NONINCREASING DEPTH FUNCTIONS OF MONOMIAL IDEALS

作者：Matsuda Kazunori^*; Suzuki Tao; Tsuchiya Akiyoshi

来源：Glasgow Mathematical Journal, 2018, 60(2): 505-511.

DOI：10.1017/S0017089517000349

Given a nonincreasing function f : Z(>= 0) \ {0} -> Z(>= 0) such that (i) f (k) - f (k + 1) <= 1 for all k >= 1 and (ii) if a = f (1) and b = lim(k ->infinity) f (k), then vertical bar f(-1)(a)vertical bar <= vertical bar f(-1)(a - 1)vertical bar <= ... <= vertical bar f(-1)(b + 1)vertical bar, a system of generators of a monomial ideal I subset of K[x(1),..., x(n)] for which depth S/I-k = f(k) for all k >= 1 is explicitly described. Furthermore, we give a characterization of triplets of integers (n, d, r) with n > 0, d >= 0 and r > 0 with the properties that there exists a monomial ideal I subset of S = K[x(1),..., x(n)] for which lim(k ->infinity) depth S/I-k = d and dstab(I) = r, where dstab(I) is the smallest integer k(0) >= 1 with depthS/I-k0 = depth S/Ik0+1 = depth S/Ik0+2 = ... .

• 出版日期2018-5