Read e-book online Commutative Algebra: with a View Toward Algebraic Geometry PDF

By David Eisenbud

ISBN-10: 1461253500

ISBN-13: 9781461253501

This can be a finished assessment of commutative algebra, from localization and first decomposition via size thought, homological equipment, unfastened resolutions and duality, emphasizing the origins of the guidelines and their connections with different components of arithmetic. The publication provides a concise remedy of Grobner foundation thought and the confident equipment in commutative algebra and algebraic geometry that circulation from it. Many workouts integrated.

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Extra resources for Commutative Algebra: with a View Toward Algebraic Geometry (Graduate Texts in Mathematics, Volume 150)

Example text

Sylvester [1864, p. 380] The work on ideals done in the 1880s, both in number-theoretic and function-theoretic contexts, seems a trifle quaint to modern readers; but the work of Hilbert just a few years later seems quite modern. In two extraordinary papers [1890, 1893], which are still a pleasure to read, Hilbert greatly advanced the theory of ideals in polynomial rings. Hilbert's motivation comes from a subject we have not yet menttoned: the theory of invariants. We shall sketch a little of this theory.

Both Dedekind's and Lasker's theories were thoroughly reformulated and axiomatized by Emmy Noether in the 1920s, initiating the modern development of commutative algebra. 2 Algebraic Curves and Function Theory L'algebre n'est qu'une geometrie ecrite; la geometrie n'est qu'une algebre figuree. ) -Sophie Germain (1776-1831) The study of algebraic curves in the early nineteenth century is in retrospect very closely related to commutative algebra, but the connection hardly began to appear until the 1870s and 1880s.

R/l ----+ R/(1 + (a)) ----+ 0, where the element a over the left-hand map indicates that it is multiplication by a. 3. One way to specify an R-module is by giving "generators and relations": For example, if we say that a module has one generator g and relations fIg = hg = ... = fng = 0, for some elements h,···, fn E R, then the module is R/(h, ... , fn). Here is an exact sequence view: An element m of a module M corresponds to a homomorphism from R to M, sending 1 to m. Thus, giving a set of elements {m a }aEA E M corresponds to giving a homomorphism cp from a direct sum G := RA of copies of R, indexed by A, to M, sending the a;th basis element to mao If the ma generate M, then cp is a surjection.

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Commutative Algebra: with a View Toward Algebraic Geometry (Graduate Texts in Mathematics, Volume 150) by David Eisenbud

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