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Clarify relationship with result in SGA2

Thanks for Remy van Dobben de Bruyn who pointed out that in the original
formulation of the proposition it wasn't clear that it implies the
result from SGA2. Now it does.
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aisejohan committed Aug 11, 2019
1 parent ec9697a commit bb46ce707cee59e2f3ec85091162e6047e6ce0c8
Showing with 62 additions and 20 deletions.
  1. +62 −20 algebraization.tex
@@ -8749,7 +8749,7 @@ \section{Invertible modules on punctured spectra, I}
identically zero, hence $\mathcal{L}$ is trivial.
\end{proof}

\begin{proposition}[Kollar]
\begin{proposition}[Koll\'ar]
\label{proposition-injective-pic}
\begin{reference}
\cite[Theorem 1.9]{Kollar-map-pic}
@@ -8758,16 +8758,17 @@ \section{Invertible modules on punctured spectra, I}
Assume
\begin{enumerate}
\item $A$ has a dualizing complex,
\item $A$ satisfies $(S_2)$,
\item $f$ is a nonzerodivisor,
\item $\text{depth}(A/fA) \geq 2$, or equivalently $\text{depth}(A) \geq 3$,
\item $\dim(A/\mathfrak p) \geq 4$ for every minimal prime of $A$.
\item if $f \in \mathfrak p \subset A$ is a prime ideal with
$\dim(A/\mathfrak p) = 2$, then $\text{depth}(A_\mathfrak p) \geq 2$.
\end{enumerate}
Let $U$, resp.\ $U_0$ be the punctured spectrum of $A$, resp.\ $A/fA$. The map
$$
\Pic(U) \to \Pic(U_0)
$$
is injective.
is injective. Finally, if (1), (2), (3), $A$ is $(S_2)$, and
$\dim(A) \geq 4$, then (4) holds.
\end{proposition}

\begin{proof}
@@ -8792,11 +8793,11 @@ \section{Invertible modules on punctured spectra, I}
$M_0 = \Gamma(X_0, \mathcal{L}_0) = \Gamma(U_0, \mathcal{L}_0)$.
Then $M$ is a finite $A$-module of depth $\geq 2$
and $M_0 \cong A/fA$, see proof of Lemma \ref{lemma-nonnegative-chi-triple}.
Observe that $H^2_\mathfrak m(M)$ is finite $A$-module by
Local Cohomology, Proposition \ref{local-cohomology-proposition-finiteness}
and assumptions (2), (5) on $A$ and the fact that
$M_\mathfrak p \cong A_\mathfrak p$
for points $\mathfrak p$ of $U$.
Note that $H^2_\mathfrak m(M)$ is finite $A$-module by
Local Cohomology, Lemma
\ref{local-cohomology-lemma-local-finiteness-for-finite-locally-free}
and the fact that $H^i_\mathfrak m(A) = 0$ for $i = 0, 1, 2$
since $\text{depth}(A) \geq 3$.
Consider the short exact sequence
$$
0 \to M/fM \to M_0 \to Q \to 0
@@ -8820,11 +8821,10 @@ \section{Invertible modules on punctured spectra, I}

