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Intervals of s-torsion pairs in extriangulated categories with negative first extensions

Part of: Abelian categories

Published online by Cambridge University Press:  05 September 2022

TAKAHIDE ADACHI ,
HARUHISA ENOMOTO  and
MAYU TSUKAMOTO
TAKAHIDE ADACHI
Affiliation:
Faculty of Global and Science Studies, Yamaguchi University, 1677-1 Yoshida, Yamaguchi 753-8541, Japan. e-mail: tadachi@yamaguchi-u.ac.jp
HARUHISA ENOMOTO
Affiliation:
Graduate School of Science, Osaka Metropolitan University, 1-1 Gakuen-cho, Naka-ku, Sakai, Osaka 599-8531, Japan. e-mail: henomoto@omu.ac.jp
MAYU TSUKAMOTO
Affiliation:
Graduate school of Sciences and Technology for Innovation, Yamaguchi University, 1677-1 Yoshida, Yamaguchi 753-8512, Japan. e-mail: tsukamot@yamaguchi-u.ac.jp
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Abstract

As a general framework for the studies of t-structures on triangulated categories and torsion pairs in abelian categories, we introduce the notions of extriangulated categories with negative first extensions and s-torsion pairs. We define a heart of an interval in the poset of s-torsion pairs, which naturally becomes an extriangulated category with a negative first extension. This notion generalises hearts of t-structures on triangulated categories and hearts of twin torsion pairs in abelian categories. In this paper, we show that an interval in the poset of s-torsion pairs is bijectively associated with s-torsion pairs in the corresponding heart. This bijection unifies two well-known bijections: one is the bijection induced by the HRS-tilt of t-structures on triangulated categories. The other is Asai–Pfeifer’s and Tattar’s bijections for torsion pairs in an abelian category, which is related to $\tau$ -tilting reduction and brick labelling.

MSC classification

Primary: 18E40: Torsion theories, radicals
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 174 , Issue 3 , May 2023 , pp. 451 - 469
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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References

Adachi, T., Iyama, O. and Reiten, I.. $\tau$ -tilting theory. Compositio Math. 150 (2014), no. 3, 415–452.CrossRefGoogle Scholar
Asai, S. and Pfeifer, C.. Wide subcategories and lattices of torsion classes. to appear in Algebr. Represent. Theory, arXiv:1905.01148.Google Scholar
Barnard, E., Carroll, A. and Zhu, S.. Minimal inclusions of torsion classes. Algebr. Comb. 2 (2019), no. 5, 879–901.Google Scholar
Belinson, A. A., Bernstein, J. and Deligne, P.. Faisceaux pervers. Analysis and topology on singular spaces, I (Luminy, 1981) Astérisque, 100 (Soc. Math. France, Paris, 1982), 5–171.Google Scholar
Beligiannis, A. and Reiten, I.. Homological and homotopical aspects of torsion theories. Mem. Amer. Math. Soc. 188 (2007), no. 883.Google Scholar
Demonet, L., Iyama, O., Reading, N., Reiten, I. and Thomas, H.. Lattice theory of torsion classes. arXiv:1711.01785v2.Google Scholar
Dyer, M.. Exact subcategories of triangulated categories. http://www.nd.edu/dyer/papers/index.html, preprint.Google Scholar
Enomoto, H. and Sakai, A.. ICE-closed subcategories and wide $\tau$ -tilting modules. Math. Z. 300 (2022), no. 1, 541–577.Google Scholar
Gorsky, M.. Semi-derived Hall algebras and tilting invariance of Bridgeland--Hall algebras. arXiv:1303.5879.Google Scholar
Happel, D., Reiten, I. and Smalø, S. O.. Tilting in abelian categories and quasitilted algebras. Mem. Amer. Math. Soc. 120 (1996), no. 575.Google Scholar
He, J.. Extensions of covariantly finite subcategories revisited. Czechoslovak Math. J. 69(144) (2019), no. 2, 403–415 CrossRefGoogle Scholar
Iyama, O., Reiten, I., Thomas, H. and Todorov, G.. Lattice structure of torsion classes for path algebras. Bull. London Math. Soc. 47 (2015), no. 4, 639–650.CrossRefGoogle Scholar
Jasso, G.. Reduction of $\tau$ -tilting modules and torsion pairs. Int. Math. Res. Not. IMRN 2015, no. 16, 7190–7237.Google Scholar
Jørgensen, P.. Abelian subcategories of triangulated categories induced by simple minded systems. Math. Z. 301 (2022), no. 1, 565–592.Google Scholar
Nakaoka, H. and Palu, Y.. Extriangulated categories, Hovey twin cotorsion pairs and model structures. Cah. Topol. Géom. Différ. Catég. 60 (2019), no. 2, 117–193.Google Scholar
Polishchuk, A.. Constant families of t-structures on derived categories of coherent sheaves. Mosc. Math. J. 7 (2007), no. 1, 109–134.Google Scholar
Saito, S.. Tilting objects in periodic triangulated categories. arXiv:2011.14096.Google Scholar
Tattar, A.. Torsion pairs and quasi-abelian categories. Algebr. Represent. Theory 24 (2021), no. 6, 1557–1581.CrossRefGoogle Scholar
Woolf, J.. Stability conditions, torsion theories and tilting. J. London Math. Soc. (2) 82 (2010), no. 3, 663–682.Google Scholar
Zhao, X.. A note on the equivalence of m-periodic derived categories. Sci. China Math. 57 (2014), no. 11, 2329–2334.CrossRefGoogle Scholar