The Stacks project

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. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0EKN. Beware of the difference between the letter 'O' and the digit '0'.