Associated primes of local cohomology after adjoining indeterminates. Part 2: The general case

Abstract

Let $A$ be a domain finitely generated as an algebra over a field, $k$ of characteristic zero, $ R=A[t_1,\dots ,t_\ell ]\quad \mbox {or}\quad A[[t_1,\dots ,t_\ell ]]$, and $I$ an ideal of $R$. If $A$ has a resolution of singularities, $Y_0$, which is the blowup of $A$ along an ideal of depth at least 2 and is covered by a finite number of open affines with $H^j(Y_0,\mathcal {O}_{Y_0})$ of finite length over $A$ for $j>0$, we prove that $\rm{Ass} _RH^i_I(R) $ is finite for every $i$. In particular, this holds when $A$ is a finite-dimensional normal domain with an isolated singularity which is a finitely generated algebra over a field of characteristic~0.