Showing papers on "Global dimension published in 2004"
TL;DR: In this paper, the closely related Gorenstein projective, Goren stein injective and 2-at dimensions of modules are studied, and a generalization of these results is given to give homological descriptions of the GORNE dimensions over arbitrary associative rings.
762 citations
TL;DR: In this article, generalizations of the definitions of one-dimensional tilting and cotilting modules are considered. But they do not consider the notions of tilting modules of finite homological dimension.
104 citations
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TL;DR: In this paper, the authors give a negative answer to the question: if the n-th Hochschild cohomology group of a finite-dimensional algebra over a field vanishes for all sufficiently large n, is the global dimension of the Hochmoor group finite?
Abstract: Dieter Happel asked the following question: If the $n$-th Hochschild cohomology group of a finite dimensional algebra $\Gamma$ over a field vanishes for all sufficiently large $n$, is the global dimension of $\Gamma$ finite? We give a negative answer to this question.
85 citations
TL;DR: In this paper, the authors studied a class of noncommutative surfaces, and their higher dimensional analogs, which come from generic subalgebras of twisted homogeneous coordinate rings of projective space.
42 citations
TL;DR: In this paper, it was shown that the finitistic dimension of a properly stratified algebra having a simple preserving duality and for which every tilting module is cotilting, equals twice the projective dimension of the characteristic tilting modules.
42 citations
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TL;DR: In this article, the authors constructed four families of Artin-Schelter regular algebras of global dimension four, which are strongly noetherian, Auslander regular and Cohen-Macaulay.
Abstract: We construct four families of Artin-Schelter regular algebras of global dimension four. Under some generic conditions, this is a complete list of Artin-Schelter regular algebras of global dimension four that are generated by two elements of degree 1. These algebras are also strongly noetherian, Auslander regular and Cohen-Macaulay. One of the main tools is Keller's higher-multiplication theorem on A-infinity Ext-algebras.
34 citations
TL;DR: In this article, the authors studied noetherian graded idealizer rings which have very different behavior on the right and left sides of the algebraically closed field k. In particular, they showed that T satisfies left χd for some d⩾2 yet fails right χ1.
29 citations
TL;DR: A possible generalization of the Serre problem (Quillen-Suslin theorem) on the freeness of projective modules over polynomial rings, which seems to be of interest in systems theory applications, is shown in this article.
28 citations
TL;DR: In this article, the authors studied the relationship between equivalence classes of Krull-Schmidt categories with additive generators M and Morita equivalences classes of semiperfect rings, which is given by C 7→ C(M, M) and the converse is given for the category pr of finitely generated projective -modules.
Abstract: We will study the relationship of quite different object in the theory of artin algebras, namely Auslander-regular rings of global dimension two, torsion theories, �-categories and almost abelian categories. We will apply our results to characterization problems of Auslander-Reiten quivers. 0.1 There exists a bijection between equivalence classes of Krull-Schmidt categories C with additive generators M and Morita-equivalence classes of semiperfect rings , which is given by C 7→ C(M, M) and the converse is given by 7→ pr for the category pr of finitely generated projective -modules. Although th is bijection itself is rather formal, it will be very fruitful to study the relationship of (A)-(D) below. The object of this paper is to study it under the assumption that is an arti n algebra.
23 citations
TL;DR: The notion of finitistic Δ-filtration dimension was introduced in this paper for algebraic groups, and it was shown that the finitist dimension for such algebras is bounded by the sum of the Δ-filtering dimension and the Weyl dimension.
Abstract: In this paper, we discuss generalizations of the concepts of good filtration dimension and Weyl filtration dimension, introduced by Friedlander and Parshall for algebraic groups, to properly stratified algebras. We introduce the notion of the finitistic Δ-filtration dimension for such algebras and show that the finitistic dimension for such an algebra is bounded by the sum of the finitistic Δ-filtration dimension and the -filtration dimension. In particular, the finitistic dimension must be finite. We also conjecture that this bound is exact when the algebra has a simple preserving duality. We give several examples of well-known algebras where this is the case, including many of the Schur algebras, and blocks of category 𝒪. We also give an explicit combinatorial formula for the global dimension in this case.
18 citations
TL;DR: It was shown in this paper that every submodule of a module with countable Noetherian dimension is countably generated, and that ωω is a universal upper bound for the lengths of all chains in Artinian modules over commutative rings.
