The Stacks project

Remark 52.23.4. Proposition 52.23.3 is a local version of [Theorem 2.10 (i), Baranovsky]. It is straightforward to deduce the global results from the local one; we will sketch the argument. Namely, suppose $(R, \mathfrak m)$ is a complete Noetherian local ring and $X \to \mathop{\mathrm{Spec}}(R)$ is a proper morphism. For $n \geq 1$ set $X_ n = X \times _{\mathop{\mathrm{Spec}}(R)} \mathop{\mathrm{Spec}}(R/\mathfrak m^ n)$. Let $Z \subset X_1$ be a closed subset of the special fibre. Set $U = X \setminus Z$ and denote $j : U \to X$ the inclusion morphism. Suppose given an object

\[ (\mathcal{F}_ n) \text{ of } \textit{Coh}(U, \mathfrak m\mathcal{O}_ U) \]

which is flat over $R$ in the sense that $\mathcal{F}_ n$ is flat over $R/\mathfrak m^ n$ for all $n$. Assume that $j_*\mathcal{F}_1$ and $R^1j_*\mathcal{F}_1$ are coherent modules. Then affine locally on $X$ we get a canonical extension of $(\mathcal{F}_ n)$ by Proposition 52.23.3 and formation of this extension commutes with localization (by Lemma 52.16.11). Thus we get a canonical global object $(\mathcal{G}_ n)$ of $\textit{Coh}(X, \mathfrak m\mathcal{O}_ X)$ whose restriction of $U$ is $(\mathcal{F}_ n)$. By Grothendieck's existence theorem (Cohomology of Schemes, Proposition 30.25.4) we see there exists a coherent $\mathcal{O}_ X$-module $\mathcal{G}$ whose completion is $(\mathcal{G}_ n)$. In this way we see that $(\mathcal{F}_ n)$ is algebraizable, i.e., it is the completion of a coherent $\mathcal{O}_ U$-module.

We add that the coherence of $j_*\mathcal{F}_1$ and $R^1j_*\mathcal{F}_1$ is a condition on the special fibre. Namely, if we denote $j_1 : U_1 \to X_1$ the special fibre of $j : U \to X$, then we can think of $\mathcal{F}_1$ as a coherent sheaf on $U_1$ and we have $j_*\mathcal{F}_1 = j_{1, *}\mathcal{F}_1$ and $R^1j_*\mathcal{F}_1 = R^1j_{1, *}\mathcal{F}_1$. Hence for example if $X_1$ is $(S_2)$ and irreducible, we have $\dim (X_1) - \dim (Z) \geq 3$, and $\mathcal{F}_1$ is a locally free $\mathcal{O}_{U_1}$-module, then $j_{1, *}\mathcal{F}_1$ and $R^1j_{1, *}\mathcal{F}_1$ are coherent modules.

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