Showing papers on "Global dimension published in 2003"
TL;DR: A constructive version of Hausdorff dimension is developed using constructive supergales, which are betting strategies that generalize the constructive supermartingales used in the theory of individual random sequences and the Kolmogorov complexity of a string is proven to be the product of its length and its dimension.
Abstract: A constructive version of Hausdorff dimension is developed using constructive supergales, which are betting strategies that generalize the constructive supermartingales used in the theory of individual random sequences. This constructive dimension is used to assign every individual (infinite, binary) sequence S a dimension, which is a real number dim(S) in the interval [0, 1]. Sequences that are random (in the sense of Martin-Lof) have dimension 1, while sequences that are decidable, Σ10, or Π10, have dimension 0. It is shown that for every Δ20-computable real number α in [0, 1] there is a Δ20 sequence S such that dim(S) = α. A discrete version of constructive dimension is also developed using termgales, which are supergale-like functions that bet on the terminations of (finite, binary) strings as well as on their successive bits. This discrete dimension is used to assign each individual string w a dimension, which is a nonnegative real number dim(w). The dimension of a sequence is shown to be the limit inferior of the dimensions of its prefixes. The Kolmogorov complexity of a string is proven to be the product of its length and its dimension. This gives a new characterization of algorithmic information and a new proof of Mayordomo's recent theorem stating that the dimension of a sequence is the limit inferior of the average Kolmogorov complexity of its first n bits. Every sequence that is random relative to any computable sequence of coin-toss biases that converge to a real number β in (0, 1) is shown to have dimension H(β), the binary entropy of β.
213 citations
07 Nov 2003
TL;DR: In this paper, it was shown that for any associative ring R, and for any left R-module M with finite projective dimension, the Gorenstein injective dimension Gid R M equals the usual injective dimensions id R M, provided that R is commutative and Noetherian.
Abstract: In this paper we prove that for any associative ring R, and for any left R-module M with finite projective dimension, the Gorenstein injective dimension Gid R M equals the usual injective dimension id R M. In particular, if Gid R R is finite, then also id R R is finite, and thus R is Gorenstein (provided that R is commutative and Noetherian).
64 citations
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TL;DR: In this paper, it was shown that if the N-homogeneous algebra A is generalized Koszul, AS-Gorenstein and of finite global dimension, then one can apply the Van den Bergh duality theorem to A, i.e., there is a Poincare duality between Hochschild homology and cohomology of A, as for N = 2.
Abstract: Koszul property was generalized to homogeneous algebras of degree N>2 in [5], and related to N-complexes in [7]. We show that if the N-homogeneous algebra A is generalized Koszul, AS-Gorenstein and of finite global dimension, then one can apply the Van den Bergh duality theorem [23] to A, i.e., there is a Poincare duality between Hochschild homology and cohomology of A, as for N=2.
57 citations
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TL;DR: In this article, the resolution dimension of functorially finite subcategories was studied, and a chain of rejective sub-categories were constructed to construct modules with endomorphisms rings of finite global dimension.
Abstract: We will study the resolution dimension of functorially finite subcategories. The subcategories with the resolution dimension zero correspond to ring epimorphisms, and rejective subcategories correspond to surjective ring morphisms. We will study a chain of rejective subcategories to construct modules with endomorphisms rings of finite global dimension. We apply these result to study a function $r_\Lambda:\mod\Lambda\to
nn_{\ge0}$ which is a natural extension of Auslander's representation dimension.
56 citations
TL;DR: In this article, the authors introduce a new class of rings called Nonnil-Noetherian rings, which are closely related to the class of Noetherian ring and show that many of the properties of nonnil-noetherians are also true for nonnil noetherians.
Abstract: Let R be a commutative ring with 1 such that Nil(R) is a divided prime ideal of R. The purpose of this paper is to introduce a new class of rings that is closely related to the class of Noetherian rings. A ring R is called a Nonnil-Noetherian ring if every nonnil ideal of R is finitely generated. We show that many of the properties of Noetherian rings are also true for Nonnil-Noetherian rings; we use the idealization construction to give examples of Nonnil-Noetherian rings that are not Noetherian rings; we show that for each n ≥ 1, there is a Nonnil-Noetherian ring with Krull dimension n which is not a Noetherian ring.
45 citations
TL;DR: In this paper, the authors introduce the double tilted algebras and prove various characterizations, including the relation between the double section and the AR-quiver of a double section with a natural property.
43 citations
TL;DR: For a Noetherian local ring, the prime ideals in the singular locus completely determine the category of finitely generated modules up to direct summands, extensions and syzygies as discussed by the authors.
Abstract: For a Noetherian local ring, the prime ideals in the singular locus completely determine the category of finitely generated modules up to direct summands, extensions and syzygies. From this some simple homological criteria are derived for testing whether an arbitrary module has finite projective dimension.
41 citations
TL;DR: In this article, the dominant numbers of noetherian rings have been studied and duality on n-Gorenstein rings has been established. But the authors focus on homological properties of rings and do not consider the duality of the dominant number.
