## 52.15 The completion functor

Let $X$ be a Noetherian scheme. Let $Y \subset X$ be a closed subscheme with quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_ X$. In this section we consider inverse systems of coherent $\mathcal{O}_ X$-modules $(\mathcal{F}_ n)$ with $\mathcal{F}_ n$ annihilated by $I^ n$ such that the transition maps induce isomorphisms $\mathcal{F}_{n + 1}/I^ n\mathcal{F}_{n + 1} \to \mathcal{F}_ n$. The category of these inverse systems was denoted

$\textit{Coh}(X, \mathcal{I})$

in Cohomology of Schemes, Section 30.23. This category is equivalent to the category of coherent modules on the formal completion of $X$ along $Y$; however, since we have not yet introduced formal schemes or coherent modules on them, we cannot use this terminology here. We are particularly interested in the completion functor

$\textit{Coh}(\mathcal{O}_ X) \longrightarrow \textit{Coh}(X, \mathcal{I}),\quad \mathcal{F} \longmapsto \mathcal{F}^\wedge$

See Cohomology of Schemes, Equation (30.23.3.1).

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). $\square$

Given a Noetherian scheme and a quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_ X$ we will say an object $(\mathcal{F}_ n)$ of $\textit{Coh}(X, \mathcal{I})$ is finite locally free if each $\mathcal{F}_ n$ is a finite locally free $\mathcal{O}_ X/\mathcal{I}^ n$-module.

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). $\square$

Lemma 52.15.3. Let $X$ be a Noetherian scheme. Let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals. The functor

$\textit{Coh}(X, \mathcal{I}) \longrightarrow \text{Pro-}\mathit{QCoh}(\mathcal{O}_ X)$

is fully faithful, see Categories, Remark 4.22.5.

Proof. Let $(\mathcal{F}_ n)$ and $(\mathcal{G}_ n)$ be objects of $\textit{Coh}(X, \mathcal{I})$. A morphism of pro-objects $\alpha$ from $(\mathcal{F}_ n)$ to $(\mathcal{G}_ n)$ is given by a system of maps $\alpha _ n : \mathcal{F}_{n'(n)} \to \mathcal{G}_ n$ where $\mathbf{N} \to \mathbf{N}$, $n \mapsto n'(n)$ is an increasing function. Since $\mathcal{F}_ n = \mathcal{F}_{n'(n)}/\mathcal{I}^ n\mathcal{F}_{n'(n)}$ and since $\mathcal{G}_ n$ is annihilated by $\mathcal{I}^ n$ we see that $\alpha _ n$ induces a map $\mathcal{F}_ n \to \mathcal{G}_ n$. $\square$

Next we add some examples of the kind of fully faithfulness result we will be able to prove using the work done earlier in this chapter.

Lemma 52.15.4. Let $I \subset \mathfrak a$ be ideals of a Noetherian ring $A$. Let $U = \mathop{\mathrm{Spec}}(A) \setminus V(\mathfrak a)$. Assume

1. $A$ is $I$-adically complete and has a dualizing complex,

2. for any associated prime $\mathfrak p \subset A$, $I \not\subset \mathfrak p$ and $\mathfrak q \in V(\mathfrak p) \cap V(\mathfrak a)$ we have $\dim ((A/\mathfrak p)_\mathfrak q) > \text{cd}(A, I) + 1$.

3. for $\mathfrak p \subset A$, $I \not\subset \mathfrak p$ with with $V(\mathfrak p) \cap V(I) \subset V(\mathfrak a)$ we have $\text{depth}(A_\mathfrak p) \geq 2$.

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. By Lemma 52.15.1 it suffices to show that

$\Gamma (U, \mathcal{O}_ U) = \mathop{\mathrm{lim}}\nolimits \Gamma (U, \mathcal{O}_ U/I^ n\mathcal{O}_ U)$

This follows immediately from Lemma 52.12.3. $\square$

Lemma 52.15.5. 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 52.15.4 applies. Assumption (1) of Lemma 52.15.4 holds. Observe that $\text{cd}(A, (f)) \leq 1$, see Local Cohomology, Lemma 51.4.3. 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 52.15.4. 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 52.15.4 and the proof is complete. $\square$

Lemma 52.15.6. Let $A$ be a Noetherian ring. Let $f \in \mathfrak a \subset A$ be an element of an ideal of $A$. Let $U = \mathop{\mathrm{Spec}}(A) \setminus V(\mathfrak a)$. Assume

1. $A$ is $f$-adically complete,

2. $H^1_\mathfrak a(A)$ and $H^2_\mathfrak a(A)$ are annihilated by a power of $f$.

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. By Lemma 52.15.1 it suffices to show that

$\Gamma (U, \mathcal{O}_ U) = \mathop{\mathrm{lim}}\nolimits \Gamma (U, \mathcal{O}_ U/I^ n\mathcal{O}_ U)$

This follows immediately from Lemma 52.12.6. $\square$

Lemma 52.15.7. 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. for every prime $\mathfrak p \subset A$, $f \not\in \mathfrak p$ and $\mathfrak q \in V(\mathfrak p) \cap V(\mathfrak a)$ we have $\text{depth}(A_\mathfrak p) + \dim ((A/\mathfrak p)_\mathfrak q) > 2$.

