Lemma 58.26.1 (0EY9)—The Stacks project

Lemma 58.26.1. Let $(A, \mathfrak m)$ be a Noetherian local ring of depth $\geq 2$ . Let $B = A[[x_1, \ldots , x_ d]]$ with $d \geq 1$ . Set $Y = \mathop{\mathrm{Spec}}(B)$ and $Y_0 = V(x_1, \ldots , x_ d)$ . For any open subscheme $V \subset Y$ with $V_0 = V \cap Y_0$ equal to $Y_0 \setminus \{ \mathfrak m_ B\}$ the restriction functor

$\textit{FÉt}_ V \longrightarrow \textit{FÉt}_{V_0}$

is fully faithful.

Proof. Set $I = (x_1, \ldots , x_ d)$ . Set $X = \mathop{\mathrm{Spec}}(A)$ . If we use the map $Y \to X$ to identify $Y_0$ with $X$ , then $V_0$ is identified with the punctured spectrum $U$ of $A$ . Pushing forward modules by this affine morphism we get

$\begin{align*} \mathop{\mathrm{lim}}\nolimits _ n \Gamma (V_0, \mathcal{O}_ V/I^ n\mathcal{O}_ V) & = \mathop{\mathrm{lim}}\nolimits _ n \Gamma (V_0, \mathcal{O}_ Y/I^ n\mathcal{O}_ Y) \\ & = \mathop{\mathrm{lim}}\nolimits _ n \Gamma (U, \mathcal{O}_ U[x_1, \ldots , x_ d]/(x_1, \ldots , x_ d)^ n) \\ & = \mathop{\mathrm{lim}}\nolimits _ n A[x_1, \ldots , x_ d]/(x_1, \ldots , x_ d)^ n \\ & = B \end{align*}$

Namely, as the depth of $A$ is $\geq 2$ we have $\Gamma (U, \mathcal{O}_ U) = A$ , see Local Cohomology, Lemma 51.8.2. Thus for any $V \subset Y$ open as in the lemma we get

$B = \Gamma (Y, \mathcal{O}_ Y) \to \Gamma (V, \mathcal{O}_ V) \to \mathop{\mathrm{lim}}\nolimits _ n \Gamma (V_0, \mathcal{O}_ Y/I^ n\mathcal{O}_ Y) = B$

which implies both arrows are isomorphisms (small detail omitted). By Algebraic and Formal Geometry, Lemma 52.15.1 we conclude that $\textit{Coh}(\mathcal{O}_ V) \to \textit{Coh}(V, I\mathcal{O}_ V)$ is fully faithful on the full subcategory of finite locally free objects. Thus we conclude by Lemma 58.17.1. $\square$

The Stacks project

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