The Stacks project

Remark 76.42.12 (Unwinding Grothendieck's existence theorem). Let $A$ be a Noetherian ring complete with respect to an ideal $I$. Write $S = \mathop{\mathrm{Spec}}(A)$ and $S_ n = \mathop{\mathrm{Spec}}(A/I^ n)$. Let $X \to S$ be a morphism of algebraic spaces that is separated and of finite type. For $n \geq 1$ we set $X_ n = X \times _ S S_ n$. Picture:

\[ \xymatrix{ X_1 \ar[r]_{i_1} \ar[d] & X_2 \ar[r]_{i_2} \ar[d] & X_3 \ar[r] \ar[d] & \ldots & X \ar[d] \\ S_1 \ar[r] & S_2 \ar[r] & S_3 \ar[r] & \ldots & S } \]

In this situation we consider systems $(\mathcal{F}_ n, \varphi _ n)$ where

  1. $\mathcal{F}_ n$ is a coherent $\mathcal{O}_{X_ n}$-module,

  2. $\varphi _ n : i_ n^*\mathcal{F}_{n + 1} \to \mathcal{F}_ n$ is an isomorphism, and

  3. $\text{Supp}(\mathcal{F}_1)$ is proper over $S_1$.

Theorem 76.42.11 says that the completion functor

\[ \begin{matrix} \text{coherent }\mathcal{O}_ X\text{-modules }\mathcal{F} \\ \text{with support proper over }A \end{matrix} \quad \longrightarrow \quad \begin{matrix} \text{systems }(\mathcal{F}_ n) \\ \text{as above} \end{matrix} \]

is an equivalence of categories. In the special case that $X$ is proper over $A$ we can omit the conditions on the supports.

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Your email address will not be published.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 08BF. Beware of the difference between the letter 'O' and the digit '0'.