A finite poset X carries a natural structure of a topological space. Fix a field k, and denote by D b (X ) the bounded derived category of sheaves of finite dimensional k-vector spaces over X . Two posets X and Y are said to be derived equivalent if D b (X ) and D b (Y ) are equivalent as triangulated categories.We give explicit combinatorial properties of X which are invariant under derived equivalence; among them are the number of points, the Z-congruency class of the incidence matrix, and the Betti numbers. We also show that taking opposites and products preserves derived equivalence.For any closed subset Y ⊆ X , we construct a strongly exceptional collection in D b (X ) and use it to show an equivalence D b (X ) D b (A) for a finite dimensional algebra A (depending on Y ). We give conditions on X and Y under which A becomes an incidence algebra of a poset.We deduce that a lexicographic sum of a collection of posets along a bipartite graph S is derived equivalent to the lexicographic sum of the same collection along the opposite S op .This construction produces many new derived equivalences of posets and generalizes other well-known ones. As a corollary we show that the derived equivalence class of an ordinal sum of two posets does not depend on the order of summands. We give an example that this is not true for three summands.
We obtain a complete derived equivalence classification of the cluster-tilted algebras of Dynkin type E. There are 67, 416, 1574 algebras in types E6, E7 and E8 which turn out to fall into 6, 14, 15 derived equivalence classes, respectively. This classification can be achieved computationally and we outline an algorithm which has been implemented to carry out this task. We also make the classification explicit by giving standard forms for each derived equivalence class as well as complete lists of the algebras contained in each class; as these lists are quite long they are provided as supplementary material to this paper. From a structural point of view the remarkable outcome of our classification is that two cluster-tilted algebras of Dynkin type E are derived equivalent if and only if their Cartan matrices represent equivalent bilinear forms over the integers which in turn happens if and only if the two algebras are connected by a sequence of "good" mutations. This is reminiscent of the derived equivalence classification of cluster-tilted algebras of Dynkin type A, but quite different from the situation in Dynkin type D where a far-reaching classification has been obtained using similar methods as in the present paper but some very subtle questions are still open.
A triangular matrix ring is defined by a triplet (R, S, M) where R and S are rings and R M S is an S-R-bimodule. In the main theorem of this paper we show that if T S is a tilting S-module, then under certain homological conditions on the S-module M S , one can extend T S to a tilting complex over inducing a derived equivalence between and another triangular matrix ring specified by (S , R, M ), where the ring S and the R-S -bimodule M depend only on M and T S , and S is derived equivalent to S. Note that no conditions on the ring R are needed. These conditions are satisfied when S is an Artin algebra of finite global dimension and M S is finitely generated. In this case, (S , R, M ) = (S, R, DM) where D is the duality on the category of finitely generated S-modules. They are also satisfied when S is arbitrary, M S has a finite projective resolution and Ext
Abstract. We provide a far reaching derived equivalence classification of the cluster-tilted algebras of Dynkin type D and suggest standard forms for the derived equivalence classes. We believe that the classification is complete, but some subtle questions remain open. We introduce another notion of equivalence called good mutation equivalence which is slightly stronger than derived equivalence but is algorithmically more tractable, and give a complete classification together with standard forms.
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