Trivial but interesting combination of results

local-cohomology.tex

@@ -5562,6 +5562,37 @@ \section{Algebraization of local cohomology, bootstrap}
 The result follows easily from Lemma \ref{lemma-bootstrap}.
 \end{proof}
 
+\begin{remark}
+\label{remark-combine}
+Let $I \subset \mathfrak a \subset A$ be ideals of a Noetherian ring $A$
+and let $M$ be a finite $A$-module. Let $s$ and $d$ be integers.
+Suppose that
+\begin{enumerate}
+\item $A, I, \mathfrak a, M$ satisfy the conditions of
+Situation \ref{situation-bootstrap} for $s$ and $d$, and
+\item $A, I, \mathfrak a, M$ satisfy the conditions of
+Lemma \ref{lemma-algebraize-local-cohomology-general}
+for $s + 1$ and $d$ with $J = \mathfrak a$.
+\end{enumerate}
+Then there exists an ideal
+$J_0 \subset \mathfrak a$ with $V(J_0) \cap V(I) = V(\mathfrak a)$
+such that for any $J' \subset J_0$ with $V(J') \cap V(I) = V(\mathfrak a)$
+the map
+$$
+H^{s + 1}_{J'}(M) \longrightarrow \lim H^{s + 1}_\mathfrak a(M/I^nM)
+$$
+is an isomorphism. Namely, we have the existence of $J_0$
+and the isomorphism
+$H^{s + 1}_{J'}(M) = H^{s + 1}(R\Gamma_\mathfrak a(M)^\wedge)$
+by Lemma \ref{lemma-algebraize-local-cohomology-general},
+we have $H^{s + 1}(R\Gamma_\mathfrak a(M)^\wedge)$ sandwiched
+between the limit and $R^1\lim H^s_\mathfrak a(M/I^nM)$ by
+Dualizing Complexes, Lemma \ref{dualizing-lemma-completion-local},
+and we have the vanishing of
+$R^1\lim H^s_\mathfrak a(M/I^nM)$ by Lemma \ref{lemma-final-bootstrap}.
+\end{remark}
+
+