Titlebook: Commutative Algebra; with a View Toward A David Eisenbud Textbook 1995 Springer Science+Business Media New York 1995 Algebraic Geometry.alg

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Dimension and Hilbert-Samuel PolynomialsIn this section we present a characterization of dimension that yields other important invariants and that is well suited to computation using techniques to be developed in Chapter 15. Throughout we shall write . for a local ring with maximal ideal m. All modules will be finitely generated .-modules unless otherwise stated.

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Commutative Algebra978-1-4612-5350-1Series ISSN 0072-5285 Series E-ISSN 2197-5612

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definition

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https://doi.org/10.1007/978-94-010-3670-2sion. The notion of flatness was first isolated by Serre [1955–56] and was then systematically developed and mined by Grothendieck. It is now a central theme in algebraic geometry and commutative algebra.

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Roots of Commutative Algebrand 1893. Three major strands of nineteenth-century activity lie behind commutative algebra and are still its primary fields of application: number theory, algebraic geometry (the algebraic aspect really begins with Riemann’s “function theory”), and invariant theory. We shall say a little about developments in each.

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Localizationechnique of . reduces many problems in commutative algebra to problems about local rings. This often turns out to be extremely useful: Most of the problems with which commutative algebra has been successful are those that can be reduced to the local case.