Finite Global Dimension Research Articles

The category of F-modules has finite global dimension

Let R be a regular ring of characteristic p>0. In [4], Hochster showed that the category of Lyubeznikʼs FR-modules has enough injectives, so that every FR-module has an injective resolution in this category. We show in this paper that under mild conditions on R, for example when R is essentially of finite type over an F-finite regular local ring, the category of F-modules has finite global dimension d+1 where d=dimR. In [4], Hochster also showed that when M and N are FR-finite FR-modules, HomFR(M,N) is finite. We show that in general ExtFR1(M,N) is not necessarily finite.

Homogeneous G-algebras II

G-algebras, or Groebner bases algebras, were considered by Levandovsky, these algebras include very important families of algebras, like the Weyl algebras and the universal enveloping algebra of a finite dimensional Lie algebra. These algebras are not graded in a natural way, but they are filtered. Going to the associated graded algebra is the standard way to study their properties. In a previous paper we studied an homogeneous version of the G-algebras, which results naturally graded, and through a des homogenization property, we obtain an ordinary G-algebra. The main results in our first paper were the following: Homogeneous G-algebras are Koszul of finite global dimension, Artin Schelter regular, noetherian and have a Poincare Birkoff basis. We give the structure of their Yoneda algebras by generators and relations. In the second part of the paper we consider the quantum polynomial ring and prove, for the finitely generated graded modules, cohomology formulas analogous to those we have in the usual polynomial ring. In the third part we use the results of the previous part to study the cohomology of the homogeneous G-algebras. Part four was dedicated to the study of the relations between an homogeneous G-algebra B and its des homogenization B/(Z-1)B=A, Applications to the study of finite dimensional Lie algebras and Weyl algebras were given. In the last sections of the paper we study the relations, at the level of modules, among the homogenized algebra B, its graded localization B_{z}, the des homogenization A=B/(Z-1)B and the Yoneda algebra B^{!}. In this paper we investigate the relations among all these algebras at the level of derived categories.

Realizing stable categories as derived categories

In this paper, we discuss a relationship between representation theory of graded self-injective algebras and that of algebras of finite global dimension. For a positively graded self-injective algebra A such that A0 has finite global dimension, we construct two types of triangle-equivalences. First we show that there exists a triangle-equivalence between the stable category of Z-graded A-modules and the derived category of a certain algebra Γ of finite global dimension. Secondly we show that if A has Gorenstein parameter ℓ, then there exists a triangle-equivalence between the stable category of Z/ℓZ-graded A-modules and a derived-orbit category of Γ, which is a triangulated hull of the orbit category of the derived category.

Vanishing of Ext, cluster tilting modules and finite global dimension of endomorphism rings

Let $R$ be a Cohen-Macaulay ring and $M$ a maximal Cohen-Macaulay $R$-module. Inspired by recent striking work by Iyama, Burban-Iyama-Keller-Reiten and van den Bergh we study the question of when the endomorphism ring of $M$ has finite global dimension via certain conditions about vanishing of Ext modules. We are able to strengthen certain results by Iyama on connections between a higher dimension version of Auslander correspondence and existence of non-commutative crepant resolutions. We also recover and extend to positive characteristics a recent Theorem by Burban-Iyama-Keller-Reiten on cluster-tilting objects in the category of maximal Cohen-Macaulay modules over reduced $1$-dimensional hypersurfaces.

On the extension problem and the nil groups of rings of finite global dimension

A vanishing result is obtained in respect of nil groups of rings of finite global dimension. Also a connection is established with the extension problem.

Elementary Superalgebras

Let Λ be a finite-dimensional superalgebra over a field K. A characterization of an elementary superalgebra Λ is given by a quiver and a weight function. It is shown that Λ is elementary if and only if its Hochschild extension is elementary. Furthermore, if Λ is elementary of finite global dimension and {e1, …, en} is a complete set of gr-primitive orthogonal idempotents of Λ, then the following equalities hold: [Formula: see text] where ΦΛ is the Coxeter matrix of Λ, tr is the trace function of a matrix, HHi(Λ) and HHi(Λ) are the i-th Hochschild homology and cohomology, respectively.

On root categories of finite-dimensional algebras

For any finite-dimensional algebra A over a field k of finite global dimension, we investigate the root category RA as the triangulated hull of the 2-periodic orbit category of A via the construction of B. Keller in “On triangulated orbit categories”. This is motivated by Ringel–Hall Lie algebras associated to 2-periodic triangulated categories. As an application, we study the Ringel–Hall Lie algebras for a class of finite-dimensional k-algebras of global dimension 2, which turns out to give an alternative answer to a question of GIM-Lie algebras by Slodowy in “Beyond Kac–Moody algebra, and inside”.

