Showing papers on "Global dimension published in 2015"

Modular categories as representations of the 3-dimensional bordism 2-category

Abstract: We show that once-extended anomalous 3-dimensional topological quantum field theories valued in the 2-category of k-linear categories are in canonical bijection with modular tensor categories equipped with a square root of the global dimension in each factor.

Non-commutative resolutions and grothendieck groups

Abstract: Let R be a noetherian normal domain. We investigate when R admits a faithful module whose endomorphism ring has finite global dimension. This can be viewed as a non- commutative desingularization of Spec(R). We show that the existence of such modules forces stringent conditions on the Grothendieck group of finitely generated modules over R. In some cases those conditions are enough to imply that Spec(R) has only rational singularities.

Noncommutative motives of Azumaya algebras

Abstract: Let be a base commutative ring, a commutative ring of coefficients, a quasi-compact quasi-separated -scheme, and a sheaf of Azumaya algebras over of rank . Under the assumption that , we prove that the noncommutative motives with -coefficients of and are isomorphic. As an application, we conclude that a similar isomorphism holds for every -linear additive invariant. This leads to several computations. Along the way we show that, in the case of finite-dimensional algebras of finite global dimension, all additive invariants are nilinvariant.

Combinatorial topology and the global dimension of algebras arising in combinatorics

Abstract: In a highly influential paper, Bidigare, Hanlon and Rockmore showed that a number of popular Markov chains are random walks on the faces of a hyperplane arrangement. Their analysis of these Markov chains took advantage of the monoid structure on the set of faces. This theory was later extended by Brown to a larger class of monoids called left regular bands. In both cases, the representation theory of these monoids played a prominent role. In particular, it was used to compute the spectrum of the transition operators of the Markov chains and to prove diagonalizability of the transition operators. In this paper, we establish a close connection between algebraic and combinatorial invariants of a left regular band: we show that certain homological invariants of the algebra of a left regular band coincide with the cohomology of order complexes of posets naturally associated to the left regular band. For instance, we show that the global dimension of these algebras is bounded above by the Leray number of the associated order complex. Conversely, we associate to every flag complex a left regular band whose algebra has global dimension precisely the Leray number of the flag complex.

On Orbifold Theory

Abstract: Let V be a simple vertex operator algebra and G a finite automorphism group of V such that V^G is regular. It is proved that every irreducible V^G-module occurs in an irreducible g-twisted V-module for some g in G. Moreover, the quantum dimensions of each irreducible V^G-module are determined and a global dimension formula for V in terms of twisted modules is obtained.

Donaldson-Thomas theory for categories of homological dimension one with potential

Abstract: The aim of the paper is twofold. Firstly, we give an axiomatic presentation of Donaldson-Thomas theory for categories of homological dimension at most one with potential. In particular, we provide rigorous proofs of all standard results concerning the integration map, wall-crossing, PT-DT correspondence, etc. following Kontsevich and Soibelman. We also show the equivalence of their approach and the one given by Joyce and Song. Secondly, we relate Donaldson-Thomas functions for such a category with arbitrary potential to those with zero potential under some mild conditions. As a result of this, we obtain a geometric interpretation of Donaldson-Thomas functions in all known realizations, i.e. mixed Hodge modules, perverse sheaves and constructible functions.

Noncommutative (Crepant) Desingularizations and the Global Spectrum of Commutative Rings

Abstract: In this paper we study endomorphism rings of finite global dimension over not necessarily normal commutative rings. These objects have recently attracted attention as noncommutative (crepant) resolutions, or NC(C)Rs, of singularities. We propose a notion of a NCCR over any commutative ring that appears weaker but generalizes all previous notions. Our results yield strong necessary and sufficient conditions for the existence of such objects in many cases of interest. We also give new examples of NCRs of curve singularities, regular local rings and normal crossing singularities. Moreover, we introduce and study the global spectrum of a ring R, that is, the set of all possible finite global dimensions of endomorphism rings of MCM R-modules. Finally, we use a variety of methods to compute global dimension for many endomorphism rings.

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 \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.

Extension Fullness of the Categories of Gelfand{Zeitlin and Whittaker Modules ?

