On the Hilbert depth of the Hilbert function of a finitely generated graded module

Abstract

Let K be a field, A a standard graded K-algebra and M a finitely generated graded A-module. Inspired by our previous works, see [2] and [3], we study the invariant called Hilbert depth of hM, that is hdepth(hM)=max{d:∑j≤k(-1)k-j(d-jk-j)hM(j)≥0 for all k≤d},\mathrm{hdepth} \left( {{h_M}} \right) = \max \left\{ {d:\sum\limits_{j \le k} {{{\left( { - 1} \right)}^{k - j}}\left( {\matrix{ {d - j} \cr {k - j} \cr } } \right){h_M}\left( j \right) \ge 0\,\, \mathrm{for}\, \mathrm{all}\, k \le d}} \right\}, where hM (−) is the Hilbert function of M , and we prove basic results regard it. Using the theory of hypergeometric functions, we prove that hdepth(hS) = n, where S = K[x1, . . . , xn].

Keywords

• graded module
• hypergeometric function
• hilbert depth
• monomial ideal
• primary 13c15, 13p10, 13f20
• secondary 05a18, 06a07