Deformations of Koszul Artin–Schelter Gorenstein algebras

We compute the Nakayama automorphism of a Poincaré–Birkhoff–Witt (PBW)-deformation of a Koszul Artin–Schelter (AS) Gorenstein algebra of finite global dimension, and give a criterion for an augmented PBW-deformation of a Koszul Calabi–Yau algebra to be Calabi–Yau. The relations between the Calabi–Yau property of augmented PBW-deformations and that of non-augmented cases are discussed. The Nakayama automorphisms of PBW-deformations of Koszul AS–Gorenstein algebras of global dimensions 2 and 3 are given explicitly. We show that if a PBW-deformation of a graded Calabi–Yau algebra is still Calabi–Yau, then it is defined by a potential under some mild conditions. Some classical results are also recovered. Our main method used in this article is elementary and based on linear algebra. The results obtained in this article will be applied in a subsequent paper (He et al., Skew polynomial algebras with coefficients in AS regular algebras, preprint, 2011).

Hochschild cohomology of Beilinson algebra of exterior algebra

Let Λn be the Beilinson algebra of exterior algebra of an n-dimensional vector space, which is derived equivalent to the endomorphism algebra \(End_{\mathcal{O}_X } (T)\) of a tilting complex T = Πi=0nOX(i) of coherent OX-modules over a projective scheme X = Pkn. In this paper we first construct a minimal projective bimodule resolution of Λn, and then apply it to calculate k-dimensions of the Hochschild cohomology groups of Λn in terms of parallel paths. Finally, we give an explicit description of the cup product and obtain a Gabriel presentation of Hochschild cohomology ring of Λn. As a consequence, we provide a class of algebras of finite global dimension whose Hochschild cohomology rings have non-trivial multiplicative structures.

The Higher Relation Bimodule

Given a finite dimensional algebra A of finite global dimension, we consider the trivial extension of A by the A − A-bimodule $\oplus_{i\ge 2} {\rm Ext}^i_A(DA,A)$ , which we call the higher relation bimodule. We first give a recipe allowing to construct the quiver of this trivial extension in case A is a string algebra and then apply it to prove that, if A is gentle, then the tensor algebra of the higher relation bimodule is gentle.

Quadratic algebras, Yang–Baxter equation, and Artin–Schelter regularity

We study two classes of quadratic algebras over a field k: the class Cn of all n-generated PBW algebras with polynomial growth and finite global dimension, and the class of quantum binomial algebras. We show that a PBW algebra A is in Cniff its Hilbert series is HA(z)=1/(1−z)n. Furthermore, the class Cn contains a unique (up to isomorphism) monomial algebra, A=k〈x1,…,xn〉/(xjxi∣1≤i<j≤n). A surprising amount can be said when A is a quantum binomial algebra, that is its defining relations are nondegenerate square-free binomials xy−cxyzt, cxy∈k×. Our main result shows that for an n-generated quantum binomial algebra A the following conditions are equivalent: (i) A is an Artin–Schelter regular PBW algebra. (ii) A is a Yang–Baxter algebra, that is the set of relations ℜ defines canonically a solution of the Yang–Baxter equation. (iii) A is a binomial skew polynomial ring, with respect to an enumeration of X. (iv) The Koszul dual A! is a quantum Grassmann algebra.

Coverings and truncations of graded self-injective algebras

Let Λ be a graded self-injective algebra. We describe its smash product Λ#kZ⁎ with the group Z, its Beilinson algebra and their relationship. Starting with Λ, we construct algebras with finite global dimension, called τ-slice algebras, we show that their trivial extensions are all isomorphic, and their repetitive algebras are the same as Λ#kZ⁎. There exist τ-mutations similar to the BGP reflections for the τ-slice algebras.

A family of Koszul self-injective algebras with finite Hochschild cohomology

This paper presents an infinite family of Koszul self-injective algebras whose Hochschild cohomology ring is finite-dimensional. Moreover, for each N ⩾ 5 we give an example where the Hochschild cohomology ring has dimension N . This family of algebras includes and generalizes the 4-dimensional Koszul self-injective local algebras of [R.-O. Buchweitz, E.L. Green, D. Madsen, Ø. Solberg, Finite Hochschild cohomology without finite global dimension, Math. Res. Lett. 12 (2005) 805–816] which were used to give a negative answer to Happel’s question, in that they have infinite global dimension but finite-dimensional Hochschild cohomology.

Faces of polytopes and Koszul algebras

Let g be a semisimple Lie algebra and V a g-semisimple module. In this paper, we study the category G of Z-graded finite-dimensional representations of g⋉V. We show that the simple objects in this category are indexed by an interval-finite poset and produce a large class of truncated subcategories which are directed and highest weight. In the case when V is a finite-dimensional g-module, we construct a family of Koszul algebras which are indexed by certain subsets of the set of weights wt(V) of V. We use these Koszul algebras to construct an infinite-dimensional graded subalgebra AΨg of the locally finite part of the algebra of invariants (EndC(V)⊗SymV)g, where V is the direct sum of all simple finite-dimensional g-modules. We prove that AΨg is Koszul of finite global dimension.

