Sincere wide τ-tilting modules

i

This article has been updated. A Correction to this article was published on .

i

The Original Article was published on

i

The Original Article was published on

i

This article has been updated

i

This article was retracted on

i

This article has been updated

Abstract

Enomoto and Sakai introduced wide τ-tilting modules, which are τ-tilting modules over functorially finite wide subcategories. They also proved that wide τ-tilting modules bijection with doubly functorially finite image-cokernel-extension-closed (ICE-closed) subcategories, which extended Adachi-Iyama-Reiten’s result. In this paper, we show that this bijection can be restricted to the support sets. As a consequence, we establish bijections between sincere wide τ-tilting modules, sincere ICE- closed subcategories, and sincere epibricks, and then we show that its number is related to the little Schröder number for Nakayama algebras.

Keywords

wide τ-tilting modules / ICE-closed subcategories / epibricks / Nakayama algebras

CHANGE HISTORY

A Correction to this paper has been published:
https://doi.org

The Retraction Note was updated to reflect reception of statements of disagreement from all authors.

This article has been retracted.Please see the Retraction Notice for more detail:
https://doi.org

References

[1] T. Adachi, O. Iyama, I. Reiten, τ-tilting theory, Compos. Math., 50 (2014), 415-452. https://doi.org/10.1112/S0010437X13007422

[2] H. Enomoto, A. Sakai, ICE-closed subcategories and wide τ-tilting modules, Math. Z., 300 (2022), 541-577. https://doi.org/10.1007/s00209-021-02796-6

[3] P. Gabriel, Des catégories abéliennes (French) , Bull. Soc. Math. France, 90 (1962), 323-448. https://doi.org/10.24033/bsmf.1583

[4] C. M. Ringel, Representations of K-species and bimodules, J. Algebra, 41 (1976), 269-302. https://doi.org/10.1016/0021-8693(76)90184-8

[5] S. Asai, Semibricks, Int. Math. Res. Not., 16 (2020), 4993-5054. https://doi.org/10.1093/imrn/rny150

[6] H. P. Gao, Semibricks over split-by-nilpotent extensions, Bull. Korean Math. Soc., 58 (2021), 183-193. https://doi.org/10.4134/BKMS.B200189

[7] H. P. Gao, A note on wide τ-tilting modules and epibricks, preprint.

[8] H. Enomoto, Monobrick, a uniform approach to torsion-free classes and wide subcategories, Adv. Math., 393 (2021). https://doi.org/10.1016/j.aim.2021.108113

[9] H. Enomoto, Rigid modules and ICE-closed subcategories in quiver representations, J. Algebra, 594 (2022), 364-388. https://doi.org/10.1016/j.jalgebra.2021.12.008

[10] I. Assem, D. Simson, A. Skowronski, Elements of the Representation Theory of Associative Algebras , London Mathematical Society Student Texts 65, Cambridge University Press, Cambridge, 2006. https://doi.org/10.1017/CBO9780511614309

[11] The On-Line Encyclopedia of Integer Sequences , Available from: http://oeis.org.