The Stacks project

52.24 Application to the completion functor

In this section we just combine some already obtained results in order to conveniently reference them. There are many (stronger) results we could state here.

Lemma 52.24.1. In Situation 52.16.1 assume

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

  2. $I = (f)$ generated by a single element,

  3. $A$ is local with maximal ideal $\mathfrak a = \mathfrak m$,

  4. one of the following is true

    1. $A_ f$ is $(S_2)$ and for $\mathfrak p \subset A$, $f \not\in \mathfrak p$ minimal we have $\dim (A/\mathfrak p) \geq 4$, or

    2. if $\mathfrak p \not\in V(f)$ and $V(\mathfrak p) \cap V(f) \not= \{ \mathfrak m\} $, then $\text{depth}(A_\mathfrak p) + \dim (A/\mathfrak p) > 3$.

Then with $U_0 = U \cap V(f)$ the completion functor

\[ \mathop{\mathrm{colim}}\nolimits _{U_0 \subset U' \subset U\text{ open}} \textit{Coh}(\mathcal{O}_{U'}) \longrightarrow \textit{Coh}(U, f\mathcal{O}_ U) \]

is an equivalence on the full subcategories of finite locally free objects.

Proof. It follows from Lemma 52.15.8 that the functor is fully faithful (details omitted). Let us prove essential surjectivity. Let $(\mathcal{F}_ n)$ be a finite locally free object of $\textit{Coh}(U, f\mathcal{O}_ U)$. By either Lemma 52.20.4 or Proposition 52.22.2 there exists a coherent $\mathcal{O}_ U$-module $\mathcal{F}$ such that $(\mathcal{F}_ n)$ is the completion of $\mathcal{F}$. Namely, for the application of either result the only thing to check is that $(\mathcal{F}_ n)$ satisfies the $(2, 3)$-inequalities. This is done in Lemma 52.20.6. If $y \in U_0$, then the $f$-adic completion of the stalk $\mathcal{F}_ y$ is isomorphic to a finite free module over the $f$-adic completion of $\mathcal{O}_{U, y}$. Hence $\mathcal{F}$ is finite locally free in an open neighbourhood $U'$ of $U_0$. This finishes the proof. $\square$

Lemma 52.24.2. In Situation 52.16.1 assume

  1. $I = (f)$ is principal,

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

  3. $f$ is a nonzerodivisor,

  4. $H^1_\mathfrak a(A/fA)$ and $H^2_\mathfrak a(A/fA)$ are finite $A$-modules.

Then with $U_0 = U \cap V(f)$ the completion functor

\[ \mathop{\mathrm{colim}}\nolimits _{U_0 \subset U' \subset U\text{ open}} \textit{Coh}(\mathcal{O}_{U'}) \longrightarrow \textit{Coh}(U, f\mathcal{O}_ U) \]

is an equivalence on the full subcategories of finite locally free objects.

Proof. The functor is fully faithful by Lemma 52.15.9. Essential surjectivity follows from Lemma 52.16.11. $\square$


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