Abstract: We prove that the categories of Gelfand{Zeitlin modules of g = gl n and Whit- taker modules associated with a semi-simple complex finite-dimensional algebra g are ex- tension full in the category of all g-modules. This is used to estimate and in some cases determine the global dimension of blocks of the categories of Gelfand{Zeitlin and Whittaker modules.

Affine Cellularity of Khovanov–Lauda–Rouquier Algebras of Finite Types

Abstract: We prove that the Khovanov-Lauda-Rouquier algebras $R_\alpha$ of finite type are (graded) affine cellular in the sense of Koenig and Xi. In fact, we establish a stronger property, namely that the affine cell ideals in $R_\alpha$ are generated by idempotents. This in particular implies the (known) result that the global dimension of $R_\alpha$ is finite.

Donaldson-Thomas invariants vs. intersection cohomology for categories of homological dimension one

Abstract: The present paper is an extension of a previous paper written in collaboration with Markus Reineke dealing with quiver representations. The aim of the paper is to generalize the theory and to provide a comprehensive theory of Donaldson-Thomas invariants for abelian categories of homological dimension one (without potential) satisfying some technical conditions. The theory will apply for instance to representations of quivers, coherent sheaves on smooth projective curves, and some coherent sheaves on smooth projective surfaces. We show that the (motivic) Donaldson-Thomas invariants satisfy the Integrality conjecture and identify the Hodge theoretic version with the (compactly supported) intersection cohomology of the corresponding moduli spaces of objects. In fact, we deal with a refined version of Donaldson-Thomas invariants which can be interpreted as classes in the Grothendieck group of some "sheaf" on the moduli space. In particular, we reproduce the intersection complex of moduli spaces using Donaldson-Thomas theory.

The Interplay of Algebra and Geometry in the Setting of Regular Algebras

Abstract: This article is based on a talk given by the author at MSRI in the workshop "Connections for Women" in January 2013, while being a part of the program "Noncommutative Algebraic Geometry and Representation Theory" at MSRI. One purpose of the exposition is to motivate and describe the geometric techniques introduced by M. Artin, J. Tate and M. Van den Bergh in the 1980s at a level accessible to graduate students. Additionally, some advances in the subject since the early 1990s are discussed, including a recent generalization of complete intersection to the noncommutative setting, and the notion of graded skew Clifford algebra and its application to classifying quadratic regular algebras of global dimension at most three. The article concludes by listing some open problems.

The derived category of a graded Gorenstein ring

Abstract: We give an exposition and generalization of Orlov's theorem on graded Gorenstein rings. We show the theorem holds for non-negatively graded rings which are Gorenstein in an appropriate sense and whose degree zero component is an arbitrary non-commutative right noetherian ring of finite global dimension. A short treatment of some foundations for local cohomology and Grothendieck duality at this level of generality is given in order to prove the theorem. As an application we give an equivalence of the derived category of a commutative complete intersection with the homotopy category of graded matrix factorizations over a related ring.

Dimension of elementary amenable groups

Abstract: This paper has three parts. It is conjectured that for every elementary amenable group G and every non-zero commutative ring k, the homological dimension hdk(G)is equal to the Hirsch length h(G) whenever G has no k-torsion. In Part I this conjecture is proved for several classes, including the abelian-by-polycyclic groups. In Part II it is shown that the elementary amenable groups of homological dimension one are colimits of systems of groups of cohomological dimension one. In Part III the deep problem of calculating the cohomological dimension of elementary amenable groups is tackled with particular emphasis on the nilpotent-by-polycyclic case, where a complete answer is obtained over Q for countable groups

Regular modules over 2-dimensional quantum Beilinson algebras of Type S

Abstract: In the study of a finite dimensional hereditary algebra of infinite representation type, understanding regular modules is essential. Recently, Herschend, Iyama and Oppermann introduced the notions of \(d\)-representation infinite algebra and \(d\)-regular module, extending the above notions to finite dimensional algebras of global dimension \(d\ge 1\). Since the Beilinson algebras of AS-regular algebras of dimension \(d+1\) are typical examples of \(d\)-representation infinite algebras, the purpose of this paper is to study the behavior of \(d\)-regular modules over such algebras. In particular, we will show that the isomorphism classes of simple 2-regular modules over a 2-representation tame quantum Beilinson algebra of Type \(S\) are parameterized by \({\mathbb {P}}^{2}\).

