5) (2013), 417–440.Google Scholar
Murfet, D. and Salarian, S., Totally acyclic complexes over Noetherian schemes, Adv. Math. 226(2) (2011): 1096–1133.Google Scholar
Nakaoka, H., General heart construction on a triangulated category (I): Unifying t-structures and cluster tilting subcategories, Appl. Categ. Struct. 19(6) (2011), 879–899.Google Scholar
Nakaoka, H., General heart construction for twin torsion pairs on triangulated categories, J. Algebra 374 (2013), 195–215.CrossRefGoogle Scholar
Nakaoka, H. and Palu, Y., Extriangulated categories, Hovey twin cotorsion pairs and model structures, Cah. Topol. Géom. Différ. Catég. 60(2) (2019), 117–193.Google Scholar
Neeman, A., Triangulated Categories, vol. 148. Annals of Mathematics Studies (Princeton University Press, Princeton, NJ, 2001).Google Scholar
Ogawa, Y., Auslander’s defects over extriangulated categories: an application for the general heart construction, J. Math. Soc. Japan 73(4) (2021), 1063–1089.CrossRefGoogle Scholar
Popescu, N., Abelian Categories with Applications to Rings and Modules (Academic Press, London-New York, 1973). London Mathematical Society Monographs, No. 3.Google Scholar
Saorín, M. and Štovíček, J., On exact categories and applications to triangulated adjoints and model structures, Adv. Math. 228(2) (2011), 968–1007.CrossRefGoogle Scholar
Sieg, D., A Homological Approach to the Splitting Theory of PLS-spaces. PhD thesis, Universität Trier, Universitätsring 15, 54296 Trier, 2010.Google Scholar
Stenström, B., Rings of Quotients (Springer-Verlag, New York-Heidelberg, 1975). Die Grundlehren der Mathematischen Wissenschaften, Band 217, An introduction to methods of ring theory.Google Scholar
Šaroch, J. and Štovíček, J., Singular compactness and definability for $\Sigma$ -cotorsion and Gorenstein modules, Selecta Math. (N.S.) 26(2) (2020), Paper No. 23.Google Scholar
Xu, A., Gorenstein modules and Gorenstein model structures, Glasg. Math. J. 59(3) (2017), 685–703.CrossRefGoogle Scholar
Yang, G. and Liu, Z. K., Cotorsion pairs and model structures on ${\text{Ch}}(R)$ , Proc. Edinb. Math. Soc. (2) 54(3) (2011), 783–797.CrossRefGoogle Scholar
Zhang, P. and Xiong, B.-L., Separated monic representations II: Frobenius subcategories and RSS equivalences, Trans. Amer. Math. Soc. 372(2) (2019), 981–1021.CrossRefGoogle Scholar