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On a Generalized Auslander-Reiten Conjecture

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It is well-known that the generalized Auslander-Reiten condition (GARC) and the symmetric Auslander condition (SAC) are equivalent, and (GARC) implies that the Auslander-Reiten condition (ARC). In this paper we explore (SAC) along with the several canonical change of rings \(R \rightarrow S\). First, we prove the equivalence of (SAC) for R and R/xR, where x is a non-zerodivisor on R, and the equivalence of (SAC) and (SACC) for rings with positive depth, where (SACC) is the symmetric Auslander condition for modules with constant rank. The latter assertion affirmatively answers a question posed by Celikbas and Takahashi. Secondly, for a ring homomorphism \(R \rightarrow S\), we prove that if S satisfies (SAC) (resp. (ARC)), then R also satisfies (SAC) (resp. (ARC)) if the flat dimension of S over R is finite. We also prove that (SAC) holds for R implies that (SAC) holds for S when R is Gorenstein and \(S=R/Q^\ell \), where Q is generated by a regular sequence of R and the length of the sequence is at least \(\ell \). This is a consequence of more general results about Ulrich ideals proved in this paper. Applying these results to determinantal rings and numerical semigroup rings, we provide new classes of rings satisfying (SAC). A relation between (SAC) and an invariant related to the finitistic extension degree is also explored.

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Acknowledgements

The authors are grateful to Olgur Celikbas. He introduced the paper [6] to the authors and gave helpful comments. The authors also thank the anonymous referee for reading this paper carefully and giving useful comments.

Funding

Souvik Dey was partly supported by the Charles University Research Center program No. UNCE/24/SCI/022 and a grant GACR 23-05148S from the Czech Science Foundation. Shinya Kumashiro was supported by JSPS KAKENHI Grant Number JP21K13766. Parangama Sarkar was supported by SERB POWER Grant with Grant No. SPG/2021/002423.

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Authors and Affiliations

  1. Faculty of Mathematics and Physics, Department of Algebra, Charles University, Sokolovská 83, 186 75, Praha, Czech Republic

    Souvik Dey

  2. Department of Mathematics, Osaka Institute of Technology, 5-16-1 Omiya, asahi-ku, 535-8585, Osaka, Japan

    Shinya Kumashiro

  3. Department of Mathematics, Indian Institute of Technology, Palakkad, India

    Parangama Sarkar

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  1. Souvik Dey
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  2. Shinya Kumashiro
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  3. Parangama Sarkar
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All authors contributed equally to the paper.

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Correspondence to Souvik Dey.

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Presented by: Christof Geiß

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The first author was supported by Charles University Research Center program No. UNCE/24/SCI/022 and a grant GACR 23-05148S from the Czech Science Foundation. The second author was supported by JSPS KAKENHI Grant Number JP21K13766. The third author was supported by SERB POWER Grant with Grant No. SPG/2021/002423.

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Dey, S., Kumashiro, S. & Sarkar, P. On a Generalized Auslander-Reiten Conjecture. Algebr Represent Theor 27, 1581–1602 (2024). https://doi.org/10.1007/s10468-024-10271-z

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