Tag Archives: krull dimension

Commutative Algebra 64

Segre Embedding Throughout this article, k is a fixed algebraically closed field. We wish to construct the product in the category of quasi-projective varieties. For our first example, let be the projective variety defined by the homogeneous equation . We define … Continue reading

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Commutative Algebra 63

Serre’s Criterion for Normality Throughout this article, fix an algebraically closed field k. In this section, A denotes a noetherian domain. We will describe Serre’s criterion, which is a necessary and sufficient condtion for A to be normal. In the … Continue reading

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Commutative Algebra 43

Catenary Rings Let us look at prime chains in greater detail. Definition. Let be a chain of prime ideals of a ring A. We say the chain is saturated if for any prime ideal of A, ; maximal if it … Continue reading

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Commutative Algebra 42

Noether Normalization Theorem Throughout this article, k is a field, not necessarily algebraically closed. Definition. Let A be a finitely generated k-algebra which is an integral domain. We say are algebraically independent over k if they are so as elements … Continue reading

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Commutative Algebra 40

More on Integrality Lemma 1. Let be an integral extension. If is an ideal and , the resulting injection is an integral extension. Proof Any element of can be written as , . Then x satisfies a monic polynomial relation: . … Continue reading

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Commutative Algebra 38

Artinian Rings The main result we wish to prove is the following. Theorem. A ring A is artinian if and only if it is noetherian and , where denotes the Krull dimension. Note Recall that means all prime ideals of A … Continue reading

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Commutative Algebra 17

Field of Fractions Throughout this article, A denotes an integral domain (which may not be a UFD). Definition. The field of fractions of A is an embedding of A into a field K, such that every element of K can be … Continue reading

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