Torsion pairs and filtrations in abelian categories with tilting objects

Abstract: Given a noetherian abelian k-category of finite homological dimension, with a tilting object T of projective dimension 2, the abelian category and the abelian category of modules over End(T)op are related by a sequence of two tilts; we give an explicit description of the torsion pairs involved. We then use our techniques to obtain a simplified proof of a theorem of Jensen–Madsen–Su, that has a three-step filtration by extension-closed subcategories. Finally, we generalize Jensen–Madsen–Su's filtration to the case where T has any finite projective dimension.

Finitistic Dimensions And Piecewise Hereditary Property Of Skew Group Algebras

Abstract: Let Λ be a finite-dimensional algebra and G be a finite group whose elements act on Λ as algebra automorphisms. Assume that Λ has a complete set E of primitive orthogonal idempotents, closed under the action of a Sylow p-subgroup S ≤ G. If the action of S on E is free, we show that the skew group algebra Λ G and Λ have the same finitistic dimension, and have the same strong global dimension if the fixed algebra ΛS is a direct summand of the ΛS-bimodule Λ. Using a homological characterization of piecewise hereditary algebras proved by Happel and Zacharia, we deduce a criterion for Λ G to be piecewise hereditary.

Graded Clifford Algebras of Prime Global Dimension with an Action of H p

Abstract: In this article, we study graded Clifford algebras with a gradation preserving action of automorphisms given by H p , the Heisenberg group of order p 3 with p prime. After reviewing results in dimensions 3 and 4, we will determine the graded Clifford algebras that are AS-regular algebras of global dimension 5 and generalize certain results to arbitrary dimension p.

On commutative rings with uniserial dimension

Abstract: In this paper, we define and study a valuation dimension for commutative rings. The valuation dimension is a measure of how far a commutative ring deviates from being valuation. It is shown that a ring R with valuation dimension has finite uniform dimension. We prove that a ring R is Noetherian (respectively, Artinian) if and only if the ring R × R has (respectively, finite) valuation dimension if and only if R has (respectively, finite) valuation dimension and all cyclic uniserial modules are Noetherian (respectively, Artinian). We show that the class of all rings of finite valuation dimension strictly lies between the class of Artinian rings and the class of semi-perfect rings.

Embeddings of algebras in derived categories of surfaces

Abstract: By a result of Orlov there always exists an embedding of the derived category of a finite-dimensional algebra of finite global dimension into the derived category of a high-dimensional smooth projective variety. In this article we give some restrictions on those algebras whose derived categories can be embedded into the bounded derived category of a smooth projective surface. This is then applied to obtain explicit results for hereditary algebras.

The global dimension of the full transformation monoid

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 Ponizovskii, 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 $n\geq 1$ and, moreover, provide an explicit minimal projective resolution of the trivial module of length $n-1$.

Local Dimension of Coatomic Modules

Abstract: We define and study local dimension for coatomic modules. Local dimension is a measure of how far a coatomic module deviates from being local. Every Noetherian module has local dimension. It is shown that a ring R with finite local dimension is semilocal. We study rings over which modules are coatomic and have local dimension. We show that, for a ring R, every right R-module is coatomic and has local dimension if and only if the free right R-module is coatomic and has local dimension, if and only if R is a semisimple Artinan ring. We obtain a characterization of right Artinian rings as those right Noetherian rings over which every finitely generated right module has finite local dimension. We show that a commutative ring R has (resp. finite) local dimension if and only if R is either Noetherian (resp. Artinian) or local.

