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ICE-closed subcategories and wide \(\tau \)-tilting modules

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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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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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Correspondence to Haruhisa Enomoto.

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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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