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
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
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
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. $\square$
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