Local Cohomology and Base Change

Abstract

Let $X \overset{f}\longrightarrow S$ be a morphism of Noetherian schemes, with $S$ reduced. For any closed subscheme $Z$ of $X$ finite over $S$, let $j$ denote the open immersion $X\setminus Z \hookrightarrow X$. Koll\'ar asked whether for any coherent sheaf $\mathcal F$ on $X\setminus Z$ and any index $r\geq 1$, the sheaf $f_*(R^rj_*\mathcal F)$ is generically free on $S$ and commutes with base change. We answer this affirmatively, by proving a related statement about local cohomology: Let $ R$ be Noetherian algebra over a Noetherian domain $A$, and let $I \subset R$ be an ideal such that $ R/I $ is finitely generated as an $A$-module. Let $M$ be a finitely generated $R$-module. Then there exists a non-zero $g \in A$ such that the local cohomology modules $H^r_I(M) \otimes_A A_g$ are free over $A_g$ and for any ring map $A\rightarrow L$ factoring through $A_g$, we have $H^r_I(M) \otimes_A L \cong H^r_{I{\otimes_A}L}(M\otimes_A L)$ for all $r$.