Showing papers on "Global dimension published in 2016"

Quantum Binary Polyhedral Groups And Their Actions On Quantum Planes

Abstract: We classify quantum analogues of actions of finite subgroups G of SL_2(k) on commutative polynomial rings k[u,v]. More precisely, we produce a classification of pairs (H,R), where H is a finite dimensional Hopf algebra that acts inner faithfully and preserves the grading of an Artin-Schelter regular algebra R of global dimension two. Remarkably, the corresponding invariant rings R^H share similar regularity and Gorenstein properties as the invariant rings k[u,v]^G in the classic setting. We also present several questions and directions for expanding this work in noncommutative invariant theory.

Nakayama automorphism and applications

Abstract: Nakayama automorphism is used to study group actions and Hopf algebra actions on Artin-Schelter regular algebras of global dimension three.

On quantum groups associated to non-Noetherian regular algebras of dimension 2

Abstract: We investigate homological and ring-theoretic properties of universal quantum linear groups that coact on Artin-Schelter regular algebras A(n) of global dimension 2, especially with central homological codeterminant (or central quantum determinant). As classified by Zhang, the algebras A(n) are connected $$\mathbb {N}$$ -graded algebras that are finitely generated by n indeterminants of degree 1, subject to one quadratic relation. In the case when the homological codeterminant of the coaction is trivial, we show that the quantum group of interest, defined independently by Manin and by Dubois-Violette and Launer, is Artin-Schelter regular of global dimension 3 and also skew Calabi-Yau (homologically smooth of dimension 3). For central homological codeterminant, we verify that the quantum groups are Noetherian and have finite Gelfand-Kirillov dimension precisely when the corresponding comodule algebra A(n) satisfies these properties, that is, if and only if $$n=2$$ . We have similar results for arbitrary homological codeterminant if we require that the quantum groups are involutory. We also establish conditions when Hopf quotients of these quantum groups, that also coact on A(n), are cocommutative.

Tilting bundles on orders on ℙ d

Abstract: We introduce a class of orders on ℙ d called Geigle-Lenzing orders and show that they have tilting bundles. Moreover we show that their module categories are equivalent to the categories of coherent sheaves on Geigle- Lenzing projective spaces introduced in [HIMO].

Stratifications of algebras with two simple modules

Abstract: Let A be a finite-dimensional algebra with two simple modules. It is shown that if the derived category of A admits a stratification with simple factors being the base field k, then A is derived equivalent to a quasi-hereditary algebra. As a consequence, if further k is algebraically closed and A has finite global dimension, then A is either derived simple or derived equivalent to a quasi-hereditary algebra. MSC 2010 classification: 16E35, 16E40, 16E45.

McKay Correspondence for semisimple Hopf actions on regular graded algebras, II

Abstract: In establishing a more general version of the McKay correspondence, we prove Auslander's theorem for actions of semisimple Hopf algebras H on noncommutative Artin-Schelter regular algebras A of global dimension two, where A is a graded H-module algebra, and the Hopf action on A is inner faithful with trivial homological determinant. We also show that each fixed ring A^H under such an action arises an analogue of a coordinate ring of a Kleinian singularity.

On the center of 3-dimensional and 4-dimensional Sklyanin algebras

Abstract: In this article, a new proof is given of the description of the center of quadratic Sklyanin algebras of global dimension three and four and the center of cubic Sklyanin algebras of global dimension three. The representation theory of the Heisenberg groups $H_2$, $H_3$ and $H_4$ will play an important role. In addition a new proof is given of Van den Bergh's result regarding noncommutative quadrics.

Relative Projective and Injective Dimensions

Abstract: We study the concepts of the 𝒫C-projective and the ℐC-injective dimensions of a module in the noncommutative case, weakening the condition of C being semidualizing. We give the relations between these dimensions and the C-relative Gorenstein dimensions (GC-projective and GC-injective dimensions) of the module. Finally, we compare, in some circumstances, the global 𝒫C-projective dimension of a ring and the global dimension of the endomorphisms ring of C.

A description of Gorenstein projective modules over the tensor products of algebras

Abstract: Let $A$ be a coherent algebra and $B$ be a finite-dimensional Gorenstein algebra over a field $k$. We describe finitely presented Gorenstein projective $A\otimes_k B$-modules in terms of their underlying onesided modules. Moreover, if the global dimension of $B$ is finite, we give a more precise description of finitely presented Gorenstein projective $A\otimes_k B$-modules.

PBW deformations of Artin–Schelter regular algebras

Abstract: We consider properties and extensions of PBW deformations of Artin–Schelter regular algebras. PBW deformations in global dimension two are classified and the geometry associated to the homogenizations of these algebras is exploited to prove that all simple modules are one-dimensional in the non-PI case. It is shown that this property is maintained under tensor products and certain skew polynomial extensions.

