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**1 - 8**of**8**### 2 DERIVED EQUIVALENCE OF SURFACE ALGEBRAS IN GENUS 0 VIA GRADED EQUIVALENCE

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### Induced and Coinduced Modules over Cluster-Tilted Algebras

, 2015

"... 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 nE. This new approach consists of using the induction functor − ⊗C B as well as the coinduction functor D(B⊗CD−). We give an explicit construction of ..."

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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 nE. This new approach consists of using the induction functor − ⊗C B as well as the coinduction functor D(B⊗CD−). We give an explicit construction of injective resolutions of pro-jective 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 = EndAT for a tilting module T over a hereditary alge-bra A, we compare the induction and coinduction functors to the Buan-Marsh-Reiten functor HomCA(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. Induced and Coinduced Modules over

### THE AG-INVARIANT FOR (m+ 2)-ANGULATIONS

"... Abstract. In this paper, we study gentle algebras that come from (m + 2)-angulations of unpunctured Riemann surfaces with boundary and marked points. We focus on calculating a derived invariant introduced by Avella-Alaminos and Geiss, generalizing previous work done when m = 1. In par-ticular, we pr ..."

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Abstract. In this paper, we study gentle algebras that come from (m + 2)-angulations of unpunctured Riemann surfaces with boundary and marked points. We focus on calculating a derived invariant introduced by Avella-Alaminos and Geiss, generalizing previous work done when m = 1. In par-ticular, we provide a method for calculating this invariant based on the the configuration of the arcs in the (m + 2)-angulation, the marked points, and the boundary components. 1.

### INDUCED AND COINDUCED MODULES IN CLUSTER-TILTED ALGEBRAS

, 2014

"... 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 n 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 ..."

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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 n 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⊗CB is a partial tilting and a τ-rigid B-module. Furthermore, if C = EndAT for a tilting module T over a hereditary algebra A, we compare the induction and coinduction functors to the Buan-Marsh-Reiten functor HomCA(T,−) from the cluster-