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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Generalized analytic independence

Author: Jacob Barshay
Journal: Proc. Amer. Math. Soc. 58 (1976), 32-36
MSC: Primary 13C15; Secondary 14M10
MathSciNet review: 0406997
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Abstract: If $ {\mathbf{a}}$ is a proper ideal of a commutative ring with unity $ A$, a set of elements $ {a_1}, \ldots ,{a_n} \in A$ is called $ {\mathbf{a}}$-independent if every form in $ A[{X_1}, \ldots ,{X_n}]$ vanishing at $ {a_1}, \ldots ,{a_n}$ has all its coefficients in $ {\mathbf{a}}$. $ \sup {\mathbf{a}}$ is defined as the maximum number of $ {\mathbf{a}}$-independent elements in $ {\mathbf{a}}$. It is shown that grade $ {\mathbf{a}} \leq \sup {\mathbf{a}} \leq {\text{height }}{\mathbf{a}}$. Examples are given to show that $ \sup {\mathbf{a}}$ need take neither of the limiting values and strong evidence is given for the conjecture that it can assume any intermediate value. Cohen-Macaulay rings are characterized by the equality of sup and grade for all ideals (or just all prime ideals). It is proven that equality of sup and height for all powers of prime ideals implies that the ring is $ {S_1}$ (the Serre condition). Finally, independence is related to the structure of certain Rees algebras.

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Keywords: Analytic independence, grade, height, sup of an ideal, Cohen-Macaulay ring, Rees algebra
Article copyright: © Copyright 1976 American Mathematical Society

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