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.
|Number of pages||24|
|Journal||Algebras and Representation Theory|
|State||Published - Apr 1 2018|
Bibliographical noteFunding 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.
© 2017, Springer Science+Business Media B.V.
- Cluster-tilted algebra
- Relation extension
ASJC Scopus subject areas
- Mathematics (all)