Skip to main content
main-content

Über dieses Buch

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original in the following aspects: - The terminology has been brought up to date (e.g. "cohomological dimension" has been replaced by the now customary "depth").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.

Inhaltsverzeichnis

Frontmatter

Chapter I. Prime Ideals and Localization

Abstract
This chapter summarizes standard results in commutative algebra. For more details, see [Bour], Chap. II, III, IV.
Jean-Pierre Serre

Chapter II. Tools

Without Abstract
Jean-Pierre Serre

Chapter III. Dimension Theory

Abstract
Let A be a ring (commutative, with a unit element).
Jean-Pierre Serre

Chapter IV. Homological Dimension and Depth

Abstract
Let A be a commutative ring (which is not assumed to be noetherian for the time being) and let x be an element of A. We denote by K(x), or sometimes K A (x), the following complex:
Jean-Pierre Serre

Chapter V. Multiplicities

Abstract
In this section, A is a commutative noetherian ring; all A-modules are assumed to be finitely generated.
Jean-Pierre Serre

Backmatter

Weitere Informationen