Lemma 52.16.8 (0EIR)—The Stacks project

Lemma 52.16.8. In Situation 52.16.1 let $(\mathcal{F}_ n)$ be an object of $\textit{Coh}(U, I\mathcal{O}_ U)$ . If $(\mathcal{F}_ n)$ canonically extends to $X$ , then

$(\widetilde{H^0(U, \mathcal{F}_ n)})$ is pro-isomorphic to an object $(\mathcal{G}_ n)$ of $\textit{Coh}(X, I \mathcal{O}_ X)$ unique up to unique isomorphism,
the restriction of $(\mathcal{G}_ n)$ to $U$ is isomorphic to $(\mathcal{F}_ n)$ , i.e., $(\mathcal{F}_ n)$ extends to $X$ ,
the inverse system $\{ H^0(U, \mathcal{F}_ n)\}$ satisfies the Mittag-Leffler condition, and
the module $M$ in (52.16.6.1) is finite over the $I$ -adic completion of $A$ and the limit topology on $M$ is the $I$ -adic topology.

Proof. The existence of $(\mathcal{G}_ n)$ in (1) follows from Definition 52.16.7. The uniqueness of $(\mathcal{G}_ n)$ in (1) follows from Lemma 52.15.3. Write $\mathcal{G}_ n = \widetilde{M_ n}$ . Then $\{ M_ n\}$ is an inverse system of finite $A$ -modules with $M_ n = M_{n + 1}/I^ n M_{n + 1}$ . By Definition 52.16.7 the inverse system $\{ H^0(U, \mathcal{F}_ n)\}$ is pro-isomorphic to $\{ M_ n\}$ . Hence we see that the inverse system $\{ H^0(U, \mathcal{F}_ n)\}$ satisfies the Mittag-Leffler condition and that $M = \mathop{\mathrm{lim}}\nolimits M_ n$ (as topological modules). Thus the properties of $M$ in (4) follow from Algebra, Lemmas 10.98.2, 10.96.12, and 10.96.3. Since $U$ is quasi-affine the canonical maps

$\widetilde{H^0(U, \mathcal{F}_ n)}|_ U \to \mathcal{F}_ n$

are isomorphisms (Properties, Lemma 28.18.2). We conclude that $(\mathcal{G}_ n|_ U)$ and $(\mathcal{F}_ n)$ are pro-isomorphic and hence isomorphic by Lemma 52.15.3. $\square$

The Stacks project

Comments (1)

Post a comment