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On a Deformation Theory of Finite Dimensional Modules Over Repetitive Algebras

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Abstract

Let Λ be a basic finite dimensional algebra over an algebraically closed field \(\Bbbk \), and let \(\widehat {\Lambda }\) be the repetitive algebra of Λ. In this article, we prove that if \(\widehat {V}\) is a left \(\widehat {\Lambda }\)-module with finite dimension over \(\Bbbk \), then \(\widehat {V}\) has a well-defined versal deformation ring \(R(\widehat {\Lambda },\widehat {V})\), which is a local complete Noetherian commutative \(\Bbbk \)-algebra whose residue field is also isomorphic to \(\Bbbk \). We also prove that \(R(\widehat {\Lambda }, \widehat {V})\) is universal provided that \(\underline {\text {End}}_{\widehat {\Lambda }}(\widehat {V})=\Bbbk \) and that in this situation, \(R(\widehat {\Lambda }, \widehat {V})\) is stable after taking syzygies. We apply the obtained results to finite dimensional modules over the repetitive algebra of the 2-Kronecker algebra, which provides an alternative approach to the deformation theory of objects in the bounded derived category of coherent sheaves over \(\mathbb {P}^{1}_{\Bbbk }\).

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Acknowledgements

The fourth author would like to express his gratitude to the other authors, faculty members, staff and students at the Instituto of Matemáticas as well as to the other people related to this work at the Universidad de Antioquia for their hospitality and support during the developing of this project. All the authors are grateful with the anonymous referee who provided many suggestions and corrections that improved the quality and the readability of this article, and who also pointed out major errors in a previous version of the statement and proof of Theorem 1.2.

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Authors and Affiliations

  1. Instituto de Matemáticas, Universidad de Antioquia, Medellín, Antioquia, Colombia

    Adriana Fonce-Camacho, Hernán Giraldo & Pedro Rizzo

  2. Department of Mathematics, Valdosta State University, Valdosta, GA, United States of America

    José A. Vélez-Marulanda

Authors
  1. Adriana Fonce-Camacho
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  2. Hernán Giraldo
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  3. Pedro Rizzo
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  4. José A. Vélez-Marulanda
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Correspondence to José A. Vélez-Marulanda.

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Presented by: Henning Krause

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This research was partially supported by CODI (Universidad de Antioquia, UdeA), COLCIENCIAS (Convocatoria Doctorados Nacionales 2016, Número 757), and the Office of Academic Affairs at the Valdosta State University.

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Fonce-Camacho, A., Giraldo, H., Rizzo, P. et al. On a Deformation Theory of Finite Dimensional Modules Over Repetitive Algebras. Algebr Represent Theor 26, 1–22 (2023). https://doi.org/10.1007/s10468-021-10083-5

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