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

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 h_M, 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 h_M (−) is the Hilbert function of M , and we prove basic results regard it. Using the theory of hypergeometric functions, we prove that hdepth(h_S) = n, where S = K[x₁, . . . , x_n].

We show that hdepth(h_S/J ) = n, if J = (f₁, . . . , f_d) ⊂ S is a complete intersection monomial ideal with deg(f_i) ≥ 2 for all 1 ≤ i ≤ d. Also, we show that hdepth(h_M̅) ≥ hdepth(h_M) for any finitely generated graded S-module M, where M̅ = M ⊗_S S[x_n+1].