On Gorenstein flat dimension

Abstract: The theory of Gorenstein flat dimension is not complete since it is not yet known whether the category 𝒢ℱ(R) of Gorenstein flat modules over a ring R is projectively resolving or not. Besides, it arises from recent investigations on this subject that there exists several ways of measuring the Gorenstein flat dimension of modules which turn out to coincide with the usual one in the case where 𝒢ℱ(R) is projectively resolving. These alternate procedures yield new invariants which enjoy very nice behavior for an arbitrary ring R. In this paper, we introduce and study one of these invariants called the cover Gorenstein flat dimension of a module M and denoted by CGfdR(M). This new entity stems from a sort of a Gorenstein flat precover of M. First, for each R-module M, we prove that GfdR(M) ≤ CGfdR(M) for each R-module M with whenever CGfdR(M) is finite. Also, we show that 𝒢ℱ(R) is projectively resolving if and only if the Gorenstein flat dimension and the introduced cover Gorenstein flat dimension coincide. In particular, if R is a right coherent ring, then CGfdR(M) = GfdR(M) for any R-module M. As a consequence, we prove that if R is a left and right GF-closed, then the Gorenstein weak global dimension of R is left–right symmetric and it is related to the cohomological invariants leftsfli(R) and rightsfli(R) by the formula

Cocycle twists of algebras

Abstract: Let $A$ be a $k$-algebra where $k$ is an algebraically closed field and $G$ be a finite abelian group for which the characteristic of $k$ does not divide $|G|$. If $G$ acts on $A$ by $k$-algebra automorphisms then the action induces a $G$-grading on $A$ which, in conjunction with a normalised 2-cocycle of the group, can be used to twist the multiplication of the algebra. Such twists can be formulated as Zhang twists as well as in the language of Hopf algebras. We investigate such cocycle twists with an emphasis on the situation where $A$ also possesses a connected graded structure and the action of $G$ respects this grading. We show that many properties are preserved under such twists; for example the strongly noetherian property, finite global dimension and Artin-Shelter regularity. The above concepts are then applied to the 4-dimensional Sklyanin algebras, $A:=A(\alpha,\beta,\gamma)$. We define an action of the Klein four-group on $A$ such that the action restricts to a special geometric factor ring $B$ (a twisted homogeneous coordinate ring). The cocycle twists of these algebras, denoted by $A^{G,\mu}$ and $B^{G,\mu}$ respectively, have very different geometric properties to their untwisted counterparts. While $A$ has point modules parameterised by an elliptic curve $E$ and four extra points, $A^{G,\mu}$ has only 20 point modules (when an automorphism associated to $E$ has infinite order). The point modules over $A$ can be used to construct fat point modules of multiplicity 2 over $A^{G,\mu}$, and there are isomorphisms among such objects corresponding to orbits of a natural action of $G$ on $E$. Furthermore, the ring $B^{G,\mu}$ can be described in terms of Artin and Stafford's classification of noncommutative curves.

Silted algebras

Abstract: We study endomorphism algebras of 2-term silting complexes in derived categories of hereditary finite dimensional algebras, or more generally of $\mathop{\rm Ext} olimits$-finite hereditary abelian categories. Module categories of such endomorphism algebras are known to occur as hearts of certain bounded $t$-structures in such derived categories. We show that the algebras occurring are exactly the algebras of small homological dimension, which are algebras characterized by the property that each indecomposable module either has injective dimension at most one, or it has projective dimension at most one.

Indecomposable decomposition and couniserial dimension

Abstract: Dimensions like Gelfand, Krull, Goldie have an intrinsic role in the study of theory of rings and modules. They provide useful technical tools for studying their structure. We define and study couniserial dimension for modules. Couniserial dimension is a measure of how far a module deviates from being uniform. Despite their different objectives, it turns out that there are certain common properties between the couniserial dimension and Krull dimension. Among others, each module having such a dimension contains a uniform submodule and has finite uniform dimension. Like all dimensions, this is an ordinal valued invariant. Every module of finite length has couniserial dimension and its value lies between the uniform dimension and the length of the module. Modules with countable couniserial dimension are shown to possess indecomposable decomposition. In particular, a von Neumann regular ring with countable couniserial dimension is semisimple artinian. If the maximal right quotient ring of a semiprime right non-singular ring \(R\) has a couniserial dimension as an \(R\)-module, then \(R\) is a semiprime right Goldie ring. As one of the applications, it follows that all right \(R\)-modules have couniserial dimension if and only if \(R\) is a semisimple artinian ring.