Induced and coinduced modules over cluster-tilted algebras

Abstract

We propose a new approach to study the relation between the module categories of a tilted algebra C and the corresponding cluster-tilted algebra B = C E. This new approach consists of using the induction functor − ⊗ C B as well as the coinduction functor D(B ⊗ C D−). We give an explicit construction of injective resolutions of projective B-modules, and as a consequence, we obtain a new proof of the 1-Gorenstein property for cluster-tilted algebras. We show that DE is a partial tilting and a τ -rigid C-module and that the induced module DE ⊗ C B is a partial tilting and a τ -rigid B-module. Furthermore, if C = End A T for a tilting module T over a hereditary algebra A, we compare the induction and coinduction functors to the Buan-Marsh-Reiten functor Hom C A (T, −) from the cluster-category of A to the module category of B. We also study the question which B-modules are actually induced or coinduced from a module over a tilted algebra.

τ-Tilting theory and τ-slices

“…They proved that every partial relation extension of a tilted algebra C has a local slice (see [4,Theorem 3]). As a consequence of the previous theorem and the explicit computation of the C-module structure E and D(E), made by Schiffler and Serhiyenko in [22], we get the following corollary, implying [2,Theorem 19] and [4,Theorem 3].…”

Cluster-tilted and quasi-tilted algebras

“…where the first isomorphism is [ScSe,Proposition 3.3] and the second is [ScSe,Proposition 4.1]. Since T has no preinjective summands, and X is regular, the only summand of τ ΩDC for which Hom(X, τ ΩDC) can be nonzero, must lie in the same tube as X.…”

“…Cluster-tilted algebras were introduced by Buan, Marsh and Reiten [BMR] and, independently in [CCS] for type A as a byproduct of the now extensive theory of cluster algebras of Fomin and Zelevinsky [FZ]. Since then, cluster-tilted algebras have been the subject of several investigations, see, for instance, [ABCP,ABS,BFPPT,BT,BOW,BMR2,KR,OS,ScSe,ScSe2].…”

Projective dimensions and extensions of modules from tilted to cluster-tilted algebras

“…[22, Proposition 3.6]. Suppose B is a split extension of C by a nilpotent bi-module E. Then, for every M ∈ mod C, there exists two short exact sequences of B-modules:…”

“…[22, Proposition 4.2]. Let C be an algebra of global dimension at most 2, and let B = C ⋉ E. Suppose M ∈ mod C, then(a) id C M ≤ 1 if and only if M ⊗ C B M. (b) pd C M ≤ 1 if and only if D(B ⊗ C DM) M. Lemma 2.21.…”