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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The Principal Ideal Theorem and Systems of Parametersassume that . = 0, and thus that . is a domain. If . ∈ ., then . = . for some ., and since . ∉ . it follows that . ∈ .; thus . = .. By Corollary 4.7, (1 - .). = 0 for some . ∈ (.). Since . is a domain, we must have . = 1, so (.) is not proper, a contradiction.

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The Dimension of Affine Rings famous results: Hilbert’s Nullstellensatz, Noether’s theorem on the finiteness of the integral closure of an affine domain, and, in the next chapter, Grothendieck’s lemma of generic freeness, with its applications to the semi-continuity of fiber dimensions.

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Associated Primes and Primary Decompositionheorists to make use of unique factorization in rings of integers in number fields other than .. When it became clear that unique factorization did not always hold, the search for the strongest available alternative began. The theory of primary decomposition is the direct result of that search. Give

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Integral Dependence and the Nullstellensatzs goal, it is often important to adjoin a solution of a polynomial equation in one variable: Given a ring . and a polynomial .(.) ∈ .[.], the ring .[.]/(.) may be thought of as the result of adjoining a root of . to . as freely as possible; the root adjoined is of course the image of ..

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Filtrations and the Artin-Rees Lemmam a sequence of ideals $R = I_0supset I_1supset I_2supset ... { m satisfying} I_iJ_jsupset I_{i+j}quad { m for all} i,j.$ A third such construction, the Rees algebra, is treated at the end of the next chapter, and sheds some light on the results we shall prove about the associated graded ring. Chapt

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Completions and Hensel’s Lemmas usually applied in the case where . is a local ring and . is the maximal ideal. If . is a polynomial ring . = .[., …, .] over a field, and . = (., …, .) is the ideal generated by the variables, then the completion is the ring .[[.,…, .]] of formal power series over .. More generally, if . is a fie