\medskip\noindent
Let prove the lemma in a special case to elucidate the rest of the proof.
Namely, assume for a moment that $A/fA$ is $(S_2)$.
Then $H^2_\mathfrak m(A/fA)$ would be a finite length module
(by Local Cohomology, Proposition \ref{local-cohomology-proposition-finiteness})
and we would have $P(1) \leq \text{length}_A H^2_\mathfrak m(A/fA)$.
The exact same argument applied to $\mathcal{L}^n$ shows that
Namely, assume for a moment that $H^2_\mathfrak m(A/fA)$ is
a finite length module. Then
we would have $P(1) \leq \text{length}_A H^2_\mathfrak m(A/fA)$.
The exact same argument applied to $\mathcal{L}^{\otimes n}$ shows that
$P(n) \leq \text{length}_A H^2_\mathfrak m(A/fA)$ for all $n$.
Thus $P$ cannot have positive degree and we win.
In the rest of the proof we will modify this argument to give
@@ -8851,15 +8851,12 @@ \section{Invertible modules on punctured spectra, I}
Local Cohomology, Lemma \ref{local-cohomology-lemma-sitting-in-degrees}
the support is contained in the set of primes $\mathfrak p \subset A$ with
$\text{depth}_{A_\mathfrak p}(M_{0, \mathfrak p}) + \dim(A/\mathfrak p) \leq 2$.
Thus it suffices to show there is no prime $\mathfrak p$ with
Thus it suffices to show there is no prime $\mathfrak p$ containing $f$ with
$\dim(A/\mathfrak p) = 2$ and
$\text{depth}_{A_\mathfrak p}(M_{0, \mathfrak p}) = 0$.
However, because $M_{0, \mathfrak p} \cong (A/fA)_\mathfrak p$
this would give $\text{depth}(A_\mathfrak p) = 1$.
By assumption (5) we have $\dim(A_\mathfrak p) + \dim(A/\mathfrak p) \geq 4$
and by assumption (2) we have
$\text{depth}(A_\mathfrak p) \geq \min(2, \dim(A_\mathfrak p))$
and we conclude $\mathfrak p$ does not exist.
this would give $\text{depth}(A_\mathfrak p) = 1$ which contradicts
assumption (4).
Choose a section $t \in \Gamma(U, \mathcal{L}^{\otimes -1})$
which does not vanish in the points $\mathfrak p_1, \ldots, \mathfrak p_r$, see
Properties, Lemma
@@ -8907,8 +8904,53 @@ \section{Invertible modules on punctured spectra, I}
with support of dimension $1$ as indicated above this length
grows linearly in $n$ by
Algebra, Lemma \ref{algebra-lemma-support-dimension-d}.

\medskip\noindent
To finish the proof we prove the final assertion. Assume
$f \in \mathfrak m \subset A$ satisfies
(1), (2), (3), $A$ is $(S_2)$, and $\dim(A) \geq 4$.
Condition (1) implies $A$ is catenary, see
Dualizing, Lemma \ref{dualizing-lemma-universally-catenary}.
Then $\Spec(A)$ is equidimensional by Local Cohomology, Lemma
\ref{local-cohomology-lemma-catenary-S2-equidimensional}.
Thus $\dim(A_\mathfrak p) + \dim(A/\mathfrak p) \geq 4$
for every prime $\mathfrak p$ of $A$. Then
$\text{depth}(A_\mathfrak p) \geq \min(2, \dim(A_\mathfrak p))
\geq \min(2, 4 - \dim(A/\mathfrak p))$ and hence (4) holds.
\end{proof}

\begin{remark}
\label{remark-compare-SGA2}
In SGA2 we find the following result. Let $(A, \mathfrak m)$ be a
Noetherian local ring. Let $f \in \mathfrak m$. Assume $A$
is a quotient of a regular ring, the element
$f$ is a nonzerodivisor, and
\begin{enumerate}
\item[(a)] if $\mathfrak p \subset A$ is a prime ideal with
$\dim(A/\mathfrak p) = 1$, then $\text{depth}(A_\mathfrak p) \geq 2$, and
\item[(b)] $\text{depth}(A/fA) \geq 3$, or equivalently
$\text{depth}(A) \geq 4$.
\end{enumerate}
Let $U$, resp.\ $U_0$ be the punctured spectrum of $A$, resp.\ $A/fA$. Then
the map
$$
\Pic(U) \to \Pic(U_0)
$$
is injective. This is \cite[Exposee XI, Lemma 3.16]{SGA2}\footnote{Condition
(a) follows from condition (b), see
Algebra, Lemma \ref{algebra-lemma-depth-localization}.}. This result
from SGA2 follows from Proposition \ref{proposition-injective-pic}
because
\begin{enumerate}
\item a quotient of a regular ring has a dualizing complex (see
Dualizing, Lemma \ref{dualizing-lemma-regular-gorenstein} and
Proposition \ref{dualizing-proposition-dualizing-essentially-finite-type}), and
\item if $\text{depth}(A) \geq 4$ then $\text{depth}(A_\mathfrak p) \geq 2$
for all primes $\mathfrak p$ with $\dim(A/\mathfrak p) = 2$, see
Algebra, Lemma \ref{algebra-lemma-depth-localization}.
\end{enumerate}
\end{remark}




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