Mathematical Journal of Okayama University volume64 issue1

2022-01 発行

Puthenpurakal, Tony J.

Publication Date

2022-01

Abstract

Let K be a field and consider the standard grading on A = K[X_{1}, ... ,X_{d}]. Let I, J be monomial ideals in A. Let I_{n}(J) = (I^{n} : J^{∞}) be the n^{th} symbolic power of I with respect to J. It is easy to see that the function f^{I} _{J} (n) = e_{0}(I_{n}(J)/I^{n}) is of quasi-polynomial type, say of period g and degree c. For n ≫ 0 say

f^{I}_{J} (n) = a_{c}(n)n^{c} + a_{c−1}(n)n^{c−1} + lower terms,

where for i = 0, ... , c, a_{i} : N → Q are periodic functions of period g and a_{c} ≠0. In [4, 2.4] we (together with Herzog and Verma) proved that dim I_{n}(J)/I^{n} is constant for n ≫ 0 and a_{c}(−) is a constant. In this paper we prove that if I is generated by some elements of the same degree and height I ≥ 2 then a_{c−1}(−) is also a constant.

f

where for i = 0, ... , c, a

Keywords

quasi-polynomials

monomial ideals

symbolic powers

Comments

Mathematics Subject Classification. Primary 13D40; Secondary 13H15.