Derived equivalences and Serre duality for Gorenstein algebras

We introduce a notion of Gorenstein algebras of codimension c and demonstrate that Serre duality theory plays an essential role in the theory of derived equivalences for Gorenstein algebras.     

Bounds on the global dimension of certain piecewise hereditary categories

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.     

Folding derived categories with Frobenius functors

Following the work [B. Deng, J. Du, Frobenius morphisms and representations of algebras, Trans. Amer. Math. Soc. 358 (2006) 3591-3622], we show that a Frobenius morphism F on an algebra A induces naturally a functor F on the (bounded) derived category D^b(A) of , and we further prove that the derived category D^b(A^F) of for the F-fixed point algebra A^F is naturally embedded as the triangulated subcategory D^b(A)^F of F-stable objects in D^b(A). When applying the theory to an algebra with finite global dimension, we discover a folding relation between the Auslander-Reiten triangles in D^b(A^F) and those in D^b(A). Thus, the AR-quiver of D^b(A^F) can be obtained by folding the AR-quiver of D^b(A). Finally, we further extend this relation to the root categories ?(A^F) of A^F and ?(A) of A, and show that, when A is hereditary, this folding relation over the indecomposable objects in ?(A^F) and ?(A) results in the same relation on the associated root systems as induced from the graph folding relation.     

Cluster tilting objects in generalized higher cluster categories

We prove the existence of an m-cluster tilting object in a generalized m-cluster category which is (m+1)-Calabi-Yau and Hom-finite, arising from an (m+2)-Calabi-Yau dg algebra. This is a generalization of the result for the m=1 case in Amiot’s Ph.D. thesis. Our results apply in particular to higher cluster categories associated to Ginzburg dg categories coming from suitable graded quivers with superpotential, and higher cluster categories associated to suitable finite-dimensional algebras of finite global dimension.     

Comparisons between singularity categories and relative stable categories of finite groups

We consider the relationship between the relative stable category of and the usual singularity category for group algebras with coefficients in a commutative noetherian ring. When the coefficient ring is self-injective we show that these categories share a common, relatively large, Verdier quotient. At the other extreme, when the coefficient ring has finite global dimension, there is a semi-orthogonal decomposition, due to Poulton, relating the two categories. We prove that this decomposition is partially compatible with the monoidal structure and study the morphism it induces on spectra.     

Grothendieck group and generalized mutation rule for 2-Calabi-Yau triangulated categories

We compute the Grothendieck group of certain 2-Calabi-Yau triangulated categories appearing naturally in the study of the link between quiver representations and Fomin-Zelevinsky cluster algebras. In this setup, we also prove a generalization of the Fomin-Zelevinsky mutation rule.     

On derived equivalences of categories of sheaves over finite posets

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 .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.     

The Grothendieck group of a cluster category

For the cluster category of a hereditary or a canonical algebra, or equivalently for the cluster category of the hereditary category of coherent sheaves on a weighted projective line, we study the Grothendieck group with respect to an admissible triangulated structure.     

The countable Telescope Conjecture for module categories

By the Telescope Conjecture for Module Categories, we mean the following claim: “Let R be any ring and (A,B) be a hereditary cotorsion pair in Mod-R with A and B closed under direct limits. Then (A,B) is of finite type.”We prove a modification of this conjecture with the word ‘finite’ replaced by ‘countable.’ We show that a hereditary cotorsion pair (A,B) of modules over an arbitrary ring R is generated by a set of strongly countably presented modules provided that B is closed under unions of well-ordered chains. We also characterize the modules in B and the countably presented modules in A in terms of morphisms between finitely presented modules, and show that (A,B) is cogenerated by a single pure-injective module provided that A is closed under direct limits. Then we move our attention to strong analogies between cotorsion pairs in module categories and localizing pairs in compactly generated triangulated categories.     

Noetherian hereditary abelian categories satisfying Serre duality

In this paper we classify -finite noetherian hereditary abelian categories over an algebraically closed field satisfying Serre duality in the sense of Bondal and Kapranov. As a consequence we obtain a classification of saturated noetherian hereditary abelian categories.

As a side result we show that when our hereditary abelian categories have no non-zero projectives or injectives, then the Serre duality property is equivalent to the existence of almost split sequences.

     

Combinatorial derived invariants for gentle algebras

We define derived equivalent invariants for gentle algebras, constructed in an easy combinatorial way from the quiver with relations defining these algebras. Our invariants consist of pairs of natural numbers and contain important information about the algebra and the structure of the stable Auslander-Reiten quiver of its repetitive algebra. As a by-product we obtain that the number of arrows of the quiver of a gentle algebra is invariant under derived equivalence. Finally, our invariants separate the derived equivalence classes of gentle algebras with at most one cycle.     

The almost split triangles for perfect complexes over gentle algebras

Throughout the paper k denotes a fixed field. All vector spaces and linear maps are k-vector spaces and k-linear maps, respectively. By Z, N, and N₊, we denote the sets of integers, nonnegative integers, and positive integers, respectively. For i,j∈Z, [i,j]:={l∈Z∣i≤l≤j} (in particular, [i,j]=∅ if i>j).     

Crystals, quiver varieties, and coboundary categories for Kac-Moody algebras

Henriques and Kamnitzer have defined a commutor for the category of crystals of a finite-dimensional complex reductive Lie algebra that gives it the structure of a coboundary category (somewhat analogous to a braided monoidal category). Kamnitzer and Tingley then gave an alternative definition of the crystal commutor, using Kashiwara's involution on Verma crystals, that generalizes to the setting of symmetrizable Kac-Moody algebras. In the current paper, we give a geometric interpretation of the crystal commutor using quiver varieties. Equipped with this interpretation we show that the commutor endows the category of crystals of a symmetrizable Kac-Moody algebra with the structure of a coboundary category, answering in the affirmative a question of Kamnitzer and Tingley.     

The Popescu-Gabriel theorem for triangulated categories

The Popescu-Gabriel theorem states that each Grothendieck abelian category is a localization of a module category. In this paper, we prove an analogue where Grothendieck abelian categories are replaced by triangulated categories which are well generated (in the sense of Neeman) and algebraic (in the sense of Keller). The role of module categories is played by derived categories of small differential graded categories. An analogous result for topological triangulated categories has recently been obtained by A. Heider.