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AuthorSerre, Jean-Pierre. author
TitleLocal Algebra [electronic resource] / by Jean-Pierre Serre
ImprintBerlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 2000
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Descript XIII, 130 p. online resource


The present book is an English translation of Algebre Locale - Multiplicites published by Springer-Verlag as no. 11 of the Lecture Notes series. The original text was based on a set of lectures, given at the College de France in 1957-1958, and written up by Pierre Gabriel. Its aim was to give a short account of Commutative Algebra, with emphasis on the following topics: a) Modules (as opposed to Rings, which were thought to be the only subject of Commutative Algebra, before the emergence of sheaf theory in the 1950s); b) H omological methods, a la Cartan-Eilenberg; c) Intersection multiplicities, viewed as Euler-Poincare characteristics. The English translation, done with great care by Chee Whye Chin, differs from the original in the following aspects: - The terminology has been brought up to date (e.g. "cohomological dimension" has been replaced by the now customary "depth"). I have rewritten a few proofs and clarified (or so I hope) a few more. - A section on graded algebras has been added (App. III to Chap. IV). - New references have been given, especially to other books on Commu- tive Algebra: Bourbaki (whose Chap. X has now appeared, after a 40-year wait) , Eisenbud, Matsumura, Roberts, .... I hope that these changes will make the text easier to read, without changing its informal "Lecture Notes" character


I. Prime Ideals and Localization -- ยง1. Notation and definitions -- ยง2. Nakayamaโ{128}{153}s lemma -- ยง3. Localization -- ยง4. Noetherian rings and modules -- ยง5. Spectrum -- ยง6. The noetherian case -- ยง7. Associated prime ideals -- ยง8. Primary decompositions -- II. Tools -- A: Filtrations and Gradings -- B: Hilbert-Samuel Polynomials -- III. Dimension Theory -- A: Dimension of Integral Extensions -- B: Dimension in Noetherian Rings -- C: Normal Rings -- D: Polynomial Rings -- IV. Homological Dimension and Depth -- A: The Koszul Complex -- B: Cohen-Macaulay Modules -- C: Homological Dimension and Noetherian Modules -- D: Regular Rings -- Appendix I: Minimal Resolutions -- Appendix II: Positivity of Higher Euler-Poincarรฉ Characteristics -- Appendix III: Graded-polynomial Algebras -- V. Multiplicities -- A: Multiplicity of a Module -- B: Intersection Multiplicity of Two Modules -- C: Connection with Algebraic Geometry -- Index of Notation

Mathematics Algebra Mathematics Algebra


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