Formulate optimal proposition

Thanks to J\'anos Koll\'ar

dualizing.tex

@@ -2384,7 +2384,7 @@ \section{Finiteness of local cohomology, I}
 be the inclusion of a nonempty open subscheme with complement $Z$. Assume
 that for all $z \in Z$ and any associated prime $\mathfrak p$ of
 the completion $\mathcal{O}_{X, z}^\wedge$
-satisfies $\dim(\mathcal{O}_{X, z}^\wedge/\mathfrak p) \geq 2$.
+we have $\dim(\mathcal{O}_{X, z}^\wedge/\mathfrak p) \geq 2$.
 Then $j_*\mathcal{O}_U$ is coherent.
 \end{lemma}
 
@@ -2441,6 +2441,24 @@ \section{Finiteness of local cohomology, I}
 Descent, Lemma \ref{descent-lemma-finite-type-descends}.
 \end{proof}
 
+\begin{remark}
+\label{remark-closure}
+Let $j : U \to X$ be an open immersion of locally Noetherian schemes.
+Let $x \in U$. Let $i_x : W_x \to U$ be the integral closed subscheme
+with generic point $x$ and let $\overline{\{x\}}$ be the closure in $X$.
+Then we have a commutative diagram
+$$
+\xymatrix{
+W_x \ar[d]_{i_x} \ar[r]_{j'} & \overline{\{x\}} \ar[d]^i \\
+U \ar[r]^j & X
+}
+$$
+We have $j_*i_{x, *}\mathcal{O}_{W_x} = i_*j'_*\mathcal{O}_{W_x}$.
+As the left vertical arrow is a closed immersion we see that
+$j_*i_{x, *}\mathcal{O}_{W_x}$ is coherent if and only of
+$j'_*\mathcal{O}_{W_x}$ is coherent.
+\end{remark}
+
 \begin{lemma}
 \label{lemma-finiteness-pushforward-general}
 Let $X$ be a locally Noetherian scheme.
@@ -2462,16 +2480,9 @@ \section{Finiteness of local cohomology, I}
 By Lemma \ref{lemma-check-finiteness-pushforward-on-associated-points}
 it suffices to prove $j_*i_{x, *}\mathcal{O}_{W_x}$
 is coherent for $x \in \text{Ass}(\mathcal{F})$.
-Then we have a commutative diagram
-$$
-\xymatrix{
-W_x \ar[d] \ar[r] & \overline{\{x\}} \ar[d] \\
-U \ar[r] & X
-}
-$$
-As the vertical arrows are closed immersions
-it suffices to prove the result for the pushforward
-along the top horizontal arrow. Moreover, the conditions
+By Remark \ref{remark-closure}
+we reduce to proving $(W_x \to \overline{\{x\}})_*\mathcal{O}_{W_x}$
+is coherent. The conditions
 (1), (2), and (3) are inherited by $W_x \subset \overline{\{x\}}$;
 this is clear for (1) and (3) and for (2) this follows from
 More on Algebra, Lemma \ref{more-algebra-lemma-formal-fibres-normal} and
@@ -2510,8 +2521,46 @@ \section{Finiteness of local cohomology, I}
 $j_*\mathcal{O}_{X' \cap U}$ corresponds to $\kappa(x)$ as an
 $\mathcal{O}_{X, z}$-module which cannot be finite as $x$ is not
 a closed point.
+
+\medskip\noindent
+In fact, the converse of Lemma \ref{lemma-sharp-finiteness-pushforward}
+holds true: given an open immersion $j : U \to X$ of integral Noetherian
+schemes and there exists a $z \in X \setminus U$ and an associated prime
+$\mathfrak p$ of the completion $\mathcal{O}_{X, z}^\wedge$
+with $\dim(\mathcal{O}_{X, z}^\wedge/\mathfrak p) = 1$,
+then $j_*\mathcal{O}_U$ is not coherent. Namely, you can pass to
+the local ring, you can enlarge $U$ to the punctured spectrum,
+you can pass to the completion, and then the argument above gives
+the nonfiniteness.
 \end{remark}
 
+\begin{proposition}[Koll\'ar]
+\label{proposition-kollar}
+Let $j : U \to X$ be an open immersion of locally Noetherian schemes
+with complement $Z$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_U$-module.
+The following are equivalent
+\begin{enumerate}
+\item $j_*\mathcal{F}$ is coherent,
+\item for $x \in \text{Ass}(\mathcal{F})$ let $W = \overline{\{x\}} \subset X$.
+Then for all $z \in Z \cap W$ and any associated prime $\mathfrak p$ of the
+completion $\mathcal{O}_{W, z}^\wedge$ we have
+$\dim(\mathcal{O}_{W, z}^\wedge/\mathfrak p) \geq 2$.
+\end{enumerate}
+\end{proposition}
+
+\begin{proof}
+If (2) holds we get (1) by a combination of
+Lemmas \ref{lemma-check-finiteness-pushforward-on-associated-points},
+Remark \ref{remark-closure}, and
+Lemma \ref{lemma-sharp-finiteness-pushforward}.
+If (2) does not hold, then $j_*i_{x, *}\mathcal{O}_{W_x}$ is not finite
+for some $x \in \text{Ass}(\mathcal{F})$ by the discussion in
+Remark \ref{remark-no-finiteness-pushforward}
+(and Remark \ref{remark-closure}).
+Thus $j_*\mathcal{F}$ is not coherent by
+Lemma \ref{lemma-check-finiteness-pushforward-on-associated-points}.
+\end{proof}
+