Injective Presentations of Induced Modules over Cluster-Tilted Algebras |
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Authors: | Email author" target="_blank">Ralf?SchifflerEmail author Khrystyna?Serhiyenko |
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Institution: | 1.Department of Mathematics,University of Connecticut,Storrs,USA;2.Department of Mathematics,University of California,Berkeley,USA |
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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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