The Stacks project

Proposition 52.22.4 (Algebraization for ideals with few generators). In Situation 52.16.1 let $(\mathcal{F}_ n)$ be an object of $\textit{Coh}(U, I\mathcal{O}_ U)$. Assume

  1. $A$ has a dualizing complex,

  2. $V(I) = V(f_1, \ldots , f_ d)$ for some $d \geq 1$ and $f_1, \ldots , f_ d \in A$,

  3. one of the following is true

    1. $(\mathcal{F}_ n)$ satisfies the $(d + 1, d + 2)$-inequalities (Definition 52.19.1), or

    2. for $y \in U \cap Y$ and a prime $\mathfrak p \subset \mathcal{O}_{X, y}^\wedge $ with $\mathfrak p \not\in V(I\mathcal{O}_{X, y}^\wedge )$ we have

      \[ \text{depth}((\mathcal{F}^\wedge _ y)_\mathfrak p) + \dim (\mathcal{O}_{X, y}^\wedge /\mathfrak p) + \delta ^ Y_ Z(y) > d + 2 \]

Then $(\mathcal{F}_ n)$ extends to $X$. In particular, if $A$ is $I$-adically complete, then $(\mathcal{F}_ n)$ is the completion of a coherent $\mathcal{O}_ U$-module.

Proof. We may assume $I = (f_1, \ldots , f_ d)$, see Cohomology of Schemes, Lemma 30.23.11. Then we see that all fibres of the blowup of $X$ in $I$ have dimension at most $d - 1$. Thus we get the extension from Lemma 52.22.3. The final statement follows from Lemma 52.16.3. $\square$

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