## Abstract

Every cluster-tilted algebra B is the relation extension C⋉ExtC2(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.

Original language | English |
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Pages (from-to) | 447-470 |

Number of pages | 24 |

Journal | Algebras and Representation Theory |

Volume | 21 |

Issue number | 2 |

DOIs | |

State | Published - Apr 1 2018 |

### Bibliographical note

Funding Information: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.

Publisher Copyright:

© 2017, Springer Science+Business Media B.V.

## Keywords

- Cluster-tilted algebra
- Coinduction
- Induction
- Relation extension

## ASJC Scopus subject areas

- Mathematics (all)