The Stacks project

Lemma 58.17.6. Let $X$ be a Noetherian scheme and let $Y \subset X$ be a closed subscheme. Let $Y_ n \subset X$ be the $n$th infinitesimal neighbourhood of $Y$ in $X$. Assume one of the following holds

1. $X$ is quasi-affine and $\Gamma (X, \mathcal{O}_ X) \to \mathop{\mathrm{lim}}\nolimits \Gamma (Y_ n, \mathcal{O}_{Y_ n})$ is an isomorphism, or

2. $X$ has an ample invertible module $\mathcal{L}$ and $\Gamma (X, \mathcal{L}^{\otimes m}) \to \mathop{\mathrm{lim}}\nolimits \Gamma (Y_ n, \mathcal{L}^{\otimes m}|_{Y_ n})$ is an isomorphism for all $m \gg 0$, or

3. for every finite locally free $\mathcal{O}_ X$-module $\mathcal{E}$ the map $\Gamma (X, \mathcal{E}) \to \mathop{\mathrm{lim}}\nolimits \Gamma (Y_ n, \mathcal{E}|_{Y_ n})$ is an isomorphism.

Then the restriction functor $\textit{FÉt}_ X \to \textit{FÉt}_ Y$ is fully faithful.