Hilbert functions for two ideals

✍ Scribed by William C. Brown

• Book ID: 102967277
• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 767 KB
• Volume: 225
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1999.8141

✦ Synopsis

Let (R, m) denote a Noetherian, local ring R with maximal ideal m. Let I and J be ideals contained in m and assume I + J is m-primary. Then for all non-negative integers r and s, the R-module [R/(I r + js)] has finite length. We denote the length of this R-module by fR[R/(F + js)] = h(r, s). The function A(r, s) is called the Hilbert function of I and J. Let 77 denote the integers and for p ~ ?7, let ?7[>_p] = {n ~ 2~[n >__ p}. In this paper, we prove the following theorem: Suppose for some non-negative integer p, there exist ordered pairs (r 1, s 1) ..... (r,, s n)

. )(I r, f3 js,)] + (ir+ 1 + js+ 1)

for all r,s >-p. Then there exists a polynomial f(x,y) ~ Q[x,y] (~ the rational numbers) such that A(r,s) =f(r,s) for all r,s ~-O. Furthermore, Ox(f) = dim(R/J), 3y(f) = dim(R/l), and cg(f) <e¢(1) +f(J). Here Ox(f) [Or(f)] denotes the degree of f as a polynomial in x[y] and O(f) denotes the total degree of f. dim(S) is the KruU dimension of the ring S and f(lI) is the analytical spread of the ideal 1.t.