Higher-dimensional Auslander-Reiten theory on maximal orthogonal subcategories

We introduce the concept of maximal (n−1)-orthogonal subcategories over Artin algebras and orders, and develop (n+1)-dimensional Auslander-Reiten theory on them. We give the n-Auslander-Reiten translation and the n-Auslander-Reiten duality, then show the existence of n-almost split sequences and n-fundamental sequences. We give some examples.     

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.     

Artin-Schelter regular algebras and categories

Motivated by constructions in the representation theory of finite dimensional algebras we generalize the notion of Artin-Schelter regular algebras of dimension n to algebras and categories to include Auslander algebras and a graded analogue for infinite representation type. A generalized Artin-Schelter regular algebra or a category of dimension n is shown to have common properties with the classical Artin-Schelter regular algebras. In particular, when they admit a duality, then they satisfy Serre duality formulas and the -category of nice sets of simple objects of maximal projective dimension n is a finite length Frobenius category.     

Generalized tilting functors and Ringel-Hall algebras

It is known from [M. Auslander, M.I. Platzeck, I. Reiten, Coxeter functors without diagrams, Trans. Amer. Math. Soc. 250 (1979) 1-46] and [C.M. Ringel, PBW-basis of quantum groups, J. Reine Angew. Math. 470 (1996) 51-85] that the Bernstein-Gelfand-Ponomarev reflection functors are special cases of tilting functors and these reflection functors induce isomorphisms between certain subalgebras of Ringel-Hall algebras. In [A. Wufu, Tilting functors and Ringel-Hall algebras, Comm. Algebra 33 (1) (2005) 343-348] the result from [C.M. Ringel, PBW-basis of quantum groups, J. Reine Angew. Math. 470 (1996) 51-85] is generalized to the tilting module case by giving an isomorphism between two Ringel-Hall subalgebras. In [J. Miyashita, Tilting Modules of Finite Projective Dimension, Math. Z. 193 (1986) 113-146] Miyashita generalized the tilting theory by introducing the tilting modules of finite projective dimension. In this paper the result in [A. Wufu, Tilting functors and Ringel-Hall algebras, Comm. Algebra 33 (1) (2005) 343-348] is generalized to the tilting modules of finite projective dimension.     

Remarks on non-commutative crepant resolutions of complete intersections

We study obstructions to existence of non-commutative crepant resolutions, in the sense of Van den Bergh, over local complete intersections.     

An Auslander-type result for Gorenstein-projective modules

An artin algebra A is said to be CM-finite if there are only finitely many, up to isomorphisms, indecomposable finitely generated Gorenstein-projective A-modules. We prove that for a Gorenstein artin algebra, it is CM-finite if and only if every its Gorenstein-projective module is a direct sum of finitely generated Gorenstein-projective modules. This is an analogue of Auslander's theorem on algebras of finite representation type [M. Auslander, A functorial approach to representation theory, in: Representations of Algebras, Workshop Notes of the Third Internat. Conference, in: Lecture Notes in Math., vol. 944, Springer-Verlag, Berlin, 1982, pp. 105-179; M. Auslander, Representation theory of artin algebras II, Comm. Algebra (1974) 269-310].     

Cluster tilting for tilted algebras

We build a connection between iterated tilted algebras with trivial cluster tilting subcategories and tilted algebras of finite type.Moreover,all tilted algebras with cluster tilting subcategories are determined in terms of quivers.As a result,we draw the quivers of Auslander’s 1-Gorenstein algebras with global dimension 2 admitting trivial cluster tilting subcategories,which implies that such algebras are of finite type but not necessarily Nakayama.     

Tilting subcategories in extriangulated categories

Extriangulated category was introduced by H.Nakaoka and Y.Palu to give a unification of properties in exact categories anjd triangulated categories.A notion of tilting(resp.,cotilting)subcategories in an extriangulated category is defined in this paper.We give a Bazzoni characterization of tilting(resp.,cotilting)subcategories and obtain an Auslander-Reiten correspondence between tilting(resp.,cotilting)subcategories and coresolving covariantly(resp.,resolving contravariantly)finite subcatgories which are closed under direct summands and satisfy some cogenerating(resp.,generating)conditions.Applications of the results are given:we show that tilting(resp.,cotilting)subcategories defined here unify many previous works about tilting modules(subcategories)in module categories of Artin algebras and in abelian categories admitting a cotorsion triples;we also show that the results work for the triangulated categories with a proper class of triangles introduced by A.Beligiannis.     

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.     

