The joint reduction number and upper bounds of Hilbert series of fiber cones (Q2862502)

Let \((R,\mathfrak m)\) be a Cohen-Macaulay local ring of dimension \(d>0,\) having infinite residue field, \(I\) an \(\mathfrak m\)-primary ideal of \(R\) and \(K\supseteq I\) another ideal. The fiber cone of \(I\) with respect to \(K\) is the standard graded algebra \(F_K(I)=\bigoplus_{n\geq 0}\frac{I^n}{KI^n}.\) For \(K=I,\) \(F_K(I)=G(I)\) is the associated graded ring of \(I.\) Let \((I^{[m]}\mid K^{[n]})\) be the multiset of ideals consisting of \(m\) copies of \(I\) and \(n\) copies of \(K.\) A sequence of elements \(a_1,\ldots,a_{d-1}\in I,\) \(a_d\in K\) is called a \textit{joint reduction} of \((I^{[d-1]}\mid K)\) if the ideal \(J=(a_1,\ldots,a_{d-1})K+a_dI\) is a reduction of \(IK\) i.e. if there exists a positive integer \(n\) such that \((IK)^{n+1}=J(IK)^n.\) Let \(L=(a_1,\ldots,a_{d})\) be a joint reduction of \((I^{[d-1]}\mid K).\) Define \(r_{L}(I\mid K)\) to be the smallest integer \(n\) (if there exists) such that \(KI^n=(a_1,\ldots,a_{d-1})KI^{n-1}+a_dI^n\) (otherwise \(r_{L}(I\mid K)=\infty\)). The smallest of all \(r_{L}(I\mid K),\) where \(L\) is varying, is the \textit{joint reduction number} of \((I^{[d-1]}\mid K)\) and it is denoted by \(r(I\mid K).\)NEWLINENEWLINEIn this paper the author shows that if depth \(G(I)\geq d-1\) and depth \(F_K(I)\geq d-2\) the joint reduction numbers \(r_{L}(I\mid K)\) are independent of \(L\) as well some lengths of certain modules.NEWLINENEWLINEIn the general case an upper bound for the Hilbert series \(F_K(I)\) is given.