×

One-sided Frobenius pairs in extriangulated categories. (English) Zbl 1502.18028

The authors introduce the notions of left Frobenius pairs and left (weak) Auslander-Buchweitz context with respect to a class proper of \(\mathbb E\)-triangles \(\xi\) in a extriangulated categories. These categories were introduced by H. Nakaoka and Y. Palu [Cah. Topol. Géom. Différ. Catég. 60, No. 2, 117–193 (2019; Zbl 1451.18021)] as a common generalization of exact and triangulated categories.
In Section 3, several properties of the left (resp. right ) \(n\)-cotorsion pairs and cotorsion pairs defined by J. He and P. Zhou [J. Algebra Appl. 21, No. 1, Article ID 2250011, 12 p. (2022; Zbl 1479.18012)] are stablished. In the last Section, a one-to-one correspondence between left Frobenius pairs, (i.e. a pair of subcategories \((\mathcal X,\omega)\) satisfying that i) \(\mathcal X\) and \(\omega\) are closed under direct summands, ii) \(\mathcal X\) is closed under \(\xi\)-extensions and cocones of \(\xi\)-deflations and iii) \(\omega\) is \(\mathcal X\)-injective) and a \(\xi\)-cogenerator of \(\mathcal X\), and left (weak) Auslander-Buchweitz contexts (i.e. two subcagories \(\mathcal A\) and \(\mathcal B\) of \(\mathcal C\) verifying that i) \((\mathcal{A}, \omega =\mathcal{A}\cap\mathcal{B})\) is a left Frobenius pair, ii) \(\mathcal B\) is closed under direct summands, \(\xi\)-extensions and cones of \(\xi\)-deflations and iii) \(\mathcal{B}\subseteq \mathcal{A}^{\wedge }\)= {objects with finite \(\mathcal{A}\)-resolutions }) is obtained.
Left Frobenius pairs for abelian categories were studied by V. Becerril et al. [J. Homotopy Relat. Struct. 14, No. 1, 1–50 (2019; Zbl 1435.18011)] and left (weak) Auslander-Buchweitz contexts for abelian categories by M. Hashimoto [Auslander-Buchweitz approximations of equivariant modules. Cambridge: Cambridge University Press (2000; Zbl 0993.13007)] and for triangulated categories by X. Ma et al. [“Left Frobenius pairs, cotorsion pairs and weak Auslander-Buchweitz contexts in triangulated categories”, Algebra Colloq (to appear)].

MSC:

18G20 Homological dimension (category-theoretic aspects)
18G25 Relative homological algebra, projective classes (category-theoretic aspects)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Adachi, T.; Tsukamoto, M., Hereditary cotorsion pairs and silting subcategories in extriangulated categories, J. Algebra, 594, 109-137 (2022) · Zbl 1485.18020
[2] Angeleri-Hügel, L.; Mendoza, O., Homological dimensions in cotorsion pairs, Illinois J. Math, 53, 251-263 (2009) · Zbl 1205.16005
[3] Auslander, M.; Buchweitz, R. O., The homological theory of maximal Cohen-Macaulay approximations, Moires Soc. Math. France, 1, 5-37 (1989) · Zbl 0697.13005
[4] Becerril, V.; Mendoza, O.; Pérez, M. A.; Santiago, V., Frobenius pairs in abelian categories, J. Homotopy Relat. Struct, 14, 1, 1-50 (2019) · Zbl 1435.18011
[5] Bühler, T., Exact categories, Expo. Math, 28, 1, 1-69 (2010) · Zbl 1192.18007
[6] Crivei, S.; Torrecillas, B., On some monic covers and epic envelopes, Arab. J. Sci. Eng, 33, 123-135 (2008) · Zbl 1185.18012
[7] Eklof, P.; Trlifaj, J., How to make \(####\) vanish, Bull. London Math. Soc, 33, 41-51 (2001) · Zbl 1030.16004
[8] Enochs, E.; Jenda, O. M. G., Relative Homological Algebra. Expositions in Math. 30 (2000), Berlin, Germany: De Gruyter, Berlin, Germany · Zbl 0952.13001
[9] Göbel, R.; Trlifaj, J., Approximations and Endomorphism Algebras of Modules. Expositions in Math. 41 (2006), Berlin, Germany: De Gruyter, Berlin, Germany · Zbl 1121.16002
[10] Hashimoto, M., Auslander-Buchweitz Approximations of Equivariant Modules. London Math. Soc. Lecture Note Ser. 282 (2000), Cambridge, UK: Cambridge University Press, Cambridge, UK · Zbl 0993.13007
[11] He, J.; Zhou, P., On the relation between n-cotorsion pairs and \(####\)-cluster tilting subcategories, J. Algebra Appl, 21, 1, 2250011 (2022) · Zbl 1479.18012
[12] Hu, J.; Zhang, D.; Zhou, P., Proper classes and Gorensteinness in extriangulated categories, J. Algebra, 551, 23-60 (2020) · Zbl 1433.18004
[13] Hu, J.; Zhang, D.; Zhou, P., Gorenstein homological dimensions for extriangulated categories, Bull. Malays. Math. Sci. Soc, 44, 4, 2235-2252 (2021) · Zbl 1477.18029
[14] Hu, J.; Zhang, D.; Zhao, T.; Zhou, P., Complete cohomology for extriangulated categories, Algebra Colloq, 28, 4, 701-720 (2021) · Zbl 1477.18019
[15] Huerta, M.; Mendoza, O.; Pérez, M. A., n-cotorsion pairs, J. Pure Appl. Algebra, 225, 5, 106556 (2021) · Zbl 1470.18018
[16] Liu, Y.; Nakaoka, H., Hearts of twin cotorsion pairs on extriangulated categories, J. Algebra, 528, 96-149 (2019) · Zbl 1419.18018
[17] Ma, X.; Zhao, T., Resolving resolution dimensions in triangulated categories, Open Math, 19, 1, 121-143 (2021) · Zbl 1484.18012
[18] Ma, X.; Zhao, T.; Huang, Z., Left Frobenius pairs, cotorsion pairs and weak Auslander-Buchweitz contexts in triangulated categories, Algebra Colloq (2021)
[19] Ma, Y., Ding, N., Zhang, Y. (2020). Auslander-Buchweitz approximation theory for extriangulated categories. arXiv: 2006.05112v2. DOI: .
[20] Nakaoka, H., Ogawa, Y., Sakai, A. (2021). Localization of extriangulated categories. arXiv: 2103.16907v2. DOI: .
[21] Nakaoka, H.; Palu, Y., Extriangulated categories, Hovey twin cotorsion pairs and model structures, Cah. Topol. Géom. Différ. Catég, 60, 2, 117-193 (2019) · Zbl 1451.18021
[22] Salce, L., Cotorsion Theories for Abelian Groups. Symposia Mathematica, Vol. XXIII, 11-32 (1979), London, UK: Academic Press, London, UK · Zbl 0426.20044
[23] Vaso, L., Gluing of n-cluster tilting subcategories for representation-directed algebras, Algebr Represent Theor, 24, 3, 715-781 (2021) · Zbl 1486.16013
[24] Zhou, P.; Zhu, B., Triangulated quotient categories revisited, J. Algebra, 502, 196-232 (2018) · Zbl 1388.18014
[25] Zhu, B.; Zhuang, X., Grothendieck groups in extriangulated categories, J. Algebra, 574, 206-232 (2021) · Zbl 1462.18003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.