Showing papers on "Global dimension published in 1998"
01 Jan 1998
TL;DR: In this article, the authors studied connected, not necessarily noetherian, regular rings of global dimension 2 and showed that connected regular rings can be connected in a non-noetherian manner.
Abstract: We study connected, not necessarily noetherian, regular rings of global dimension 2.
61 citations
TL;DR: In this article, it was shown that any complex over a commutative noetherian ring with finite Krull dimension has a flat cover and a DG-flat cover, where DG is the dimension of the DG.
59 citations
TL;DR: In this paper, the Auslander-Buchsbaum and Bass Theorem were generalized to non-commutative ℕ-algebras over fields, and it was shown that under weak (so-called χ−) conditions, such a non-complementary algebra is Artin-Schelter regular.
Abstract: Two results, the Auslander–Buchsbaum and Bass Theorems from the homological theory of commutative noetherian rings, are generalized to important classes of non-commutative ℕ-graded algebras over fields. As a corollary to the generalized Auslander–Buchsbaum Theorem, it is found that under weak (so-called χ−) conditions, a non-commutative ℕ-graded connected noetherian algebra of finite global dimension is in fact Artin–Schelter regular.
57 citations
Book•
01 Nov 1998TL;DR: Weyl algebras were introduced by S. P. Smith as discussed by the authors and they have been used for torus invariant dimension theory for finite global dimension Finite dimensional representations An example can be found in
Abstract: Introduction Notations and conventions A certain class of rings Some constructions The algebras introduced by S. P. Smith The Weyl algebras Rings of differential operators for torus invariants Dimension theory for $B^\chi$ Finite global dimension Finite dimensional representations An example References.
57 citations
TL;DR: In this article, the authors developed techniques for calculating dimensions of generalized Weyl algebras, including the Krull dimension in the sense of Rentschler-Gabriel.
28 citations
01 Jan 1998
TL;DR: In this paper, the authors consider the category of left modules over right coherent rings of finite weak global dimension and propose an idempotent radical on its category of modules which can be used to analyze the structure of the flat envelopes and of the ring itself.
Abstract: The category of left modules over right coherent rings of finite weak global dimension has several nice features. For example, every left module over such a ring has a flat cover (Belshoff, Enochs, Xu) and, if the weak global dimension is at most two, every left module has a flat envelope (Asensio, Martfnez). We will exploit these features of this category to study its objects. In particular, we will consider orthogonal complements (relative to the extension functor) of several classes of modules in this category. In the case of a commutative ring we describe an idempotent radical on its category of modules which, when the weak global dimension does not exceed 2, can be used to analyze the structure of the flat envelopes and of the ring itself.
11 citations
TL;DR: In this paper, the authors consider graded connected Gorenstein algebras with respect to the evaluation map and prove that if ev(G) not equal 0, then the global dimension of G is finite.
Abstract: We consider graded connected Gorenstein algebras with respect to the evaluation map ev(G) = Ext(G)(k,epsilon) :: Ext(G)(k,G) --> Ext(G)(k,k). We prove that if ev(G) not equal 0, then the global dimension of G is finite.
11 citations
TL;DR: In this article, it was shown that the Krull dimension is the Gelfand-Kirillov dimension, where GK and fil.dim are the gelfand and filter dimensions, respectively.
8 citations
TL;DR: In this paper, it was shown that the right noetherianness is a necessary condition for a Malcev-Neumann ring to have right Krull dimension, and that uniform dimension is a sufficient condition for right noetherian ring construction.
Abstract: It is known that if R is a right noetherian ring, then the corresponding Malcev-Neumann ring R * ((G)) is right noetherian and r.K.dim(R) = r.K.dim(R * ((G))). We prove that the right noethe-rianness is a necessary condition for a Malcev-Neumann ring to have right Krull dimension. Also, we consider uniform dimension of Malcev-Neumann rings. The results obtained are applied to some other ring constructions
7 citations
01 Jan 1998
TL;DR: In this article, the authors studied modules of the highest injective, projective and flat dimension over a Goresntein ring and proved that the last injective term En(M) has essential socle.
Abstract: We will study modules of the highest injective, projective and flat dimension over a Goresntein ring. Let R be a Gorenstein ring of self-injective dimension n and 0 → RR → E0 → · · · → En → 0 a minimal injective resolution. Then it is shown in [F-I] that the flat dimension and projective dimension of En is n, the highest dimension. In this note, we shall prove that if M is a left R-module of injective dimension n, then the last injective term En(M) in a minimal injective resolution of M has projective and flat dimension n, and any indecomposable summand of En(M) embeds in En. As a consequence, we obtain that if R is Auslander-Gorenstein, then En(M) has essential socle.
7 citations
TL;DR: In this paper, conditions under which the classical ring of quotients of A is a π-projective A-module are determined, and a criterion for a right hereditary right Noetherian prime ring to be serial is obtained.
Abstract: Let A be a hereditary Noetherian prime ring that is not right primitive. A complete description of π-injective A-modules is obtained. Conditions under which the classical ring of quotients of A is a π-projective A-module are determined. A criterion for a right hereditary right Noetherian prime ring to be serial is obtained.
01 Jun 1998
TL;DR: Classes of posets that are (co)generated by a poset and to define and investigate their global (dual) Krull dimension, which are then very easily applied to Grothendieck categories.
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TL;DR: The theory of modules over commutative rings was studied in this article, where Auslander dual, k-torsionless modules, and k-th syzygies were discussed.
Abstract: In these expository notes I discuss several concepts and results in the theory of modules over commutative rings, revolving around the Gorenstein dimension of modules. Some of the related notions are the Auslander dual, k-torsionless modules, and k-th syzygies.