Abstract: It is shown that a module M has countable Noetherian dimension if and only if the lengths of ascending chains of submodules of M has a countable upper bound. This shows in particular that every submodule of a module with countable Noetherian dimension is countably generated. It is proved that modules with Noetherian dimension over locally Noetherian rings have countable Noetherian dimension. We also observe that ωω is a universal upper bound for the lengths of all chains in Artinian modules over commutative rings.
TL;DR: It is shown that the module of splines on a polyhedral complex can be viewed as the syzygy module of its dual graph with edges weighted by powers of linear forms, and it can be used to compute the homological dimension and the Hilbert series of the module.
Abstract: The module of splines on a polyhedral complex can be viewed as the syzygy module of its dual graph with edges weighted by powers of linear forms. When the assignment of linear forms to edges meets certain conditions, we can decompose the graph into disjoint cycles without changing the isomorphism class of the syzygy module. Thus we can use this decomposition to compute the homological dimension and the Hilbert series of the module. We provide alternate proofs of some results of Schenck and Stillman, extending those results to the polyhedral case. We also provide examples which illustrate the role that geometry plays in determining the syzygy module.
TL;DR: In this article, it was shown that a nitely generated elementary amenable group G of cohomological dimension 2 possesses a nite 2D model for BG and in particular that G is nitely presented and the trivial ZG-module Z has a 2D resolution by nitely
Abstract: A group has homological dimension 1 if it is locally free. We prove the converse provided that G satises the Atiyah Conjecture about L 2 -Betti numbers. We also show that a nitely generated elementary amenable group G of cohomological dimension 2 possesses a nite 2dimensional model for BG and in particular that G is nitely presented and the trivial ZG-module Z has a 2-dimensional resolution by nitely
21 Apr 2004
TL;DR: In this paper, the authors investigated two long-standing conjectures on the Krull dimension of integer-valued polynomial rings and of polynomials in the context of (locally) essential domains.
Abstract: This note investigates two long-standing conjectures on the Krull dimension of integer-valued polynomial rings and of polynomial rings in the context of (locally) essential domains.
TL;DR: In this paper, the authors consider quasi-hereditary algebras with simple preserving duality and show that the global dimension of the algebra is twice its ∇-filtration dimension.
TL;DR: In this paper, it was shown that ΛG falls into the class of rings studied by Brown, Hajarnavis and MacEacharn in [1], where they considered various properties of Noetherian rings R of finite global dimension, including the Krull(Gabriel-Rentschler) dimension K(R) which measures how far R is from being Artinian.
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TL;DR: In this article, it was shown that for a finite R$-module of positive depth, the ring has property H. The notion of homological dimension of a module, denoted H-dim, was introduced and proved in a Noetherian local ring.
Abstract: Let $(R,\m)$ be a Noetherian local ring. Consider the notion of homological dimension of a module, denoted H-dim, for H= Reg, CI, CI$_*$, G, G$^*$ or CM. We prove that, if for a finite $R$-module $M$ of positive depth, $\Hd_R({\m}^iM)$ is finite for some $i \geq \reg(M)$, then the ring $R$ has property H.
TL;DR: The strong global dimension of a finite dimensional algebra A is defined as the maximum of the width of indecomposable bounded differential complexes of finite dimensional projective algebras as discussed by the authors.
Abstract: The strong global dimension of a finite dimensional algebra A is the maximum of the width of indecomposable bounded differential complexes of finite dimensional projective A-modules. We prove that the strong global dimension of a finite dimensional radical square zero algebra A over an algebraically closed field is finite if and only if A is piecewise hereditary. Moreover, we discuss results concerning the finiteness of the strong global dimension of algebras and the related problem on the density of the push-down functors associated to the canonical Galois coverings of the trivial extensions of algebras by their repetitive algebras.
01 Jan 2004
TL;DR: In this paper, the authors generalized Parker's notion of good filtration dimensions to the setting of standardly stratified algebras and proved that the projective dimension of the characteristic module of A is a good dimension.
Abstract: ∇-good filtration dimensions of modules and of algebras are introduced by Parker for quasi-hereditary algebras. These concepts are now generalized to the setting of standardly stratified algebras. Let A be a standardly stratified algebra. The -good filtration dimension of A is proved to be the projective dimension of the characteristic module of A. Several characterizations of -good filtration dimensions and -good filtration dimensions are given for properly stratified algebras. Finally we give an application of these results to the global dimensions of quasi-hereditary algebras with exact Borel subalgebra.