Two remarks on witten'squantum enveloping algebra

In this note we describe two simple tricks derived from work of M. Artin , J. Tate and M. Van den Bergh [2] and apply them to the quantum enveloping algebra discovered by E. Witten [ll] encoding duality in three dimensional Chern-Simons gauge theory with structure group SU(2). In particular, we give a more conceptual interpretation of the dichotony in the representation theory and determine all finite dimensional simple representations.     

Generic noncommutative surfaces

We study a class of noncommutative surfaces, and their higher dimensional analogs, which come from generic subalgebras of twisted homogeneous coordinate rings of projective space. Such rings provide answers to several open questions in noncommutative projective geometry. Specifically, these rings R are the first known graded algebras over a field k which are noetherian but not strongly noetherian: in other words, R⊗_kB is not noetherian for some choice of commutative noetherian extension ring B. This answers a question of Artin, Small, and Zhang. The rings R are also maximal orders, but they do not satisfy all of the χ conditions of Artin and Zhang. In particular, they satisfy the χ₁ condition but not χ_i for i?2, answering a question of Stafford and Zhang and a question of Stafford and Van den Bergh. Finally, we show that the noncommutative scheme R-proj has finite global dimension.     

A note on relative tilting modules

Bazzoni had given a simple characterization of infinitely generated n-tilting modules. Though her method is even inapplicable to classical n-tilting modules over Artin algebras, we show in this note that a similar characterization does hold for (finitely generated) relative n-tilting modules introduced by Auslander and Solberg for Artin algebras, by using a different method. We also present some applications.     

Maximal modifications and Auslander–Reiten duality for non-isolated singularities

We first generalize classical Auslander–Reiten duality for isolated singularities to cover singularities with a one-dimensional singular locus. We then define the notion of CT modules for non-isolated singularities and we show that these are intimately related to noncommutative crepant resolutions (NCCRs). When R has isolated singularities, CT modules recover the classical notion of cluster tilting modules but in general the two concepts differ. Then, wanting to generalize the notion of NCCRs to cover partial resolutions of \(\operatorname{Spec}R\) , in the main body of this paper we introduce a theory of modifying and maximal modifying modules. Under mild assumptions all the corresponding endomorphism algebras of the maximal modifying modules for three-dimensional Gorenstein rings are shown to be derived equivalent. We then develop a theory of mutation for modifying modules which is similar but different to mutations arising in cluster tilting theory. Our mutation works in arbitrary dimension, and in dimension three the behavior of our mutation strongly depends on whether a certain factor algebra is artinian.     

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.     

相对整体维数有限的扩张

代数的扩张是指两个代数之间保持单位元的同态映射,设f:B→A是代数的扩张,扩张f的相对整体维数是指所有A-模的相对投射维数的上确界.我们给出了扩张的相对整体维数有限的一个充分必要条件,作为应用,还获得了Hochschild的文[Relative homological algebra, Trans. Am. Math. Soc.,1956,82:246-269]中一个结果的简洁证明.     

Classifying tame blocks and related algebras up to stable equivalences of Morita type

We contribute to the classification of finite dimensional algebras under stable equivalence of Morita type. More precisely we give a classification of Erdmann’s algebras of dihedral, semi-dihedral and quaternion type and obtain as byproduct the validity of the Auslander-Reiten conjecture for stable equivalences of Morita type between two algebras, one of which is of dihedral, semi-dihedral or quaternion type.     

The representation dimension of domestic weakly symmetric algebras

Auslander’s representation dimension measures how far a finite dimensional algebra is away from being of finite representation type. In [1], M. Auslander proved that a finite dimensional algebra A is of finite representation type if and only if the representation dimension of A is at most 2. Recently, R. Rouquier proved that there are finite dimensional algebras of an arbitrarily large finite representation dimension. One of the exciting open problems is to show that all finite dimensional algebras of tame representation type have representation dimension at most 3. We prove that this is true for all domestic weakly symmetric algebras over algebraically closed fields having simply connected Galois coverings.     

Cellular resolutions of noncommutative toric algebras from superpotentials

This paper constructs cellular resolutions for classes of noncommutative algebras, analogous to those introduced by Bayer and Sturmfels (1998) [2] in the commutative case. To achieve this we generalise the dimer model construction of noncommutative crepant resolutions of three-dimensional toric algebras by associating a superpotential and a notion of consistency to toric algebras of arbitrary dimension. For abelian skew group algebras and algebraically consistent dimer model algebras, we introduce a cell complex Δ in a real torus whose cells describe uniformly all maps in the minimal projective bimodule resolution of A. We illustrate the general construction of Δ for an example in dimension four arising from a tilting bundle on a smooth toric Fano threefold to highlight the importance of the incidence function on Δ.