# stacks/stacks-project

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.
 @@ -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}