Symmetric Reductions

Abstract

In this paper, we introduce a special kind of reduction S, called a symmetric reduction, of a non-basic ideal I in a local ring (R, M). (By definition, S = (b ₁ − u _{1, i} b _i ,…, b _i−1 − u _{i−1, i} b _i , b _i+1 − u _{i+1, i} b _i ,…, b _g  − u _{g, i} b _i )R for i = 1,…, g, where b ₁,…, b _g is an arbitrary basis of I and the u _{i, j} are certain units in R such that for i ≠ j in {1,…, g}; note that S depends on the basis, but it has the nice property that I = (b _i , S)R for i = 1,…, g.) Our main results show that if I is a non-basic ideal in R and if R has an infinite residue field, then: (1) I has proper such reductions S; (2) there exists a positive integer s(I) such that: (a) for each minimal basis of I, I has a symmetric reduction S depending on this basis such that S I ^s(I) = I ^s(I)+1; and, (b) if H is any proper reduction of I and H I ⁿ  = I ⁿ⁺¹, then n ≥ s(I) (so, in particular, s(I) ≤ r(I), the reduction number of I); and, (3) r(I) ≤ s(S ₀) + … + s(S _k−1) (where I = S ₀ ⊃ S ₁ ⊃ … ⊃ S _k is a chain of ideals such that S _i is a symmetric reduction of S _i−1 for i = 1,…, k and S _k is a minimal reduction of I), and the equality holds for certain ideals I and certain such chains of symmetric reductions of I.

Key Words:

• Analytic spread
• Analytically independent elements
• Basic ideal
• Form
• Integrally dependent elements
• Local ring
• Noetherian ring
• Reduction number
• Reduction of an ideal

AMS (MOS) Subject Classification Numbers:

• Primary 13A15
• 13B20
• Secondary 13H99

Acknowledgments

Notes

^#Communicated by C. Maxson.