Uniform annihilators of local cohomology of extended Rees rings (Q1703107)

Let \(A\) be a noetherian ring, \(I\) an ideal of \(A\), and \(t\) an indeterminate over \(A\). Consider \(A[t,t^{-1}]\) as a graded ring in the natural way with \(\deg(t)=1\). The extended Rees ring of \(A\) associated to \(I\) is the graded subring \(R=A[t,t^{-1}I]\) of the ring \(A[t,t^{-1}]\). An element \(a\in A\) is said to be a uniform local cohomological annihilator of \(A\), if it is not lying in any minimal prime ideal of \(A\), and for any maximal ideal \(\mathfrak{m}\) of \(A\), \(a\) kills the \(i\)-th local cohomology module \(H_{\mathfrak{m}}^i(A)\) for \(i\lneqq ht(\mathfrak{m})\). The main result of this paper is the following: Theorem 1.1. Let \((A,\mathfrak{m})\) be a \(d\)-dimensional noetherian local ring with infinite residue field. If \(A\) has a uniform local cohomological annihilator \(a\), then there exists a positive integer \(n\) such that \((at)^n\) is a uniform local cohomological annihilator of the extended Rees ring \(A[t,t^{-1}I]\) for every \(\mathfrak{m}\)-primary ideal \(I\) of \(A\).