Derived equivalences, restriction to self-injective subalgebras and invariance of homological dimensions

Abstract: Derived equivalences between finite dimensional algebras do, in general, not pass to centraliser (or other) subalgebras, nor do they preserve homological invariants of the algebras, such as global or dominant dimension. We show that, however, they do so for large classes of algebras described in this article. Algebras $A$ of $ u$-dominant dimension at least one have unique largest non-trivial self-injective centraliser subalgebras $H_A$. A derived restriction theorem is proved: A derived equivalence between $A$ and $B$ implies a derived equivalence between $H_A$ and $H_B$. Two methods are developed to show that global and dominant dimension are preserved by derived equivalences between algebras of $ u$-dominant dimension at least one with anti-automorphisms preserving simples, and also between almost self-injective algebras. One method is based on identifying particular derived equivalences preserving homological dimensions, while the other method identifies homological dimensions inside certain derived categories. In particular, derived equivalent cellular algebras have the same global dimension. As an application, the global and dominant dimensions of blocks of quantised Schur algebras with $n \geq r$ are completely determined.

Homological dimensions for co-rank one idempotent subalgebras

Abstract: Let $k$ be an algebraically closed field and $A$ be a (left and right) Noetherian associative $k$-algebra. Assume further that $A$ is either positively graded or semiperfect (this includes the class of finite dimensional $k$-algebras, and $k$-algebras that are finitely generated modules over a Noetherian central Henselian ring). Let $e$ be a primitive idempotent of $A$, which we assume is of degree $0$ if $A$ is positively graded. We consider the idempotent subalgebra $\Gamma = (1-e)A(1-e)$ and $S_e$ the simple right $A$-module $S_e = eA/e{\rm rad}A$, where ${\rm rad}A$ is the Jacobson radical of $A$, or the graded Jacobson radical of $A$ if $A$ is positively graded. In this paper, we relate the homological dimensions of $A$ and $\Gamma$, using the homological properties of $S_e$. First, if $S_e$ has no self-extensions of any degree, then the global dimension of $A$ is finite if and only if that of $\Gamma$ is. On the other hand, if the global dimensions of both $A$ and $\Gamma$ are finite, then $S_e$ cannot have self-extensions of degree greater than one, provided $A/{\rm rad}A$ is finite dimensional.

The Global Dimension of the full Transformation Monoid (with an Appendix by V. Mazorchuk and B. Steinberg)

Abstract: The representation theory of the symmetric group has been intensively studied for over 100 years and is one of the gems of modern mathematics. The full transformation monoid \(\mathfrak {T}_{n}\) (the monoid of all self-maps of an n-element set) is the monoid analogue of the symmetric group. The investigation of its representation theory was begun by Hewitt and Zuckerman in 1957. Its character table was computed by Putcha in 1996 and its representation type was determined in a series of papers by Ponizovskiĭ, Putcha and Ringel between 1987 and 2000. From their work, one can deduce that the global dimension of \(\mathbb {C}\mathfrak {T}_{n}\) is n−1 for n = 1, 2, 3, 4. We prove in this paper that the global dimension is n−1 for all n ≥ 1 and, moreover, we provide an explicit minimal projective resolution of the trivial module of length n−1. In an appendix with V. Mazorchuk we compute the indecomposable tilting modules of \(\mathbb {C}\mathfrak T_{n}\) with respect to Putcha’s quasi-hereditary structure and the Ringel dual (up to Morita equivalence).

On α -almost Artinian modules

Abstract: Abstract In this article we introduce and study the concept of α-almost Artinian modules. We show that each α-almost Artinian module M is almost Artinian (i.e., every proper homomorphic image of M is Artinian), where α ∈ {0,1}. Using this concept we extend some of the basic results of almost Artinian modules to α-almost Artinian modules. Moreover we introduce and study the concept of α-Krull modules. We observe that if M is an α-Krull module then the Krull dimension of M is either α or α + 1.

Artinian Serial Modules over Commutative (or, Left Noetherian) Rings are at Most One Step Away from Being Noetherian

Abstract: We introduce and study the concept of dual perfect dimension which is a Krull-like dimension extension of the concept of acc on finitely generated submodules. We observe some basic facts for modules with this dimension, which are similar to the basic properties of modules with Noetherian dimension. For Artinian serial modules, we show that these two dimensions coincide. Consequently, we prove that the Noetherian dimension of non-Noetherian Artinian serial modules over the rings of the title is 1.

