Lemma 61.28.7 (09BZ)—The Stacks project

Lemma 61.28.7. Let $X$ be a weakly contractible affine scheme. Let $\Lambda$ be a Noetherian ring and let $I \subset \Lambda$ be an ideal. Let $\mathcal{F}$ be a derived complete sheaf of $\Lambda$ -modules on $X_{pro\text{-}\acute{e}tale}$ with $\mathcal{F}/I\mathcal{F}$ a locally constant sheaf of $\Lambda /I$ -modules of finite type. Then there exists an integer $t$ and a surjective map

$(\underline{\Lambda }^\wedge )^{\oplus t} \to \mathcal{F}$

Proof. Since $X$ is weakly contractible, there exists a finite disjoint open covering $X = \coprod U_ i$ such that $\mathcal{F}/I\mathcal{F}|_{U_ i}$ is isomorphic to the constant sheaf associated to a finite $\Lambda /I$ -module $M_ i$ . Choose finitely many generators $m_{ij}$ of $M_ i$ . We can find sections $s_{ij} \in \mathcal{F}(X)$ restricting to $m_{ij}$ viewed as a section of $\mathcal{F}/I\mathcal{F}$ over $U_ i$ . Let $t$ be the total number of $s_{ij}$ . Then we obtain a map

$\alpha : \underline{\Lambda }^{\oplus t} \longrightarrow \mathcal{F}$

which is surjective modulo $I$ by construction. By Lemma 61.20.1 the derived completion of $\underline{\Lambda }^{\oplus t}$ is the sheaf $(\underline{\Lambda }^\wedge )^{\oplus t}$ . Since $\mathcal{F}$ is derived complete we see that $\alpha$ factors through a map

$\alpha ^\wedge : (\underline{\Lambda }^\wedge )^{\oplus t} \longrightarrow \mathcal{F}$

Then $\mathcal{Q} = \mathop{\mathrm{Coker}}(\alpha ^\wedge )$ is a derived complete sheaf of $\Lambda$ -modules by Proposition 61.21.1. By construction $\mathcal{Q}/I\mathcal{Q} = 0$ . It follows from Lemma 61.28.6 that $\mathcal{Q} = 0$ which is what we wanted to show. $\square$

The Stacks project

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