Injective Presentations of Induced Modules over Cluster-Tilted Algebras

Ralf Schiffler; Khrystyna Serhiyenko

doi:10.1007/s10468-017-9721-0

Injective Presentations of Induced Modules over Cluster-Tilted Algebras

Ralf Schiffler, Khrystyna Serhiyenko

Mathematics

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

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⊗_CB 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
Pages (from-to)	447-470
Number of pages	24
Journal	Algebras and Representation Theory
Volume	21
Issue number	2
DOIs	https://doi.org/10.1007/s10468-017-9721-0
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)

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10.1007/s10468-017-9721-0

Cite this

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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⊗CB 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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AU - Serhiyenko, Khrystyna

N1 - 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.

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N2 - 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⊗CB 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.

AB - 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⊗CB 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.

KW - Cluster-tilted algebra

KW - Coinduction

KW - Induction

KW - Relation extension

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Injective Presentations of Induced Modules over Cluster-Tilted Algebras

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Keywords

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