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. This follows from Lemma 52.15.6 and Local Cohomology, Proposition 51.10.1. $\square$

Lemma 52.15.8. Let $I \subset \mathfrak a \subset A$ be ideals of a Noetherian ring $A$. Let $U = \mathop{\mathrm{Spec}}(A) \setminus V(\mathfrak a)$. Let $\mathcal{V}$ be the set of open subschemes of $U$ containing $U \cap V(I)$ ordered by reverse inclusion. Assume

1. $A$ is $I$-adically complete and has a dualizing complex,

2. for any associated prime $\mathfrak p \subset A$ with $I \not\subset \mathfrak p$ and $V(\mathfrak p) \cap V(I) \not\subset V(\mathfrak a)$ and $\mathfrak q \in V(\mathfrak p) \cap V(\mathfrak a)$ we have $\dim ((A/\mathfrak p)_\mathfrak q) > \text{cd}(A, I) + 1$.

Then the completion functor

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

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

Proof. By Lemma 52.15.2 it suffices to show that

$\mathop{\mathrm{colim}}\nolimits _\mathcal {V} \Gamma (V, \mathcal{O}_ V) = \mathop{\mathrm{lim}}\nolimits \Gamma (U, \mathcal{O}_ U/I^ n\mathcal{O}_ U)$

This follows immediately from Proposition 52.12.2. $\square$

Lemma 52.15.9. Let $A$ be a Noetherian ring. Let $f \in \mathfrak a \subset A$ be an element of an ideal of $A$. Let $U = \mathop{\mathrm{Spec}}(A) \setminus V(\mathfrak a)$. Let $\mathcal{V}$ be the set of open subschemes of $U$ containing $U \cap V(f)$ ordered by reverse inclusion. Assume

1. $A$ is $f$-adically complete,

2. $f$ is a nonzerodivisor,

3. $H^1_\mathfrak a(A/fA)$ is a finite $A$-module.

Then the completion functor

$\mathop{\mathrm{colim}}\nolimits _\mathcal {V} \textit{Coh}(\mathcal{O}_ V) \longrightarrow \textit{Coh}(U, f\mathcal{O}_ U), \quad \mathcal{F} \longmapsto \mathcal{F}^\wedge$

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

Proof. By Lemma 52.15.2 it suffices to show that

$\mathop{\mathrm{colim}}\nolimits _\mathcal {V} \Gamma (V, \mathcal{O}_ V) = \mathop{\mathrm{lim}}\nolimits \Gamma (U, \mathcal{O}_ U/I^ n\mathcal{O}_ U)$

This follows immediately from Lemma 52.12.5. $\square$

Lemma 52.15.10. Let $I \subset \mathfrak a \subset A$ be ideals of a Noetherian ring $A$. Let $U = \mathop{\mathrm{Spec}}(A) \setminus V(\mathfrak a)$. Let $\mathcal{V}$ be the set of open subschemes of $U$ containing $U \cap V(I)$ ordered by reverse inclusion. Let $\mathcal{F}$ and $\mathcal{G}$ be coherent $\mathcal{O}_ V$-modules for some $V \in \mathcal{V}$. The map

$\mathop{\mathrm{colim}}\nolimits _{V' \geq V} \mathop{\mathrm{Hom}}\nolimits _ V(\mathcal{G}|_{V'}, \mathcal{F}|_{V'}) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _{\textit{Coh}(U, I\mathcal{O}_ U)}(\mathcal{G}^\wedge , \mathcal{F}^\wedge )$

is bijective if the following assumptions hold:

1. $A$ is $I$-adically complete and has a dualizing complex,

2. if $x \in \text{Ass}(\mathcal{F})$, $x \not\in V(I)$, $\overline{\{ x\} } \cap V(I) \not\subset V(\mathfrak a)$ and $z \in \overline{\{ x\} } \cap V(\mathfrak a)$, then $\dim (\mathcal{O}_{\overline{\{ x\} }, z}) > \text{cd}(A, I) + 1$.

Proof. We may choose coherent $\mathcal{O}_ U$-modules $\mathcal{F}'$ and $\mathcal{G}'$ whose restriction to $V$ is $\mathcal{F}$ and $\mathcal{G}$, see Properties, Lemma 28.22.5. We may modify our choice of $\mathcal{F}'$ to ensure that $\text{Ass}(\mathcal{F}') \subset V$, see for example Local Cohomology, Lemma 51.15.1. Thus we may and do replace $V$ by $U$ and $\mathcal{F}$ and $\mathcal{G}$ by $\mathcal{F}'$ and $\mathcal{G}'$. Set $\mathcal{H} = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ U}(\mathcal{G}, \mathcal{F})$. This is a coherent $\mathcal{O}_ U$-module. We have

$\mathop{\mathrm{Hom}}\nolimits _ V(\mathcal{G}|_ V, \mathcal{F}|_ V) = H^0(V, \mathcal{H}) \quad \text{and}\quad \mathop{\mathrm{lim}}\nolimits H^0(U, \mathcal{H}/\mathcal{I}^ n\mathcal{H}) = \mathop{\mathrm{Mor}}\nolimits _{\textit{Coh}(U, I\mathcal{O}_ U)} (\mathcal{G}^\wedge , \mathcal{F}^\wedge )$

See Cohomology of Schemes, Lemma 30.23.5. Thus if we can show that the assumptions of Proposition 52.12.2 hold for $\mathcal{H}$, then the proof is complete. This holds because $\text{Ass}(\mathcal{H}) \subset \text{Ass}(\mathcal{F})$. See Cohomology of Schemes, Lemma 30.11.2. $\square$

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