Abstract
In this paper, we study ICE-closed (= Image-Cokernel-Extension-closed) subcategories of an abelian length category using torsion classes. To each interval \([\mathcal {U},\mathcal {T}]\) in the lattice of torsion classes, we associate a subcategory \(\mathcal {T} \cap \mathcal {U}^\perp \) called the heart. We show that every ICE-closed subcategory can be realized as a heart of some interval of torsion classes, and give a lattice-theoretic characterization of intervals whose hearts are ICE-closed. In particular, we prove that ICE-closed subcategories are precisely torsion classes in some wide subcategories. For an artin algebra, we introduce the notion of wide \(\tau \)-tilting modules as a generalization of support \(\tau \)-tilting modules. Then we establish a bijection between wide \(\tau \)-tilting modules and doubly functorially finite ICE-closed subcategories, which extends Adachi–Iyama–Reiten’s bijection on torsion classes. For the hereditary case, we discuss the Hasse quiver of the poset of ICE-closed subcategories by introducing a mutation of rigid modules.
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References
Adachi, T., Iyama, O., Reiten, I.: \(\tau \)-tilting theory. Compos. Math. 150(3), 415–452 (2014)
Asai, S.: Semibricks. Int. Math. Res. Not. 16, 4993–5054 (2020)
Asai, S., Pfeifer, C.: Wide subcategories and lattices of torsion classes. arXiv:1905.01148
Assem, I., Simson, D., Skowroński, A.: Elements of the Representation Theory of Associative Algebras. Vol. 1. Techniques of Representation Theory. London Mathematical Society Student Texts, vol. 65. Cambridge University Press, Cambridge (2006)
Auslander, M., Reiten, I., Smalø, S.O.: Representation Theory of Artin Algebras. Cambridge Studies in Advanced Mathematics, vol. 36. Cambridge University Press, Cambridge (1995)
Auslander, M., Smalø, S.O.: Preprojective modules over Artin algebras. J. Algebra 66, 66–122 (1980)
Auslander, M., Smalø, S.O.: Almost split sequences in subcategories. J. Algebra 69(2), 426–454 (1981)
Baumann, P., Kamnitzer, J., Tingley, P.: Affine Mirković–Vilonen polytopes. Publ. Math. Inst. Hautes Études Sci. 120, 113–205 (2014)
Brenner, S., Butler, M.C.R.: Generalizations of the Bernstein–Gelfand–Ponomarev reflection functors. In: Representation Theory, II (Proceedings of the Second International Conference, Carleton University, Ottawa, Ontario, 1979). Lecture Notes in Mathematics, vol. 832, pp. 103–169. Springer, Berlin (1980)
Bridgeland, T.: Scattering diagrams, Hall algebras and stability conditions. Algebraic Geom. 4(5), 523–561 (2017)
Demonet, L., Iyama, O., Jasso, G.: \(\tau \)-tilting finite algebras, bricks, and g-vectors. Int. Math. Res. Not. 3, 852–892 (2019)
Demonet, L., Iyama, O., Reading, N., Reiten, I., Thomas, H.: Lattice theory of torsion classes. arXiv:1711.01785
Dickson, S.E.: A torsion theory for Abelian categories. Trans. Am. Math. Soc. 121, 223–235 (1966)
Enomoto, H.: Bruhat inversions in Weyl groups and torsion-free classes over preprojective algebras. Commun. Algebra 49(5), 2156–2189 (2021)
Enomoto, H.: Monobrick, a uniform approach to torsion-free classes and wide subcategories. arXiv:2005.01626
Enomoto, H.: Rigid modules and ICE-closed subcategories in quiver representations. arXiv:2005.05536
Gentle, R., Todorov, G.: Extensions, kernels and cokernels of homologically finite subcategories. In: Representation Theory of Algebras (Cocoyoc, 1994). CMS Conference Proceedings, vol. 18, pp. 227–235. American Mathematical Society, Providence (1996)
Happel, D., Reiten, I., Smalø, S.O.: Tilting in abelian categories and quasitilted algebras. Mem. Am. Math. Soc. 120(575) (1996)
Hovey, M.: Classifying subcategories of modules. Trans. Am. Math. Soc. 353(8), 3181–3191 (2001)
Ingalls, C., Thomas, H.: Noncrossing partitions and representations of quivers. Compos. Math. 145(6), 1533–1562 (2009)
Jasso, G.: Reduction of \(\tau \)-tilting modules and torsion pairs. Int. Math. Res. Not. 16, 7190–7237 (2015)
Kalck, M.: A remark on Leclerc’s Frobenius categories. In: Homological Bonds Between Commutative Algebra and Representation Theory. Research Perspectives CRM Barcelona
Marks, F., Št’ovíček, J.: Torsion classes, wide subcategories and localisations. Bull. Lond. Math. Soc. 49(3), 405–416 (2017)
Ringel, C.M.: Representations of \(K\)-species and bimodules. J. Algebra 41(2), 269–302 (1976)
Sikko, S.A., Smalø, S.O.: Extensions of homologically finite subcategories. Arch. Math. (Basel) 60(6), 517–526 (1993)
Smalø, S.O.: Torsion theories and tilting modules. Bull. Lond. Math. Soc. 16(5), 518–522 (1984)
Tattar, A.: Torsion pairs and quasi-abelian categories. Algebras Represent. Theory arXiv:1907.10025(to appear)
Acknowledgements
The authors would like to thank their supervisor Hiroyuki Nakaoka for his encouragement and useful comments. They would also like to thank Osamu Iyama for helpful discussions. H. Enomoto is supported by JSPS KAKENHI Grant Number JP21J00299.
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Enomoto, H., Sakai, A. ICE-closed subcategories and wide \(\tau \)-tilting modules. Math. Z. 300, 541–577 (2022). https://doi.org/10.1007/s00209-021-02796-6
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DOI: https://doi.org/10.1007/s00209-021-02796-6