40 citations
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TL;DR: In this article, the authors give a homological-algebra flavored definition of an induction map in group homology associated to a topological coupling, and show that the real cohomology rings of quasi-isometric nilpotent groups are isomorphic as graded rings.
Abstract: Building upon work of Y. Shalom we give a homological-algebra flavored definition of an induction map in group homology associated to a topological coupling. As an application we obtain estimates of the (co)homological dimension of groups G and H, where G embeds uniformly into H and the (co)homological dimension of G is finite. Another consequence of our results is that the Hirsch ranks of quasi-isometric solvable groups coincide. Further, it is shown that the real cohomology rings of quasi-isometric nilpotent groups are isomorphic as graded rings. On the analytic side, we apply the induction technique to Novikov-Shubin invariants of amenable groups, which can be seen as homological invariants, and show their invariance under quasi-isometry.
39 citations
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TL;DR: A survey on the relation between homological properties of the Frobenius endomorphism and finiteness of various homological dimensions of the ring or of modules over it, such as global dimension and projective dimension, can be found in this paper.
Abstract: This is a survey on the relation between homological properties of the Frobenius endomorphism and finiteness of various homological dimensions of the ring or of modules over it, such as global dimension and projective dimension. We begin with Kunz's surprising result in 1969 that the regularity of a Noetherian local ring is equivalent to the flatness of its Frobenius endomorphism, as well as the subsequent generalizations to the module setting by Peskine and Szpiro and continue up through the recent flurry of results in the last five years. An attempt is made to include proofs whenever feasible.
32 citations
TL;DR: In this paper, the authors proved that every serial ring R has the isolation property: every isolated point in any theory of modules over R is isolated by a minimal pair, and they calculated the Krull-Gabriel dimension of the module category over serial rings.
Abstract: We prove that every serial ring R has the isolation property: every isolated point in any theory of modules over R is isolated by a minimal pair. Using this we calculate the Krull–Gabriel dimension of the module category over serial rings. For instance, we show that this dimension cannot be equal to 1.
TL;DR: In this paper, the core of a projective dimension one module is computed explicitly in terms of fitting ideals, and the core is shown to recover previous work by R. Mohan on integrally closed torsionfree modules over a two-dimensional regular local ring.
Abstract: The core of a projective dimension one module is computed explicitly in terms of Fitting ideals. In particular, our formula recovers previous work by R. Mohan on integrally closed torsionfree modules over a two-dimensional regular local ring.
TL;DR: In this paper, it was proved that if n is even, then P has a free summand of rank one if it maps onto an ideal I of A [T ] of height n which is generated by n elements.
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TL;DR: In this article, it was shown that the global dimensions of the crossed product and the weak dimensions of Hopf Hopf algebra are the same for finite-dimensional semisimple and cosemisimple Hopf algebras.
Abstract: We obtain that the global dimensions of $R$ and the crossed product $R # _\sigma H$ are the same; meantime, their weak dimensions are also the same, when $H$ is finite-dimensional semisimple and cosemisimple Hopf algebra.
TL;DR: In this article, it was shown that a submodule of an unstable Noetherian R-module over a noetherian ring R has a 𝒫*-invariant primary decomposition.
Abstract: We show that a 𝒫* submodule of an unstable Noetherian R-module over a Noetherian ring R has a 𝒫*-invariant primary decomposition.
TL;DR: In this article, the existence of a bijective correspondence between τ-torsion-free injective modules and the τ-closed prime ideals was studied for a ring with τ-Gabriel dimension.
TL;DR: In this article, it was shown that if R is a left Noetherian ring, then here, for a noetherian left module A, len A is its ordinal valued length as defined by Gulliksen.
Abstract: We show that if R is a left Noetherian ring, then Here, for a Noetherian left module A, len A is its ordinal valued length as defined by Gulliksen (Gulliksen, (1973) T H A theory of length for noetherian modules J Pure Appl Algebra 3:159–170), and ⊗ is the natural product on ordinal numbers
TL;DR: In this article, a wide class of domestic finite dimensional algebras over an algebraically closed field which are obtained from the hereditary alges of Euclidean type, n≥1, by iterated one-point extensions by two-ray modules are introduced.
Abstract: In the paper, we introduce a wide class of domestic finite dimensional algebras over an algebraically closed field which are obtained from the hereditary algebras of Euclidean type , n≥1, by iterated one-point extensions by two-ray modules. We prove that these algebras are domestic and their Auslander-Reiten quivers admit infinitely many nonperiodic connected components with infinitely many orbits with respect to the action of the Auslander-Reiten translation. Moreover, we exhibit a wide class of almost sincere domestic simply connected algebras of large global dimensions.
TL;DR: In this article, the classification of central extensions of three dimensional Artin-Schelter regular algebras to four dimensional Algebraic regular algesia is complete.
