Effective generic freeness and applications to local cohomology (Q6641559)

Let \(A\) be a Noetherian domain and \(R\) a finitely generated \(A\)-algebra. \textit{Grothendieck's Generic Freeness Lemma} states that for any finitely generated \(R\)-module \(M\), there exists a nonzero element \(a\) in \(A\) such that \(M\otimes_AA_a\) is a free \(A_a\)-module.\N\NLocal cohomology modules \(\text{H}_I^i(R)\), where \(I\) is an ideal of \(R\), are typically not finitely generated. As the main contribution of this paper, the authors extend Grothendieck's Generic Freeness Lemma to local cohomology modules \(\text{H}_I^i(R)\). More precisely, they prove the following:\N\N\vspace{0.2cm} {Theorem.} Let \(A\) be Noetherian domain containing a field \(\mathbb{K}\), \(R\) a smooth \(A\)-algebra, and \(I\) an ideal of \(R\). Assume that\N\begin{itemize}\N\item[(a)] \(\mathbb{K}\) is a field of characteristic zero, or\N\item[(b)] \(\mathbb{K}\) is a field of positive characteristic and the regular locus \(\text{Reg}(A)\) contains a nonempty open subset of \(\text{Spec}(A)\).\N\end{itemize}\NThen, there exists a nonzero element \(a\) in \(A\) such that \(\text{H}_I^i(R)\otimes_AA_a\) is a free \(A_a\)-module for all \(i\geq 0\).