Topic
Global dimension
About: Global dimension is a research topic. Over the lifetime, 2029 publications have been published within this topic receiving 38266 citations.
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01 Jan 1993
TL;DR: In this article, the authors present a self-contained introduction to the homological and combinatorial aspects of the theory of Cohen-Macaulay rings, Gorenstein rings, local cohomology, and canonical modules.
Abstract: In the last two decades Cohen-Macaulay rings and modules have been central topics in commutative algebra. This book meets the need for a thorough, self-contained introduction to the homological and combinatorial aspects of the theory of Cohen-Macaulay rings, Gorenstein rings, local cohomology, and canonical modules. A separate chapter is devoted to Hilbert functions (including Macaulay's theorem) and numerical invariants derived from them. The authors emphasize the study of explicit, specific rings, making the presentation as concrete as possible. So the general theory is applied to Stanley-Reisner rings, semigroup rings, determinantal rings, and rings of invariants. Their connections with combinatorics are highlighted, e.g. Stanley's upper bound theorem or Ehrhart's reciprocity law for rational polytopes. The final chapters are devoted to Hochster's theorem on big Cohen-Macaulay modules and its applications, including Peskine-Szpiro's intersection theorem, the Evans-Griffith syzygy theorem, bounds for Bass numbers, and tight closure. Throughout each chapter the authors have supplied many examples and exercises which, combined with the expository style, will make the book very useful for graduate courses in algebra. As the only modern, broad account of the subject it will be essential reading for researchers in commutative algebra.
2,760 citations
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01 Jan 1960
TL;DR: In this paper, the authors propose a theory of homology and cohomology theories of groups and moniods, and derive derived functors from homology functors, including Tensor products, groups of homomorphisms, and projective and injective modules.
Abstract: Preface 1. Generalities concerning modules 2. Tensor products and groups of homomorphisms 3. Categories and functors 4. Homology functors 5. Projective and injective modules 6. Derived functors 7. Torsion and extension functors 8. Some useful identities 9. Commutative Noetherian rings of finite global dimension 10. Homology and cohomology theories of groups and moniods Notes References Index.
2,253 citations
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TL;DR: In this article, Kaplansky showed that a commutative ring R is left T-nilpotent if, given any sequence {at} of elements in N, there exists an re such that ai • • • an = 0.
Abstract: Introduction. If P is a ring and M a left P-module, then homological algebra attaches three dimensions to M, projective, weak, and injective(1)By taking the supremum of one of these dimensions as M ranges over all left P-modules, one obtains one of the left \"global\" dimensions of R. Auslander and Buchsbaum [3] and, subsequently, Serre [14], found it relevant and fruitful, in the study of commutative Noetherian rings, to introduce a \"finitistic global\" dimension defined by restricting the supremum of projective dimensions to finitely generated modules of finite projective dimension. The impressive theory developed by these authors prompted Kaplansky to consider, for general commutative rings, a similar finitistic dimension (waiving the restriction, in the above, to finitely generated modules), and, in a seminar at the University of Chicago (1958), he proved the theorem below, characterizing commutative rings R for which this dimension vanishes. This result is the origin of the present paper. Definition. If N is an ideal in a ring P, we say that N is left T-nilpotent i\"T\" ior transfinite) if, given any sequence {at} of elements in N, there exists an re such that ai • • • an = 0. iRight T-nilpotence requires instead that a„ • • • oi = 0.)
1,003 citations
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01 Jan 1989TL;DR: The 2004 introduction to noncommutative noetherian rings is intended to be accessible to anyone with a basic background in abstract algebra as mentioned in this paper, and it can be used as a second-year graduate text, or as a self-contained reference.
Abstract: This 2004 introduction to noncommutative noetherian rings is intended to be accessible to anyone with a basic background in abstract algebra. It can be used as a second-year graduate text, or as a self-contained reference. Extensive explanatory discussion is given, and exercises are integrated throughout. Various important settings, such as group algebras, Lie algebras, and quantum groups, are sketched at the outset to describe typical problems and provide motivation. The text then develops and illustrates the standard ingredients of the theory: e.g., skew polynomial rings, rings of fractions, bimodules, Krull dimension, linked prime ideals. Recurring emphasis is placed on prime ideals, which play a central role in applications to representation theory. This edition incorporates substantial revisions, particularly in the first third of the book, where the presentation has been changed to increase accessibility and topicality. Material includes the basic types of quantum groups, which then serve as test cases for the theory developed.
985 citations
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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