Mathematical Journal of Okayama University volume64 issue1
2022-01 発行

Symbolic powers of monomial ideals

Puthenpurakal, Tony J.
Publication Date
2022-01
Abstract
Let K be a field and consider the standard grading on A = K[X1, ... ,Xd]. Let I, J be monomial ideals in A. Let In(J) = (In : J) be the nth symbolic power of I with respect to J. It is easy to see that the function fI J (n) = e0(In(J)/In) is of quasi-polynomial type, say of period g and degree c. For n ≫ 0 say

fIJ (n) = ac(n)nc + ac−1(n)nc−1 + lower terms,

where for i = 0, ... , c, ai : N → Q are periodic functions of period g and ac ≠0. In [4, 2.4] we (together with Herzog and Verma) proved that dim In(J)/In is constant for n ≫ 0 and ac(−) 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 ac−1(−) is also a constant.
Keywords
quasi-polynomials
monomial ideals
symbolic powers
Comments
Mathematics Subject Classification. Primary 13D40; Secondary 13H15.