Efficient generation of maximal ideals in polynomial rings

Abstract

The cardinality of a minimal basis of an ideal I is denoted ν ( I ) \nu (I) . Let A be a polynomial ring in n > 0 n > 0 variables with coefficients in a noetherian (commutative with 1 ≠ 0 1 \ne 0 ) ring R, and let M be a maximal ideal of A. In general ν ( M A M ) + 1 ⩾ ν ( M ) ⩾ ν ( M A M ) \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 {A_M} is not regular (valid even if A is not a polynomial ring), (2) M ∩ R M \cap R is maximal in R and (3) n > 1 n > 1 . Equality may fail for n = 1 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 > 1 . In case n = 1 n = 1 and dim ⁡ ( R ) = 2 \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, ν ( M ) ⩽ max { ν ( M A M ) , dim ⁡ ( A ) } \nu (M) \leqslant \max \{ \nu (M{A_M}),\dim (A)\} . Corollary to (3) is that for R regular and n > 1 n > 1 , every maximal ideal of A is generated by a regular sequence—a result well known (for all n ⩾ 1 n \geqslant 1 ) if R is a field (and somewhat less well known for R a Dedekind domain).