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.