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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Efficient generation of maximal ideals in polynomial rings


Authors: E. D. Davis and A. V. Geramita
Journal: Trans. Amer. Math. Soc. 231 (1977), 497-505
MSC: Primary 13F20; Secondary 13E05
MathSciNet review: 0472800
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Abstract: The cardinality of a minimal basis of an ideal I is denoted $ \nu (I)$. Let A be a polynomial ring in $ n > 0$ variables with coefficients in a noetherian (commutative with $ 1 \ne 0$) ring R, and let M be a maximal ideal of A. In general $ \nu (M{A_M}) + 1 \geqslant \nu (M) \geqslant \nu (M{A_M})$. This paper is concerned with the attaining of equality with the lower bound. It is shown that equality is attained in each of the following cases: (1) $ {A_M}$ is not regular (valid even if A is not a polynomial ring), (2) $ M \cap R$ is maximal in R and (3) $ n > 1$. Equality may fail for $ n = 1$, even for R of dimension 1 (but not regular), and it is an open question whether equality holds for R regular of dimension $ > 1$. In case $ n = 1$ and $ \dim (R) = 2$ the attaining of equality is related to questions in the K-theory of projective modules. Corollary to (1) and (2) is the confirmation, for the case of maximal ideals, of one of the Eisenbud-Evans conjectures; namely, $ \nu (M) \leqslant \max \{ \nu (M{A_M}),\dim (A)\} $. Corollary to (3) is that for R regular and $ n > 1$, every maximal ideal of A is generated by a regular sequence--a result well known (for all $ n \geqslant 1$) if R is a field (and somewhat less well known for R a Dedekind domain).


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DOI: https://doi.org/10.1090/S0002-9947-1977-0472800-5
Keywords: Polynomial ring, maximal ideal, number of generators, Hilbert rings, regular sequences, projective modules, $ {K_0}$, seminormal ring
Article copyright: © Copyright 1977 American Mathematical Society