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On triangle equivalences of stable categories

Part of: Commutative algebra: Homological methods Homological methods Abelian categories Homological algebra

Published online by Cambridge University Press:  26 January 2019

Zhenxing Di ,
Zhongkui Liu  and
Jiaqun Wei
Zhenxing Di
Affiliation:
Department of Mathematics, Northwest Normal University, Lanzhou730070, People's Republic of China ([email protected]; [email protected])
Zhongkui Liu
Affiliation:
Department of Mathematics, Northwest Normal University, Lanzhou730070, People's Republic of China ([email protected]; [email protected])
Jiaqun Wei*
Affiliation:
Department of Mathematics, Northwest Normal University, Lanzhou730070, People's Republic of China and Institute of Mathematics, School of Mathematics Sciences, Nanjing Normal University, Nanjing210023, People's Republic of China ([email protected])
*
*Corresponding author.
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Abstract

We apply the Auslander–Buchweitz approximation theory to show that the Iyama and Yoshino's subfactor triangulated category can be realized as a triangulated quotient. Applications of this realization go in three directions. Firstly, we recover both a result of Iyama and Yang and a result of the third author. Secondly, we extend the classical Buchweitz's triangle equivalence from Iwanaga–Gorenstein rings to Noetherian rings. Finally, we obtain the converse of Buchweitz's triangle equivalence and a result of Beligiannis, and give characterizations for Iwanaga–Gorenstein rings and Gorenstein algebras.

Keywords

Presilting subcategorysubfactor triangulated categoryprojective Frobenius subcategoryGorenstein projective module

MSC classification

Primary: 13D05: Homological dimension
Secondary: 16E35: Derived categories 18G25: Relative homological algebra, projective classes 18E30: Derived categories, triangulated categories
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 2 , April 2020 , pp. 955 - 974
Copyright
Copyright © Royal Society of Edinburgh 2019

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Footnotes

In Memory of Professor Ragnar-Olaf Buchweitz

References

1Aihara, T. and Iyama, O.. Silting mutation in triangulated categories. J. Lond. Math. Soc. 85 (2012), 633–668.CrossRefGoogle Scholar
2Auslander, M. and Buchweitz, R.-O.. The Homological theory of maximal Cohen-Macaulay approximations. Mm. Soc. Math. France 38 (1989), 5–37.CrossRefGoogle Scholar
3Auslander, M. and Reiten, I.. Applications of contravariantly finite subcategories. Adv. Math. 86 (1991), 111–152.CrossRefGoogle Scholar
4Auslander, M. and Smalø, S.O.. Preprojective modules over Artin algebras. J. Algebra 66 (1980), 61–122.CrossRefGoogle Scholar
5Beligiannis, A.. The homological theory of contravariantly finite subcategories: Auslander-Buchweitz contexts, Gorenstein categories and (co)stabilization. Comm. Algebra 28 (2000), 4547–4596.CrossRefGoogle Scholar
6Bergh, P. A., Jørgensen, D. A. and Oppermann, S.. The Gorenstein defect category. Q. J. Math. 66 (2015), 459–471.CrossRefGoogle Scholar
7Bondarko, M. V.. Weight structures vs. t-structures; weight filtrations, spectral sequences, and complexes (for motives and in general). J. K-Theory 6 (2010), 387–504.CrossRefGoogle Scholar
8Buchweitz, R.-O.. Maximal Cohen-Macaulay modules and Tate-cohomology over Gorenstein rings. unpublished manuscript, http://hdl.handle.net/1807/16682, 1986.Google Scholar
9Chen, X. W.. Relative singularity categories and Gorenstein projective modules. Math. Nachr. 284 (2011), 199–212.CrossRefGoogle Scholar
10Christensen, L. W.. Gorenstein dimensions. Lecture Notes in Math (Berlin: Springer-Verlag, 2000).CrossRefGoogle Scholar
11Christensen, L. W., Frankild, A. and Holm, H.. On Gorenstein projective, injective and flat dimensions – A functorial description with applications. J. Algebra 302 (2006), 231–279.CrossRefGoogle Scholar
12Emmanouil, I.. On the finiteness of Gorenstein homological dimensions. J. Algebra 372 (2012), 376–396.CrossRefGoogle Scholar
13Enochs, E. E. and Jenda, O. M. G.. Relative homological algebra, vol. 2, de Gruyter Expositions in Mathematics, vol. 54 (New York: Walter de Gruyter, 2000).CrossRefGoogle Scholar
14Happel, D.. On gorenstein algebras. Progress in Mathematics, vol. 95, pp. 389–404 (Basel: Birkh auser Verlag, 1991).Google Scholar
15Happel, D.. Triangulated categories in the representation theory of finite-dimensional algebras. London Mathematical Society Lecture Note Series, vol. 119 (Cambridge: Cambridge University Press, 1988).CrossRefGoogle Scholar
16Holm, H.. Gorenstein homological dimensions. J. Pure Appl. Algebra 189 (2004), 167–193.CrossRefGoogle Scholar
17Iyama, O. and Yang, D.. Silting reduction and Calabi-Yau reduction of triangulated categories. Trans. Amer. Math. Soc. 370 (2018), 7861–7898.CrossRefGoogle Scholar
18Iyama, O. and Yoshino, Y.. Mutations in triangulated categories and rigid Cohen-Macaulay modules. Invent. Math. 172 (2008), 117–168.CrossRefGoogle Scholar
19Krause, H.. The stable derived category of a Noetherian scheme. Compos. Math. 141 (2005), 1128–1162.CrossRefGoogle Scholar
20Mendoza Hernández, O., Sáenz Valadez, E., Santiago Vargas, V. and Souto Salorio, M.. Auslander-Buchweitz approximation theory for triangulated categories. Appl. Categ. Structures 21 (2013a), 119–139.CrossRefGoogle Scholar
21Mendoza Hernández, O., Sáenz Valadez, E., Santiago Vargas, V. and Souto Salorio, M.. Auslander-Buchweitz context and co-t-structures. Appl. Categ. Structures 21 (2013b), 417–440.CrossRefGoogle Scholar
22Nicolás, P., Saorín, M. and Zvonareva, A.. Silting theory in triangulated categories with coproducts. J. Pure Appl. Algebra, In Press (2018), https://doi.org/10.1016/j.jpaa.2018.07.016.CrossRefGoogle Scholar
23Oppermann, S.. Quivers for silting mutation. Adv. Math. 307 (2017), 684–714.CrossRefGoogle Scholar
24Oppermann, S., Psaroudakis, C. and Stai, T.. Change of rings and singularity categories. (2018), ArXiv: 1801.07995.Google Scholar
25Orlov, D.. Triangulated categories of singularities and D-branes in Landau-Ginzburg models. Proc. Steklov Inst. Math. 246 (2004), 227–248.Google Scholar
26Pauksztello, D.. Compact corigid objects in triangulated categories and co-t-structures. Cent. Eur. J. Math. 6 (2008), 25–42.CrossRefGoogle Scholar
27Rickard, J.. Derived categories and stable equivalence. J. Pure Appl. Algebra 61 (1989), 303–317.CrossRefGoogle Scholar
28Veliche, O.. Gorenstein projective dimension for complexes. Trans. Amer. Math. Soc. 358 (2006), 1257–1283.CrossRefGoogle Scholar
29Wei, J.. Semi-tilting complexes. Israel J. Math. 194 (2013), 871–893.CrossRefGoogle Scholar
30Wei, J.. Relative singularity categories, Gorenstein objects and silting theory. J. Pure Appl. Algebra 222 (2018), 2310–2322.CrossRefGoogle Scholar