Dimensions of triangulated categories via Koszul objects

Lower bounds for the dimension of a triangulated category are provided. These bounds are applied to stable derived categories of Artin algebras and of commutative complete intersection local rings. As a consequence, one obtains bounds for the representation dimensions of certain Artin algebras.     

On the Derived Categories and Quasitilted Algebras

In this paper, we present some results on the bounded derived category of Artin algebras, and in particular on the indecomposable objects in these categories, using homological properties. Given a complex X ^*, we consider the set and we define the application . We give relationships between some homological properties of an algebra and the respective application l. On the other hand, using homological properties again, we determine two subcategories of the bounded derived category of an algebra, which turn out to be the bounded derived categories of quasi-tilted algebras. As a consequence of these results we obtain new characterizations of quasi-tilted and quasi-tilted gentle algebras. Presented by Raymundo Bautista.     

Relative derived equivalences and relative homological dimensions

Let A be a small abelian category. For a closed subbifunctor F of Ext_A¹(-,-), Buan has generalized the construction of Verdier's quotient category to get a relative derived category, where he localized with respect to F-acyclic complexes. In this paper, the homological properties of relative derived categories are discussed, and the relation with derived categories is given. For Artin algebras, using relative derived categories, we give a relative version on derived equivalences induced by F-tilting complexes. We discuss the relationships between relative homological dimensions and relative derived equivalences.     

Lifting and Restricting Recollement Data

We study the problem of lifting and restricting TTF triples (equivalently, recollement data) for a certain wide type of triangulated categories. This, together with the parametrizations of TTF triples given in Nicolás and Saorín (Parametrizing recollement data for triangulated categories. To appear in J. Algebra), allows us to show that many well-known recollements of right bounded derived categories of algebras are restrictions of recollements in the unbounded level, and leads to criteria to detect recollements of general right bounded derived categories. In particular, we give in Theorem 1 necessary and sufficient conditions for a right bounded derived category of a differential graded (=dg) category to be a recollement of right bounded derived categories of dg categories. Theorem 2 considers the case of dg categories with cohomology concentrated in non-negative degrees. In Theorem 3 we consider the particular case in which those dg categories are just ordinary algebras.     

Gorenstein derived categories

Gorenstein derived categories are defined, and the relation with the usual derived categories is given. The bounded Gorenstein derived categories of Gorenstein rings and of finite-dimensional algebras are explicitly described via the homotopy categories of Gorenstein-projective modules, and some applications are obtained. Gorenstein derived equivalences between CM-finite Gorenstein algebras are discussed.     

Representation Dimension and Quasi-hereditary Algebras

We study Auslander's representation dimension of Artin algebras, which is by definition the minimal projective dimension of coherent functors on modules which are both generators and cogenerators. We show the following statements: (1) if an Artin algebra A is stably hereditary, then the representation dimension of A is at most 3. (2) If two Artin algebras are stably equivalent of Morita type, then they have the same representation dimension. Particularly, if two self-injective algebras are derived equivalent, then they have the same representation dimension. (3) Any incidence algebra of a finite partially ordered set over a field has finite representation dimension. Moreover, we use results on quasi-hereditary algebras to show that (4) the Auslander algebra of a Nakayama algebra has finite representation dimension.     

Geometricity for derived categories of algebraic stacks

We prove that the dg category of perfect complexes on a smooth, proper Deligne–Mumford stack over a field of characteristic zero is geometric in the sense of Orlov, and in particular smooth and proper. On the level of triangulated categories, this means that the derived category of perfect complexes embeds as an admissible subcategory into the bounded derived category of coherent sheaves on a smooth, projective variety. The same holds for a smooth, projective, tame Artin stack over an arbitrary field.     

Jordan–Hölder theorems for derived module categories of piecewise hereditary algebras

A Jordan–Hölder theorem is established for derived module categories of piecewise hereditary algebras, in particular for representations of quivers and for hereditary abelian categories of a geometric nature. The resulting composition series of derived categories are shown to be independent of the choice of bounded or unbounded derived module categories, and also of the choice of finitely generated or arbitrary modules.     

Derived equivalences and Cohen-Macaulay Auslander algebras

Let A and B be Artin R-algebras of finite Cohen-Macaulay type. Then we prove that, if A and B are standard derived equivalent, then their Cohen-Macaulay Auslander algebras are also derived equivalent. And we show that Gorenstein projective conjecture is an invariant under standard derived equivalence between Artin R-algebras.     

Generalized Auslander-Reiten Conjecture and Derived Equivalences

In this note, we prove that the generalized Auslander-Reiten conjecture is preserved under derived equivalences between Artin algebras.     

