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The Homotopy Category of Monomorphisms Between Projective Modules

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Abstract

Let \((S, {\mathfrak {n}})\) be a commutative noetherian local ring and \(\omega \in {\mathfrak {n}}\) be non-zerodivisor. This paper deals with the behavior of the category \({\textsf{Mon}}(\omega , \mathcal {P})\) consisting of all monomorphisms between finitely generated projective S-modules with cokernels annihilated by \(\omega \). We introduce a homotopy category \({\textsf{H}}{\textsf{Mon}}(\omega , \mathcal {P})\), which is shown to be triangulated. It is proved that this homotopy category embeds into the singularity category of the factor ring \(R=S/{(\omega )}\). As an application, not only the existence of almost split sequences ending at indecomposable non-projective objects of \({\textsf{Mon}}(\omega , \mathcal {P})\) is proved, but also the Auslander–Reiten translation, \(\tau _{{\textsf{Mon}}}(-)\), is completely recognized. Particularly, it will be observed that any non-projective object of \({\textsf{Mon}}(\omega , \mathcal {P})\) with local endomorphism ring is invariant under the square of the Auslander–Reiten translation.

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Acknowledgements

The authors are grateful to the referee for reading the paper very carefully and giving a lot of valuable suggestions kindly and patiently.

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Authors and Affiliations

  1. Department of Mathematics, Gonbad Kavous University, Gonbad Kavus, 4971799151, Iran

    Abdolnaser Bahlekeh

  2. School of Mathematics, Institute for Research in Fundamental Science (IPM), P.O. Box 19395-5746, Tehran, Iran

    Fahimeh Sadat Fotouhi & Shokrollah Salarian

  3. Department of Pure Mathematics, Faculty of Mathematics and Statistics, University of Isfahan, P.O. Box 81746-73441, Isfahan, Iran

    Armin Nateghi & Shokrollah Salarian

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  1. Abdolnaser Bahlekeh
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  2. Fahimeh Sadat Fotouhi
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  3. Armin Nateghi
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  4. Shokrollah Salarian
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Correspondence to Shokrollah Salarian.

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The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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Communicated by Rosihan M. Ali.

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This work is based upon research funded by Iran National Science Foundation (INSF) under project No. 4001480. The research of the second author was in part supported by a grant from IPM.

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Bahlekeh, A., Fotouhi, F.S., Nateghi, A. et al. The Homotopy Category of Monomorphisms Between Projective Modules. Bull. Malays. Math. Sci. Soc. 46, 91 (2023). https://doi.org/10.1007/s40840-023-01483-5

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  • DOI: https://doi.org/10.1007/s40840-023-01483-5

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