Affine cellular algebras

Graham and Lehrer have defined cellular algebras and developed a theory that allows in particular to classify simple representations of finite dimensional cellular algebras. Many classes of finite dimensional algebras, including various Hecke algebras and diagram algebras, have been shown to be cellular, and the theory due to Graham and Lehrer successfully has been applied to these algebras. We will extend the framework of cellular algebras to algebras that need not be finite dimensional over a field. Affine Hecke algebras of type A and infinite dimensional diagram algebras like the affine Temperley–Lieb algebras are shown to be examples of our definition. The isomorphism classes of simple representations of affine cellular algebras are shown to be parameterised by the complement of finitely many subvarieties in a finite disjoint union of affine varieties. In this way, representation theory of non-commutative algebras is linked with commutative algebra. Moreover, conditions on the cell chain are identified that force the algebra to have finite global cohomological dimension and its derived category to admit a stratification; these conditions are shown to be satisfied for the affine Hecke algebra of type A if the quantum parameter is not a root of the Poincaré polynomial.

Global dimension of Noetherian serial rings

The global dimension of Noetherian serial rings is studied. It is proved that if an indecomposable serial ring has infinite global dimension then it is Artinian and its quiver is a simple cycle. Using methods of the theory of right serial quivers, we give an upper estimate on the Loewy length of Artinian rings of finite global dimension. Applications to the calculation of the global dimension of tiled orders of width 2 are given.

On a common generalization of Koszul duality and tilting equivalence

We propose a new definition of Koszulity for graded algebras where the degree zero part has finite global dimension, but is not necessarily semi-simple. The standard Koszul duality theorems hold in this setting.

Ampleness of two-sided tilting complexes

In this paper we define the notion of ampleness for two-sided tilting complexes over finite dimensional algebras and prove its basic properties. We call a finite dimensional k-algebra A of finite global dimension Fano if (A∗[−d])−1 is ample for some d ≥ 0. For example geometric algebras in the sense of Bondal-Polishchuk are Fano. We give a characterization of representation type of a quiver from a noncommutative algebro-geometric view point, that is, a finite acyclic quiver has finite representation type if and only if its path algebra is fractional Calabi-Yau, and a finite acyclic quiver has infinite representation type if and only if its path algebra is Fano. 0 Introduction Let X be a nonsingular projective variety over a field k and let ωX be its canonical bundle. Then the functor SX := −⊗X ωX [dimX] : D(cohX) −→ D(cohX) is the Serre functor ,i.e., HomX(G ·,F ·)∗ is functorially isomorphic to HomX(F ·, SX(G ·)) for F ·,G · ∈ D(cohX). By this fact, from a noncommutative (or categorical) algebro-geometric view point, one thinks of a triangulated category T as the derived category of coherent sheaves of some ”space” X and of the Serre functor ST of T (if exists) as the derived tensor product of ” dimX”-shifted ”canonical bundle” ωX . From this view point, the notion of Calabi-Yau algebra ( and Calabi-Yau category ) is defined and studied extensively by many researchers. In this paper we introduce the notion of ampleness for two-sided tilting complexes over finite dimensional k-algebras. Let A be a finite dimensional k-algebra of finite global dimension. Definition 0.1 (Definition 2.6). A two-sided tilting complex σ over A is called very ample if H(σ) = 0 for i ≥ 1 and σ is pure for n A 0. σ is called ample if σ is pure for n A 0. In Section 2, we justify this definition by using the theory of noncommutative projective schemes due to Artin-Zhang [AZ] and Polishchuk [Po]. In the theory of noncommutative projective schemes , for a graded coherent ring R over k we attach an imaginary geometric object projR = ( cohprojR, R, (1) ) . An abelian category cohprojR is considered as the category of coherent sheaves on projR. (See Section 1.) In Section 2 we show that the following facts hold. If σ is a very ample tilting complex over A, then the tensor algebra T := TA(H (σ)) of H(σ) over A is a graded connected coherent ring over A and there is a t-structure D defined by σ in Perf A and its heart H is equivalent to cohprojT . Moreover the following Theorem holds. Theorem 0.2 (Theorem 2.8). Let A be a finite dimensional k-algebra of finite global dimension and let σ be a very ample two-sided tilting complex. Then there is a natural equivalence of triangulated categories D (mod-A) ∼ −→ D (cohprojT ) . where T := TA(H (σ)) is the tensor algebra of H(σ) over A.

Combinatorial classification of piecewise hereditary algebras

We use the characteristic polynomial of the Coxeter matrix of an algebra to complete the combinatorial classification of piecewise hereditary algebras which Happel gave in terms of the trace of the Coxeter matrix. We also give a cohomological interpretation of the coefficients (other than the trace) of the characteristic polynomial of the Coxeter matrix of any finite dimensional algebra with finite global dimension.

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.