The Stacks project

Proposition 52.23.1. In Situation 52.16.1 let $(\mathcal{F}_ n)$ be an object of $\textit{Coh}(U, I\mathcal{O}_ U)$. Assume

  1. there exist $f_1, \ldots , f_ d \in I$ such that for $y \in U \cap Y$ the ideal $I\mathcal{O}_{X, y}$ is generated by $f_1, \ldots , f_ d$ and $f_1, \ldots , f_ d$ form a $\mathcal{F}_ y^\wedge $-regular sequence,

  2. $H^0(U, \mathcal{F}_1)$ and $H^1(U, \mathcal{F}_1)$ are finite $A$-modules.

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

Proof. We will prove this by verifying hypotheses (a), (b), and (c) of Lemma 52.16.10. For every $n$ we have a short exact sequence

\[ 0 \to I^ n\mathcal{F}_{n + 1} \to \mathcal{F}_{n + 1} \to \mathcal{F}_ n \to 0 \]

Since $f_1, \ldots , f_ d$ forms a regular sequence (and hence quasi-regular, see Algebra, Lemma 10.69.2) on each of the “stalks” $\mathcal{F}_ y^\wedge $ and since we have $I\mathcal{F}_ n = (f_1, \ldots , f_ d)\mathcal{F}_ n$ for all $n$, we find that

\[ I^ n\mathcal{F}_{n + 1} = \bigoplus \nolimits _{e_1 + \ldots + e_ d = n} \mathcal{F}_1 \cdot f_1^{e_1} \ldots f_ d^{e_ d} \]

by checking on stalks. Using the assumption of finiteness of $H^0(U, \mathcal{F}_1)$ and induction, we first conclude that $M_ n = H^0(U, \mathcal{F}_ n)$ is a finite $A$-module for all $n$. In this way we see that condition (c) of Lemma 52.16.10 holds. We also see that

\[ \bigoplus \nolimits _{n \geq 0} H^1(U, I^ n\mathcal{F}_{n + 1}) \]

is a finite graded $R = \bigoplus I^ n/I^{n +1}$-module. By Cohomology, Lemma 20.35.1 we conclude that condition (a) of Lemma 52.16.10 is satisfied. Finally, condition (b) of Lemma 52.16.10 is satisfied because $\bigoplus H^0(U, I^ n\mathcal{F}_{n + 1})$ is a finite graded $R$-module and we can apply Cohomology, Lemma 20.35.3. $\square$

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