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Algebras and Representation Theory

, Volume 21, Issue 2, pp 447–470 | Cite as

Injective Presentations of Induced Modules over Cluster-Tilted Algebras

  • Ralf Schiffler
  • Khrystyna Serhiyenko
Article
  • 28 Downloads

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.

Keywords

Cluster-tilted algebra Induction Coinduction Relation extension 

Mathematics Subject Classification (2010)

16G20 16G70 13F60 

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Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ConnecticutStorrsUSA
  2. 2.Department of MathematicsUniversity of CaliforniaBerkeleyUSA

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