The Stacks project

Example 52.28.6. Let $k$ be a field and let $X$ be a proper variety over $k$. Let $Y \subset X$ be an effective Cartier divisor such that $\mathcal{O}_ X(Y)$ is ample and denote $\mathcal{I} \subset \mathcal{O}_ X$ the corresponding sheaf of ideals. Let $(\mathcal{E}_ n)$ an object of $\textit{Coh}(X, \mathcal{I})$ with $\mathcal{E}_ n$ finite locally free. Here are some special cases of Proposition 52.28.5.

1. If $X$ is a curve or a surface, we don't learn anything.

2. If $X$ is a Cohen-Macaulay threefold, then $(\mathcal{E}_ n)$ is the completion of a coherent $\mathcal{O}_ X$-module $\mathcal{E}$.

3. More generally, if $\dim (X) \geq 3$ and $X$ is $(S_3)$, then $(\mathcal{E}_ n)$ is the completion of a coherent $\mathcal{O}_ X$-module $\mathcal{E}$.

Of course, if $\mathcal{E}$ exists, then $\mathcal{E}$ is finite locally free in an open neighbourhood of $Y$.