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TL;DR: In this paper, the authors define three new homological dimensions -Cohen-Macaulay injective, projective, and flat dimension -which inhabit a theory similar to that of classical injective and projective dimension.
Abstract: We define three new homological dimensions - Cohen-Macaulay injective, projective, and flat dimension - which inhabit a theory similar to that of classical injective, projective, and flat dimension. Finiteness of the new dimensions characterizes Cohen-Macaulay rings with dualizing modules.
TL;DR: In this article, the relationship between weak and global dimension, strong n-coherence, and the (n, d)-property of a commutative ring and its sub-rings is studied.
Abstract: In this paper we study the relationship between some homological properties such as the weak and the global dimension, the strong n-coherence, and the (n, d)-property, where n and d are two integers, of a commutative ring and its subrings retract A special application is the transfer of these properties from a commutative ring to its fixed subring with respect to a subgroup of its group of automorphisms It concludes with a discussion of the scopes and limits of our results
TL;DR: In this article, an approach to cohomological dimension theory based on in-nite symmetric products and on the general theory of dimension called the extension di-mension is presented.
Abstract: We present an approach to cohomological dimension theory based on in- nite symmetric products and on the general theory of dimension called the extension di- mension. The notion of the extension dimension ext-dim(X) was introduced by A. N. Dran- ishnikov (9) in the context of compact spaces and CW complexes. This paper investigates extension types of innite symmetric products SP(L). One of the main ideas of the paper is to treat ext-dim(X) SP(L) as the fundamental concept of cohomological dimension theory instead of dimG(X) n. In a subsequent paper (18) we show how properties of innite symmetric products lead naturally to a calculus of graded groups which implies most of the classical results on the cohomological dimension. The basic notion in (18) is that of homological dimension of a graded group which allows for simultaneous treatment of cohomological dimension of compacta and extension properties of CW complexes. We introduce cohomology of X with respect to L (dened as homotopy groups of the function space SP(L) X ). As an application of our results we characterize all countable groups G so that the Moore space M(G;n) is of the same extension type as the Eilenberg{ MacLane space K(G;n). Another application is a characterization of innite symmetric products of the same extension type as a compact (or nite-dimensional and countable) CW complex. Contents
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TL;DR: In this paper, the authors studied under which conditions invertible modules over a commutative S-algebra give rise to elements in the algebraic Picard group of invertibly graded modules over the coefficient ring by taking homotopy groups.
Abstract: The aim of this note is to understand under which conditions invertible modules over a commutative S-algebra in the sense of Elmendorf, Kriz, Mandell and May give rise to elements in the algebraic Picard group of invertible graded modules over the coefficient ring by taking homotopy groups. If a connective commutative S-algebra R has coherent localizations (R_*)_m for every maximal ideal m in R_*, then for every invertible R-module U, U_* is an invertible graded R_*-module. In some non-connective cases we can carry the result over under the additional assumption that the commutative S-algebra has `residue fields' for all maximal ideals m in R_* if the global dimension of R_* is small or if R is 2-periodic with underlying Noetherian complete local regular ring R_0.
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TL;DR: In this paper, necessary and sufficient conditions are given for the completed group algebras of a compact p-adic analytic group with coefficient ring the padic integers or the field of p elements to be prime, semiprime and a domain.
Abstract: Necessary and sufficient conditions are given for the completed group algebras of a compact p-adic analytic group with coefficient ring the p-adic integers or the field of p elements to be prime, semiprime and a domain. Necessary and sufficient conditions for the localisation at semiprime ideals related to the augmentation ideals of closed normal subgroups are found. Some information is obtained about the Krull and global dimensions of the localisations. The results extend and complete work of A. Neumann and J. Coates et al.
TL;DR: In this article, the concept of m -adic pn-basis is introduced, and it is shown that R/Rpn has an m-adic Pn-Basis for every n (n=1,2,…) if and only if R is a regular local ring.
TL;DR: It is proved thatany rigid left Noetherian ring is either a domain or isomorphic to some ring ℤ p n of integers modulo a prime power p n .
Abstract: We prove thatany rigid left Noetherian ring is either a domain
or isomorphic to some ring ℤ p n of integers modulo
a prime power p n .
TL;DR: In this paper, a new homological dimension, called GCM-dimension, was defined for any finitely generated module M over a local Noetherian ring R. This dimension is finer than CM-dimension.