The Stacks project

Lemma 61.25.5. Let $i : Z \to X$ be a closed immersion of schemes. Let $U \to X$ be an object of $X_{pro\text{-}\acute{e}tale}$ such that

  1. $U$ is affine and weakly contractible, and

  2. every point of $U$ specializes to a point of $U \times _ X Z$.

Then $i_{pro\text{-}\acute{e}tale}^{-1}\mathcal{F}(U \times _ X Z) = \mathcal{F}(U)$ for all abelian sheaves on $X_{pro\text{-}\acute{e}tale}$.

Proof. Since pullback commutes with restriction, we may replace $X$ by $U$. Thus we may assume that $X$ is affine and weakly contractible and that every point of $X$ specializes to a point of $Z$. By Lemma 61.25.2 part (1) it suffices to show that $v(Z) = X$ in this case. Thus we have to show: If $A$ is a w-contractible ring, $I \subset A$ an ideal contained in the Jacobson radical of $A$ and $A \to B \to A/I$ is a factorization with $A \to B$ ind-étale, then there is a unique section $B \to A$ compatible with maps to $A/I$. Observe that $B/IB = A/I \times R$ as $A/I$-algebras. After replacing $B$ by a localization we may assume $B/IB = A/I$. Note that $\mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)$ is surjective as the image contains $V(I)$ and hence all closed points and is closed under specialization. Since $A$ is w-contractible there is a section $B \to A$. Since $B/IB = A/I$ this section is compatible with the map to $A/I$. We omit the proof of uniqueness (hint: use that $A$ and $B$ have isomorphic local rings at maximal ideals of $A$). $\square$


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