The Stacks project

Lemma 86.26.1. Let $A \to B$ be a ring homomorphism of Noetherian rings inducing an isomorphism on $I$-adic completions for some ideal $I \subset A$ (for example if $B$ is the $I$-adic completion of $A$). Then base change defines an equivalence of categories between the category (86.26.0.1) for $(A, I)$ with the category (86.26.0.1) for $(B, IB)$.

Proof. Set $X = \mathop{\mathrm{Spec}}(A)$ and $T = V(I)$. Set $X_1 = \mathop{\mathrm{Spec}}(B)$ and $T_1 = V(IB)$. By Theorem 86.23.4 (in fact we only need the affine case treated in Lemma 86.23.3) the category (86.26.0.1) for $X$ and $T$ is equivalent to the the category of rig-├ętale morphisms $W \to X_{/T}$ of locally Noetherian formal algebraic spaces. Similarly, the the category (86.26.0.1) for $X_1$ and $T_1$ is equivalent to the category of rig-├ętale morphisms $W_1 \to X_{1, /T_1}$ of locally Noetherian formal algebraic spaces. Since $X_{/T} = \text{Spf}(A^\wedge )$ and $X_{1, /T_1} = \text{Spf}(B^\wedge )$ (Formal Spaces, Lemma 85.10.6) we see that these categories are equivalent by our assumption that $A^\wedge \to B^\wedge $ is an isomorphism. We omit the verification that this equivalence is given by base change. $\square$


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