Local Cohomology and Base Change

Abstract

Let $X\stackrel{f}{⟶}S$ 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 $X\setminus Z\hookrightarrow X$. Kollár asked whether for any coherent sheaf $\mathcal{F}$ on $X\setminus Z$ and any index $r\ge 1$, the sheaf $f_{\ast }(R^r{j}_{\ast }\mathcal{F})$ 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 $H_I^r(M){\otimes }_A{A}_g$ are free over $A_g$ and for any ring map $A\to L$ factoring through $A_g$, we have $H_I^r(M){\otimes }_{A}L\cong H_{I{\otimes }_{A}L}^r(M{\otimes }_{A}L)$ for all $r$.