Essentially everything in these notes can be found, in one form or another, in the memoir "Stable module theory" by M. Auslander and M. Bridger (Mem. A.M.S., no. 94, 1969). The only difference is in presentation. In the Auslander-Bridger memoir many of the results are proved in the most general setting, e.g. over possibly non-commutative, non-Noetherian rings. The techniques used are quite abstract and unfamiliar to many commutative algebraists. Much space is devoted to the theory of satellites of functors which are exact only in the middle, etc. While such a degree of generality has many advantages, it does make the memoir difficult to read for the non-expert. My goal in writing these notes was to develop the theory in the context of commutative Noetherian rings, and to show that, in this important special case, the theory is fairly elementary and easy to build. As a practical matter, then, I wrote the notes using Matsumura's "Commutative ring theory" as the only pre-requisite; and indeed, my hope is that these notes can be read just like an extra chapter in Matsumura's book.
TL;DR: In this paper, the concept of a pseudo-balanced submodule was introduced for a class of torsion-free balanced-projective R-modules. But the concept was not applied to the case of pure-essential submodules.
01 Jan 1998
TL;DR: In this article, the authors consider certain regular algebras of global dimension four that map surjectively onto the two-Veronese of a regular algebra of global dimensions three on two generators.
01 Jan 1998
TL;DR: In this paper, the authors weaken the conditions of Sieklucki's Theorem by using cohomological local connectivity and co-homological dimension based on Alexander-Spanier co-ho-mology with compact supports in a countable principal ideal domain.
Abstract: We weaken the conditions of Sieklucki's The orem by using cohomological local connectivity and co homological dimension based on Alexander-Spanier coho mology with compact supports in a countable principal ideal domain L. We could also use Cech cohomology. The theorem states that if X is a el en locally compact separable metric space with dimL X = nand {X).} ).EA is an uncountable collection of closed subsets 'of X with dimL X,x == n for all A, then there are two distinct in dices 11, A E A such that dimL(XJ.l n X,x) == n. The proof combines the fact the family of submodules of a finitely generated L-module is countable with a Mayer-Vietoris argument.
TL;DR: In this paper, the Jacobson radical of a right noetherian ring of Krull dimension is shown to be 0 for some natural number n. The Jacobson radicals are defined in terms of the powers of any ordinal.
Abstract: Let R be a right noetherian ring of Krull dimension \( \kappa \) and denote by J its Jacobson radical. Using a new definition of the powers \( J^\alpha (\alpha \) any ordinal) it is shown that \( J^{\omega \kappa +n}=0 \) for some natural number n.
TL;DR: In this paper, it was shown that for any Banach A -bimodule X, the second continuous Hochschild cohomology group H 2 (A, X) of A with coefficients in X is defined, and there is a natural correspondence between the elements of this group and equivalence classes of singular, admissible extensions of A by X.
Abstract: Let A be a C * -algebra. For each Banach A -bimodule X , the second continuous Hochschild cohomology group H 2 (A, X) of A with coefficients in X is defined (see [6]); there is a natural correspondence between the elements of this group and equivalence classes of singular, admissible extensions of A by X. Specifically this means that H 2 (A, X) ≠ {0} for some X if and only if there exists a Banach algebra B with Jacobson radical R such that R 2 = {0}, R is complemented as a Banach space, and B/R ≅ A , but B has no strong Wedderburn decomposition; i.e., there is no closed subalgebra C of B such that B ≅ C © R. In turn this is equivalent to db A ≥ 2, where db A is the homological bidimension of A ; i.e., the homological dimension of A # , the unitization of A , as an, A -bimodule [6, III. 5.15]. This paper is concerned with the following basic question, which was posed in [7].
TL;DR: The main aim of as mentioned in this paper is to show that an AB5 ∗ module whose small submodules have Krull dimensions has a radical having Krull dimension, and the proof uses the notion of dual Goldie dimension.
TL;DR: In this article, it was shown that for every natural number n, there exists a unital semisimple Banach star algebra A and a closed star subalgebra B of the centre of A, different from C, such that the global B-homology of A can be computed.
Abstract: We prove that, for every natural number n, there exists a unital semisimple Banach star algebra Aand a closed star subalgebra Bof the centre of A, different from C, such that the global B-homologic...
TL;DR: In this article, a ring with krull dimension one is presented. But it is not a ring of rings with a ring, it is a one-dimensional ring with dimension one.
Abstract: (1998). Rings with krull dimension one. Communications in Algebra: Vol. 26, No. 7, pp. 2147-2158.
TL;DR: In this paper, Enochs and Jenda showed that the negative-weak-global dimension of R α is bounded by the upper bound of the negative weak-global dimensions of the rings R α.
Abstract: Using definitions and properties by E.E. Enochs [1], V. K. Akatsa [2], E.E. Enochs and O.Mg Jenda [3], we prove first, that, if (R α)α∈I is a direct system of coherent rings, so that, for every α ∈ I, the limit R is a rα-flat module, then the negative-weak-global dimension of R is bounded by the upper bound of the negative-weak-global dimensions of the rings R α Then, if is a product of rings so that every R-module admits a flat envelope, the negative-weak-global dimension of R is bounded by the upper bound of the negative weak-global dimensions of the rings Ri . We study a class of inverse systems of modules and prove that inverse limit of FP-injective modules whose set of index is N and morphisms are onto is a module of negative-weak dimension zero Finally, using results of L. FUCHS [4] we give examples of rings of which we compute the homogical dimension, the weak-global dimension, the negative-weak-global dimension and the pure-global dimension.