The Stacks project

Lemma 52.15.1. Let $X$ be a Noetherian scheme and let $Y \subset X$ be a closed subscheme. Let $Y_ n \subset X$ be the $n$th infinitesimal neighbourhood of $Y$ in $X$. Consider the following conditions

  1. $X$ is quasi-affine and $\Gamma (X, \mathcal{O}_ X) \to \mathop{\mathrm{lim}}\nolimits \Gamma (Y_ n, \mathcal{O}_{Y_ n})$ is an isomorphism,

  2. $X$ has an ample invertible module $\mathcal{L}$ and $\Gamma (X, \mathcal{L}^{\otimes m}) \to \mathop{\mathrm{lim}}\nolimits \Gamma (Y_ n, \mathcal{L}^{\otimes m}|_{Y_ n})$ is an isomorphism for all $m \gg 0$,

  3. for every finite locally free $\mathcal{O}_ X$-module $\mathcal{E}$ the map $\Gamma (X, \mathcal{E}) \to \mathop{\mathrm{lim}}\nolimits \Gamma (Y_ n, \mathcal{E}|_{Y_ n})$ is an isomorphism, and

  4. the completion functor $\textit{Coh}(\mathcal{O}_ X) \to \textit{Coh}(X, \mathcal{I})$ is fully faithful on the full subcategory of finite locally free objects.

Then (1) $\Rightarrow $ (2) $\Rightarrow $ (3) $\Rightarrow $ (4) and (4) $\Rightarrow $ (3).

Proof. Proof of (3) $\Rightarrow $ (4). If $\mathcal{F}$ and $\mathcal{G}$ are finite locally free on $X$, then considering $\mathcal{H} = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{G}, \mathcal{F})$ and using Cohomology of Schemes, Lemma 30.23.5 we see that (3) implies (4).

Proof of (2) $\rightarrow $ (3). Namely, let $\mathcal{L}$ be ample on $X$ and suppose that $\mathcal{E}$ is a finite locally free $\mathcal{O}_ X$-module. We claim we can find a universally exact sequence

\[ 0 \to \mathcal{E} \to (\mathcal{L}^{\otimes p})^{\oplus r} \to (\mathcal{L}^{\otimes q})^{\oplus s} \]

for some $r, s \geq 0$ and $0 \ll p \ll q$. If this holds, then using the exact sequence

\[ 0 \to \mathop{\mathrm{lim}}\nolimits \Gamma (\mathcal{E}|_{Y_ n}) \to \mathop{\mathrm{lim}}\nolimits \Gamma ((\mathcal{L}^{\otimes p})^{\oplus r}|_{Y_ n}) \to \mathop{\mathrm{lim}}\nolimits \Gamma ((\mathcal{L}^{\otimes q})^{\oplus s}|_{Y_ n}) \]

and the isomorphisms in (2) we get the isomorphism in (3). To prove the claim, consider the dual locally free module $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{E}, \mathcal{O}_ X)$ and apply Properties, Proposition 28.26.13 to find a surjection

\[ (\mathcal{L}^{\otimes -p})^{\oplus r} \longrightarrow \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{E}, \mathcal{O}_ X) \]

Taking duals we obtain the first map in the exact sequence (it is universally injective because being a surjection is universal). Repeat with the cokernel to get the second. Some details omitted.

Proof of (1) $\Rightarrow $ (2). This is true because if $X$ is quasi-affine then $\mathcal{O}_ X$ is an ample invertible module, see Properties, Lemma 28.27.1.

We omit the proof of (4) $\Rightarrow $ (3). $\square$

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