arXiv subjects
Publications and source records attributed to Sefi Ladkani.
We relate the notions of BB-tilting and perverse derived equivalence at a vertex. Based on these notions, we define mutations of algebras, leading to derived equivalent ones. We present applications to endomorphism algebras of cluster-tilting objects in 2-Calabi-Yau categories and to algebras of global dimension at most 2.
A triangular matrix ring A is defined by a triplet (R,S,M) where R and S are rings and M is an S-R-bimodule. In the main theorem of this paper we show that if T is a tilting S-module, then under certain homological conditions on M as an S-module, one can extend T to a tilting complex over A inducing a derived equivalence between A and another triangular matrix ring specified by (S',R,M'), where the ring S' and the R-S'-bimodule M' depend only on T and M, 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 is finitely generated as an S-module. 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 has a finite projective resolution as an S-module, and $\Ext^n_S(M,S)=0$ for all $n>0$. In this case, $(S',R,M')=(S,R,\Hom_S(M,S))$.
We give bounds on the global dimension of a finite length, piecewise hereditary category in terms of quantitative connectivity properties of its graph of indecomposables. We use this to show that the global dimension of a finite dimensional, piecewise hereditary algebra A cannot exceed 3 if A is an incidence algebra of a finite poset or more generally, a sincere algebra. This bound is tight.
We give a full description of all the canonical algebras over an algebraically closed field that are derived equivalent to incidence algebras of finite posets. These are the canonical algebras whose number of weights is either 2 or 3.
We show that for two quivers without oriented cycles related by a BGP reflection, the posets of their cluster tilting objects are related by a simple combinatorial construction, which we call a flip-flop. We deduce that the posets of cluster tilting objects of derived equivalent path algebras of quivers without oriented cycles are universally derived equivalent. In particular, all Cambrian lattices corresponding to the various orientations of the same Dynkin diagram are universally derived equivalent.
We show that for two quivers without oriented cycles related by a BGP reflection, the posets of their tilting modules are related by a simple combinatorial construction, which we call flip-flop. We deduce that the posets of tilting modules of derived equivalent path algebras of quivers without oriented cycles are universally derived equivalent.
We show that for piecewise hereditary algebras, the periodicity of the Coxeter transformation implies the non-negativity of the Euler form. Contrary to previous assumptions, the condition of piecewise heredity cannot be omitted, even for triangular algebras, as demonstrated by incidence algebras of posets. We also give a simple, direct proof, that certain products of reflections, defined for any square matrix A with 2 on its main diagonal, and in particular the Coxeter transformation corresponding to a generalized Cartan matrix, can be expressed as $-A_{+}^{-1} A_{-}^t$, where A_{+}, A_{-} are closely associated with the upper and lower triangular parts of A.
By using only combinatorial data on two posets X and Y, we construct a set of so-called formulas. A formula produces simultaneously, for any abelian category A, a functor between the categories of complexes of diagrams over X and Y with values in A. This functor induces a triangulated functor between the corresponding derived categories. This allows us to prove, for pairs X, Y of posets sharing certain common underlying combinatorial structure, that for any abelian category A, regardless of its nature, the categories of diagrams over X and Y with values in A are derived equivalent.
A finite poset X carries a natural structure of a topological space. Fix a field k, and denote by D(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(X) and D(Y) are equivalent as triangulated categories. We give explicit combinatorial properties of a poset 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. Then we construct, for any closed subset Y of X, a strongly exceptional collection in D(X) and use it to show an equivalence between D(X) and the bounded derived category of 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 is derived equivalent to the lexicographic sum of the same collection along the opposite graph. 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.