Remark 76.42.12 (08BF): Unwinding Grothendieck's existence theorem

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

$\mathcal{F}_ n$ is a coherent $\mathcal{O}_{X_ n}$-module,
$\varphi _ n : i_ n^*\mathcal{F}_{n + 1} \to \mathcal{F}_ n$ is an isomorphism, and
$\text{Supp}(\mathcal{F}_1)$ 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.

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