Cluster tilting modules and noncommutative projective schemes

Abstract: In this paper, we study the relationship between equivalences of noncommutative projective schemes and cluster tilting modules. In particular, we prove the following result. Let $A$ be an AS-Gorenstein algebra of dimension $d\geq 2$ and ${\mathsf{tails}\,} A$ the noncommutative projective scheme associated to $A$. If $\operatorname{gldim}({\mathsf{tails}\,} A)< \infty$ and $A$ has a $(d-1)$-cluster tilting module $X$ satisfying that its graded endomorphism algebra is $\mathbb N$-graded, then the graded endomorphism algebra $B$ of a basic $(d-1)$-cluster tilting submodule of $X$ is a two-sided noetherian $\mathbb N$-graded AS-regular algebra over $B_0$ of global dimension $d$ such that ${\mathsf{tails}\,} B$ is equivalent to ${\mathsf{tails}\,} A$.

On Modules over Infinite Group Rings

Abstract: Let R be a commutative ring and Γ be an infinite discrete group. The algebraic K-theory of the group ring R[Γ] is an important object of computation in geometric topology and number theory. When the group ring is Noetherian, there is a companion G-theory of R[Γ] which is often easier to compute. However, it is exceptionally rare that the group ring is Noetherian for an infinite group. In this paper, we define a version of G-theory for any finitely generated discrete group. This construction is based on the coarse geometry of the group. It has some expected properties such as independence from the choice of a word metric. We prove that, whenever R is a regular Noetherian ring of finite global homological dimension and Γ has finite asymptotic dimension and a finite model for the classifying space BΓ, the natural Cartan map from the K-theory of R[Γ] to G-theory is an equivalence. On the other hand, our G-theory is indeed better suited for computation as we show in a separate paper. Some results and constructions in this paper might be of independent interest as we learn to construct projective resolutions of finite type for certain modules over group rings.

Relative Gorenstein global dimension

Abstract: We study the relative Gorenstein projective global dimension of a ring with respect to a weakly Wakamatsu tilting module C. We prove that this relative global dimension is finite if and only if the injective dimension of every module in Add(C) and the 𝒫C-projective dimension of every injective module are both finite (indeed these three dimensions have a common upper bound). When RC satisfies some extra conditions we prove that the relative Gorenstein projective global dimension of R is always bounded above by the 𝒫C-projective global dimension of R, these two dimensions being equal when the class of all C-Gorenstein projective R-modules is contained in the Bass class of R relative to C. Of course we also give the dual results concerning the relative Gorenstein injective global dimension.

On Pure Derived Categories

Abstract: We investigate the properties of pure derived categories of module categories, and show that pure derived categories share many nice properties of classical derived categories. In particular, we show that bounded pure derived categories can be realized as certain homotopy categories. We introduce the pure projective (resp. injective) dimension of complexes in pure derived categories, and give some criteria for computing these dimensions in terms of the properties of pure projective (resp. injective) resolutions and pure derived functors. As a consequence, we get some equivalent characterizations for the finiteness of the pure global dimension of rings. Finally, pure projective (resp. injective) resolutions of unbounded complexes are considered.

Derived invariants for surface cut algebras II: the punctured case

Abstract: For each algebra of global dimension 2 arising from the quiver with potential associated to a triangulation of an unpunctured surface, Amiot-Grimeland have defined an integer-valued function on the first singular homology group of the surface, and have proved that two such algebras of global dimension 2 are derived equivalent precisely when there exists an automorphism of the surface that makes their associated functions coincide. In the present paper we generalize the constructions and results of Amiot-Grimeland to the setting of arbitrarily punctured surfaces. As an application, we show that there always is a derived equivalence between any two algebras of global dimension 2 arising from the quivers with potential of (valency at least 2) triangulations of arbitrarily punctured polygons. While in the unpunctured case the quiver with potential of any triangulation admits cuts yielding algebras of global dimension at most 2, in the case of punctured surfaces the QPs of some triangulations do not admit cuts, and even when they do, the global dimension of the corresponding degree-0 algebra may exceed 2. In this paper we give a combinatorial characterization of each of these two situations.

Endomorphism algebras of 2-term silting complexes

Abstract: We study possible values of the global dimension of endomorphism algebras of 2-term silting complexes. We show that for any algebra $A$ whose global dimension $\mathop{\rm gl. dim} olimits A\leq 2$ and any 2-term silting complex $\mathbf{P}$ in the bounded derived category ${D^b(A)}$ of $A$, the global dimension of $\mathop{\rm End} olimits_{D^b(A)}(\mathbf{P})$ is at most 7. We also show that for each $n>2$, there is an algebra $A$ with $\mathop{\rm gl. dim} olimits A=n$ such that ${D^b(A)}$ admits a 2-term silting complex $\mathbf{P}$ with $\mathop{\rm gl. dim} olimits \mathop{\rm End} olimits_{D^b(A)}(\mathbf{P})$ infinite.