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.     

Auslander correspondence

We study Auslander correspondence from the viewpoint of higher-dimensional analogue of Auslander-Reiten theory [O. Iyama, Higher dimensional Auslander-Reiten theory on maximal orthogonal subcategories, Adv. Math. 210 (1) (2007) 22-50 (this issue)] on maximal orthogonal subcategories. We give homological characterizations of higher dimensional analogue of Auslander algebras in terms of global dimension, Auslander-type conditions and so on. Especially we give an answer to a question of M. Artin [M. Artin, Maximal orders of global dimension and Krull dimension two, Invent. Math. 84 (1) (1986) 195-222]. They are also closely related to Auslander's representation dimension of Artin algebras [M. Auslander, Representation dimension of Artin algebras, in: Lecture Notes, Queen Mary College, London, 1971] and Van den Bergh's non-commutative crepant resolutions of Gorenstein singularities [M. Van den Bergh, Non-commutative crepant resolutions, in: The Legacy of Niels Henrik Abel, Springer, Berlin, 2004, pp. 749-770].     

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.     

A functorial approach to categorical resolutions

Using a relative version of Auslander's formula, we give a functorial approach to show that the bounded derived category of every Artin algebra admits a categorical resolution. This, in particular, implies that the bounded derived categories of Artin algebras of finite global dimension determine bounded derived categories of all Artin algebras. Hence, this paper can be considered as a typical application of functor categories,introduced in representation theory by Auslander(1971), to categorical resolutions.     

关于有限维数猜想的一些新进展

在Artin代数的表示理论中,有一个著名的有限维数猜想：任意给定一个Artin代数,它的有限维数都是有限的．这个猜想已有45年的历史,至今悬而未决．本文主要综述它的一些历史发展情况,并介绍关于有限维数猜想的一些最新进展．     

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 theory of two families of quantum projective 3-spaces

Stephenson and Vancliff recently introduced two families of quantum projective 3-spaces (quadratic and Artin–Schelter regular algebras of global dimension 4) which have the property that the associated automorphism of the scheme of point modules is finite order, and yet the algebra is not finite over its center. This is in stark contrast to theorems of Artin, Tate, and Van den Bergh in global dimension 3. We analyze the representation theory of these algebras. We classify all of the finite-dimensional simple modules and describe some zero-dimensional elements of Proj, i.e., so called fat point modules. In particular, we observe that the shift functor on zero-dimensional elements of Proj, which is closely related to the above automorphism, actually has infinite order.     

Representation Dimension as a Relative Homological Invariant of Stable Equivalence

Over an Artin algebra Λ many standard concepts from homological algebra can be relativized with respect to a contravariantly finite subcategory of mod-Λ, which contains the projective modules. The main aim of this article is to prove that the resulting relative homological dimensions of modules are preserved by stable equivalences between Artin algebras. As a corollary, we see that Auslander’s notion of representation dimension is invariant under stable equivalence (a result recently obtained independently by Guo). We then apply these results to the syzygy functor for self-injective algebras of representation dimension three, where we bound the number of simple modules in terms of the number of indecomposable nonprojective summands of an Auslander generator.      

Homologies d'algèbres Artin–Schelter régulières cubiques

Hochschild homology of cubic Artin–Schelter regular algebras of type A with generic coefficients is computed. We follow the method used by Van den Bergh (K-Theory 8 (1994) 213–230) in the quadratic case, by considering these algebras as deformations of a polynomial algebra, with remarkable Poisson brackets. A new quasi-isomorphism is introduced. De Rham cohomology, cyclic and periodic cyclic homologies, and Hochschild cohomology are also computed. To cite this article: N. Marconnet, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Hochschild cohomology and “smoothness” in monoidal categories

We introduce and investigate the properties of Hochschild cohomology of algebras in an abelian monoidal category M. We show that the second Hochschild cohomology group of an algebra in M classifies extensions of A up to an equivalence. We characterize algebras of Hochschild dimension 0 (separable algebras), and of Hochschild dimension ≤1 (formally smooth algebras). Several particular cases and applications are included in the last section of the paper.     

Finitistic Dimensions of Artin Algebras with Two Simples and a Conjecture of Marczinzik

Let Λ be an Artin algebra with a unique non-injective indecomposable projective module. In this situation, Marczinzik conjectured that the dominant dimension of Λ agrees with its finitistic dimension. In this paper, we give a proof of a stronger statement. As a byproduct, we obtain excellent control over the finitistic dimensions of Artin algebras with two simples and positive dominant dimension, and also establish the Gorenstein symmetry conjecture for all algebras under consideration.

     

GRADED RINGS OF LOCAL FINITE REPRESENTATION TYPE

Abstract

Graded Artin algebras whose category of graded modules is locally of finite representation type are introduced. The representation theory of such algebras is studied. In the hereditary case and in the stably equivalent to hereditary case, such algebras are classified.     

Group actions on algebras and the graded Lie structure of Hochschild cohomology

Hochschild cohomology governs deformations of algebras, and its graded Lie structure plays a vital role. We study this structure for the Hochschild cohomology of the skew group algebra formed by a finite group acting on an algebra by automorphisms. We examine the Gerstenhaber bracket with a view toward deformations and developing bracket formulas. We then focus on the linear group actions and polynomial algebras that arise in orbifold theory and representation theory; deformations in this context include graded Hecke algebras and symplectic reflection algebras. We give some general results describing when brackets are zero for polynomial skew group algebras, which allow us in particular to find noncommutative Poisson structures. For abelian groups, we express the bracket using inner products of group characters. Lastly, we interpret results for graded Hecke algebras.     

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).     

Restricted Lie algebras with bounded cohomology and related classes of algebras

We determine the structure of restricted Lie algebras with bounded cohomology over arbitrary fields of prime characteristic. As a byproduct a classification of the serial restricted Lie algebras and the restricted Lie algebras of finite representation type is obtained. In addition, we derive complete information on the finite dimensional indecomposable restricted modules of these algebras over algebraically closed fields.     

Deformations of Loday-type algebras and their morphisms

We study formal deformations of multiplication in an operad. This closely resembles Gerstenhaber's deformation theory for associative algebras. However, this applies to various algebras of Loday-type and their twisted analogs. We explicitly describe the cohomology of these algebras with coefficients in a representation. Finally, deformation of morphisms between algebras of the same Loday-type is also considered.     

Igusa–Todorov functions for Artin algebras

In this paper we study the behavior of the Igusa–Todorov functions for Artin algebras A with finite injective dimension, and Gorenstein algebras as a particular case. We show that the ?-dimension and ψ-dimension are finite in both cases. Also we prove that monomial, gentle and cluster tilted algebras have finite ?-dimension and finite ψ-dimension.     

Global Homological Dimension of Radical Banach Algebras of Power Series

We show that the global dimension of a broad class of radical Banach algebras of power series is at least 3 and obtain applications to cohomology groups.     

On the First Hochschild Cohomology of Admissible Algebras

The aim of this paper is to investigate the first Hochschild cohomology of admissible algebras which can be regarded as a generalization of basic algebras.For this purpose,the authors study differential operators on an admissible algebra.Firstly,differential operators from a path algebra to its quotient algebra as an admissible algebra are discussed.Based on this discussion,the first cohomology with admissible algebras as coefficient modules is characterized,including their dimension formula.Besides,for planar quivers,the fc-linear bases of the first cohomology of acyclic complete monomial algebras and acyclic truncated quiver algebras are constructed over the field fc of characteristic 0.