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 nth infinitesimal neighbourhood of Y in X. Let \mathcal{V} be the set of open subschemes V \subset X containing Y ordered by reverse inclusion.
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,
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,
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
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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