The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Lemma 49.15.5. Let $A$ be a Noetherian ring. Let $f \in \mathfrak a$ be an element of an ideal of $A$. Let $U = \mathop{\mathrm{Spec}}(A) \setminus V(\mathfrak a)$. Assume

  1. $A$ has a dualizing complex and is complete with respect to $f$,

  2. $A_ f$ is $(S_2)$ and for every minimal prime $\mathfrak p \subset A$, $f \not\in \mathfrak p$ and $\mathfrak q \in V(\mathfrak p) \cap V(\mathfrak a)$ we have $\dim ((A/\mathfrak p)_\mathfrak q) \geq 3$.

Then the completion functor

\[ \textit{Coh}(\mathcal{O}_ U) \longrightarrow \textit{Coh}(U, I\mathcal{O}_ U), \quad \mathcal{F} \longmapsto \mathcal{F}^\wedge \]

is fully faithful on the full subcategory of finite locally free objects.

Proof. We will show that Lemma 49.15.4 applies. Assumption (1) of Lemma 49.15.4 holds. Observe that $\text{cd}(A, (f)) \leq 1$, see Local Cohomology, Lemma 48.3.3. Since $A_ f$ is $(S_2)$ we see that every associated prime $\mathfrak p \subset A$, $f \not\in \mathfrak p$ is a minimal prime. Thus we get assumption (2) of Lemma 49.15.4. If $\mathfrak p \subset A$, $f \not\in \mathfrak p$ satisfies $V(\mathfrak p) \cap V(I) \subset V(\mathfrak a)$ and if $\mathfrak q \in V(\mathfrak p) \cap V(f)$ is a generic point, then $\dim ((A/\mathfrak p)_\mathfrak q) = 1$. Then we obtain $\dim (A_\mathfrak p) \geq 2$ by looking at the minimal primes $\mathfrak p_0 \subset \mathfrak p$ and using that $\dim ((A/\mathfrak p_0)_\mathfrak q) \geq 3$ by assumption. Thus $\text{depth}(A_\mathfrak p) \geq 2$ by the $(S_2)$ assumption. This verifies assumption (3) of Lemma 49.15.4 and the proof is complete. $\square$


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