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Injective Presentations of Induced Modules over Cluster-Tilted Algebras

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Abstract

Every cluster-tilted algebra B is the relation extension \(C\ltimes \textup {Ext}^{2}_{C}(DC,C)\) of a tilted algebra C. A B-module is called induced if it is of the form M⊗ C B for some C-module M. We study the relation between the injective presentations of a C-module and the injective presentations of the induced B-module. Our main result is an explicit construction of the modules and morphisms in an injective presentation of any induced B-module. In the case where the C-module, and hence the B-module, is projective, our construction yields an injective resolution. In particular, it gives a module theoretic proof of the well-known 1-Gorenstein property of cluster-tilted algebras.

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Authors and Affiliations

  1. Department of Mathematics, University of Connecticut, Storrs, CT, 06269-3009, USA

    Ralf Schiffler

  2. Department of Mathematics, University of California, Berkeley, CA, 94720-3840, USA

    Khrystyna Serhiyenko

Authors
  1. Ralf Schiffler
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  2. Khrystyna Serhiyenko
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Corresponding author

Correspondence to Ralf Schiffler.

Additional information

Presented by Henning Krause.

The authors were supported by the NSF CAREER grant DMS-1254567 and by the University of Connecticut. The second author was also supported by the NSF Postdoctoral fellowship MSPRF-1502881.

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Schiffler, R., Serhiyenko, K. Injective Presentations of Induced Modules over Cluster-Tilted Algebras. Algebr Represent Theor 21, 447–470 (2018). https://doi.org/10.1007/s10468-017-9721-0

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  • DOI: https://doi.org/10.1007/s10468-017-9721-0

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