Efficient generation of maximal ideals in polynomial rings

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).