The Stacks project

Lemma 52.15.2. Let $X$ be a Noetherian scheme and let $Y \subset X$ be a closed subscheme with ideal sheaf $\mathcal{I} \subset \mathcal{O}_ X$. Let $Y_ n \subset X$ be the $n$th infinitesimal neighbourhood of $Y$ in $X$. Let $\mathcal{V}$ be the set of open subschemes $V \subset X$ containing $Y$ ordered by reverse inclusion.

  1. $X$ is quasi-affine and

    \[ \mathop{\mathrm{colim}}\nolimits _\mathcal {V} \Gamma (V, \mathcal{O}_ V) \longrightarrow \mathop{\mathrm{lim}}\nolimits \Gamma (Y_ n, \mathcal{O}_{Y_ n}) \]

    is an isomorphism,

  2. $X$ has an ample invertible module $\mathcal{L}$ and

    \[ \mathop{\mathrm{colim}}\nolimits _\mathcal {V} \Gamma (V, \mathcal{L}^{\otimes m}) \longrightarrow \mathop{\mathrm{lim}}\nolimits \Gamma (Y_ n, \mathcal{L}^{\otimes m}|_{Y_ n}) \]

    is an isomorphism for all $m \gg 0$,

  3. for every $V \in \mathcal{V}$ and every finite locally free $\mathcal{O}_ V$-module $\mathcal{E}$ the map

    \[ \mathop{\mathrm{colim}}\nolimits _{V' \geq V} \Gamma (V', \mathcal{E}|_{V'}) \longrightarrow \mathop{\mathrm{lim}}\nolimits \Gamma (Y_ n, \mathcal{E}|_{Y_ n}) \]

    is an isomorphism, and

  4. the completion functor

    \[ \mathop{\mathrm{colim}}\nolimits _\mathcal {V} \textit{Coh}(\mathcal{O}_ V) \longrightarrow \textit{Coh}(X, \mathcal{I}), \quad \mathcal{F} \longmapsto \mathcal{F}^\wedge \]

    is fully faithful on the full subcategory of finite locally free objects (see explanation above).

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

Proof. Observe that $\mathcal{V}$ is a directed set, so the colimits are as in Categories, Section 4.19. The rest of the argument is almost exactly the same as the argument in the proof of Lemma 52.15.1; we urge the reader to skip it.

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

Proof of (2) $\Rightarrow $ (3). Let $\mathcal{L}$ be ample on $X$ and suppose that $\mathcal{E}$ is a finite locally free $\mathcal{O}_ V$-module for some $V \in \mathcal{V}$. We claim we can find a universally exact sequence

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

for some $r, s \geq 0$ and $0 \ll p \ll q$. If this is true, then the isomorphism in (2) will imply the isomorphism in (3). To prove the claim, recall that $\mathcal{L}|_ V$ is ample, see Properties, Lemma 28.26.14. Consider the dual locally free module $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ V}(\mathcal{E}, \mathcal{O}_ V)$ and apply Properties, Proposition 28.26.13 to find a surjection

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

(it is universally injective because being a surjection is universal). Taking duals we obtain the first map in the exact sequence. 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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