Composite of irreducible morphisms in the bounded derived category

We study necessary and sufficient conditions for the existence of n irreducible morphisms in the bounded derived category of an Artin algebra, with non-zero composite in the n+1-power of the radical. In the case of , the bounded derived category of an Ext-finite hereditary k-category with tilting object, such irreducible morphisms exist if and only if H is derived equivalent to a wild hereditary algebra or to a wild canonical algebra. We also characterize the cluster tilted algebras having such irreducible morphisms.     

Classification of discrete derived categories

The main aim of the paper is to classify the discrete derived categories of bounded complexes of modules over finite dimensional algebras.     

Adjoint Functors and Representation Dimensions

We study the global dimensions of the coherent functors over two categories that are linked by a pair of adjoint functors. This idea is then exploited to compare the representation dimensions of two algebras. In particular, we show that if an Artin algebra is switched from the other, then they have the same representation dimension.     

Representation dimension and finitely generated cohomology

We consider selfinjective Artin algebras whose cohomology groups are finitely generated over a central ring of cohomology operators. For such an algebra, we show that the representation dimension is strictly greater than the maximal complexity occurring among its modules. This provides a unified approach to computing lower bounds for the representation dimension of group algebras, exterior algebras and Artin complete intersections. We also obtain new examples of classes of algebras with arbitrarily large representation dimension.     

Maximal Green Sequences for Preprojective Algebras

Maximal green sequences were introduced as combinatorical counterpart for Donaldson-Thomas invariants for 2-acyclic quivers with potential by B. Keller. We take the categorical notion and introduce maximal green sequences for hearts of bounded t-structures of triangulated categories that can be tilted indefinitely. We study the case where the heart is the category of modules over the preprojective algebra of a quiver without loops. The combinatorical counterpart of maximal green sequences for Dynkin quivers are maximal chains in the Hasse quiver of basic support τ-tilting modules. We show that a quiver has a maximal green sequence if and only if it is of Dynkin type. More generally, we study module categories for finite-dimensional algebras with finitely many bricks.     

Brauer-Thrall Type Theorems for Derived Module Categories

The numerical invariants (global) cohomological length, (global) cohomological width, and (global) cohomological range of a complex (an algebra) are introduced. Cohomological range leads to the concepts of derived bounded algebra and strongly derived unbounded algebra naturally. The first and second Brauer-Thrall type theorems for the bounded derived category of a finite-dimensional algebra over an algebraically closed field are obtained. The first Brauer-Thrall type theorem says that derived bounded algebras are just derived finite algebras. The second Brauer-Thrall type theorem says that an algebra is either derived discrete or strongly derived unbounded, but not both. Moreover, piecewise hereditary algebras and derived discrete algebras are characterized as the algebras of finite global cohomological width and the algebras of finite global cohomological length respectively.     

Automorphismes des algèbres cubiques

Manin associated to a quadratic algebra (quantum space) the quantum matrix group of its automorphisms. This Note aims to demonstrate that Manin's construction can be extended for quantum spaces which are non-quadratic homogeneous algebras. The Artin–Schelter classification of regular algebras of global dimension three contains two types of algebra: quadratic and cubic. Ewen and Ogievetsky classified the quantum matrix groups which are deformations of GL(3) corresponding to the quadratic algebras in the Artin–Schelter classification. In this Note we consider the cubic Artin–Schelter algebras as quantum spaces and construct Hopf algebras of their automorphisms. To cite this article: T. Popov, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

On generalized symmetries and structure of modular categories

Pursuing a generalization of group symmetries of modular categories to category symmetries in topological phases of matter, we study linear Hopf monads. The main goal is a generalization of extension and gauging group symmetries to category symmetries of modular categories, which include also categorical Hopf algebras as special cases. As an application, we propose an analogue of the classification of finite simple groups to modular categories, where we define simple modular categories as the prime ones without any nontrivial normal algebras.     

On the Analytic Type of Artin Algebras

In this paper we give a complete analytic classification of Artin Gorenstein almost stretched algebras, i.e., Artin Gorenstein algebras with Hilbert function {1, n, 2,…, 2, 1,…, 1}.     

Piecewise Hereditary Triangular Matrix Algebras

For any positive integer N,we clearly describe all finite-dimensional algebras A such that the upper triangular matrix algebras TN(A) are piecewise hereditary.Consequently,we describe all finite-dimensional algebras A such that their derived categories of N-complexes are triangulated equivalent to derived categories of hereditary abelian categories,and we describe the tensor algebras A (×) K[X]/(XN) for which their singularity categories are triangulated orbit categories of the derived categories of hereditary abelian categories.