Dependence of Hilbert coefficients (Q2634742)

Let \(M\) be a finitely generated \(d\)-dimensional module over a local Noetherian ring \((A,\mathfrak{m})\). Let \(I \subset A\) denote an ideal such that the length \(\ell_A(M/IM)\) is finite. Then the Hilbert-Samuel function \(H_{I,M}(n) = \ell_A(M/I^{n+1}M)\) becomes for \(n \gg 0\) a polynomial \(P_{I,M}(n) = \sum_{i=0}^{d} (-1)^ie_i(I,M) \binom{n+d-i}{d-i}\), the Hilbert polynomial. The numbers \(e_i(I,M), i = 0,\ldots,d,\) are the Hilbert coefficients of \(M\) with respect to \(I\) and are important invariants, investigated in a large number of research papers. \textit{V. Trivedi} [Manuscr. Math. 94, No. 4, 485--499 (1997; Zbl 0893.13003)] proved that \((-1)^{i-1}e_i(I,M)\) is bounded above by a function depending on \(e_0(I,M), \ldots, e_{i-1}(I,M)\) and \(i\) for all \(i \geq 1\). Let \(\text{depth}_A M = t\). As a main result of the present article it is shown that the last \(t\) Hilbert coefficients \(e_{d-t+1}(I,M), \ldots,e_d(I,M)\) are bounded below and above in terms of the first \(d-t+1\) Hilbert coefficients \(e_0(I,M), \ldots,e_{d-t}(I,M)\) and \(d\). The bounds are exponential. The authors deal -- more generally -- with \(I\)-good filtrations.