Abstract: This paper completes the classification of central extensions of three dimensional Artin-Schelter regular algebras to four dimensional Artin-Schelter regular algebras. Let A be an AS regular algebra of global dimension three and let D be an extension of A by a central graded element z, i.e., D/⟨z⟩ = A. If A is generated by elements of degree one, those algebras D which are again AS regular have been classified in Le Bruyn et al. (Le Bruyn L., Smith, S. P., Van den Bergh, M. (1996). Central extensions of three dimensional Artin-Schelter regular algebras. Math. Zeitschrift 222:171–212.) and Cassidy (Cassidy, T. (1999). Global dimension 4 extensions of Artin-Schelter regular algebras. J. Algebra 220:225–254.). If A is not generated by elements of degree one, then A falls under a classification due to Stephenson (Stephenson, D. R. (1996). Artin-Schelter regular algebras of global dimension three. J. Algebra 183(1):55–73 and Stephenson, D. R. (1997). Algebras associated to elliptic curves. Trans. Amer...
TL;DR: In this paper, all coordinates in two variables over a Noetherian Q -domain of Krull dimension one are proved to be stably tame and some results concerning stable tameness of polynomials in general are shown.
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05 Dec 2003
TL;DR: In this article, the authors give a homological-algebra flavored definition of an induction map in group homology associated to a topological coupling, and show that the real cohomology rings of quasi-isometric nilpotent groups are isomorphic as graded rings.
Abstract: Building upon work of Y. Shalom we give a homological-algebra flavored definition of an induction map in group homology associated to a topological coupling. As an application we obtain estimates of the (co)homological dimension of groups G and H, where G embeds uniformly into H and the (co)homological dimension of G is finite. Another consequence of our results is that the Hirsch ranks of quasi-isometric solvable groups coincide. Further, it is shown that the real cohomology rings of quasi-isometric nilpotent groups are isomorphic as graded rings. On the analytic side, we apply the induction technique to Novikov-Shubin invariants of amenable groups, which can be seen as homological invariants, and show their invariance under quasi-isometry.
TL;DR: In this article, the authors introduce the notion of (n, d)-Krull rings, which is a generalization of the concept of Krull dimension rings, and establish the transfer of this notion to trivial extension and to the direct product notions.
Abstract: In this paper, we introduce the notion of “(n, d)-Krull rings” which is in some way a generalization of the notion of “Krull dimension rings”. The richness of this notion resides in its ability to classify some rings having infinite Krull dimension. We establish the transfer of this notion to the trivial extension and to the direct product notions. We conclude with a brief discussion of the scopes and limits of our results.
TL;DR: The Local to Global Dimension of the Sacred (L2GDS) as discussed by the authors is an extension of the L2GMS. Museum International: Vol. 55, No. 2, pp. 18-24, 2003
Abstract: (2003). Local to Global Dimension of the Sacred. Museum International: Vol. 55, No. 2, pp. 18-24.
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TL;DR: In this article, the authors studied the relationship of quite different object in the theory of artin algebras, namely Auslander-regular rings of global dimension two, torsion theories, $\tau$-categories and almost abelian categories.
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, $\tau$-categories and almost abelian categories. We will apply our results to characterization problems of Auslander-Reiten quivers.
TL;DR: Weakly regular local rings are Auslander-regular and Macaulay regularity as mentioned in this paper, and weakly regular rings are also regular Lie algebras of finite dimension.
Abstract: For a (left and right) noetherian semilocal ring R we analyse a regularity concept (called weak regularity) based on the equation gld R = dim R. Examples are regular Cohen-Macaulay orders over a regular local ring, localized enveloping algebras of finite dimensional Lie algebras, and the regular rings classified in Rump (2001b). We prove that weakly regular rings are Auslander-regular and Macaulay.
TL;DR: In this article, it was shown that these injective covers over commutative noetherian rings with global dimension at most 2 have properties analogous to those of the flat envelopes over these rings.
Abstract: In (3), Del Valle, Enochs and Mart´inez studied flat en- velopes over rings and they showed that over rings as in the title these are very well behaved. If we replace flat with injective and envelope with the dual notion of a cover we then have the injective covers. In this article we show that these injective covers over the commutative noetherian rings with global dimension at most 2 have properties analogous to those of the flat envelopes over these rings.
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TL;DR: In this paper, it was shown that the global dimension of a semiprimary ring is finite if, and only if, there is a $m>0$ such that $Ext_A^n(S,S)=0, for all simple $A$-modules $S$ and all $n\geq m$
Abstract: Suppose that $A$ is a semiprimary ring satisfying one of the two conditions: 1) its Yoneda ring is generated in finite degrees; 2) its Loewy length is less or equal than three. We prove that the global dimension of $A$ is finite if, and only if, there is a $m>0$ such that $Ext_A^n(S,S)=0$, for all simple $A$-modules $S$ and all $n\geq m$.
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TL;DR: In this paper, the main topic is the use of the theory of local Chern characters to answer some questions on modules of finite homological dimension, based on talks given at the conference on cycles in Morelia, Mexico in June 2003.
Abstract: We describe several applications of the theory of cycles to questions in Commutative Algebra. The main topic is the use of the theory of local Chern characters to answer some questions on modules of finite homological dimension. This paper is based on talks given at the conference on cycles in Morelia, Mexico in June 2003.