The Stacks project

Lemma 85.11.1. Let $(A, I)$ be a pair consisting of a Noetherian ring and an ideal $I$. Let $A^\wedge $ be the $I$-adic completion of $A$. Then base change defines an equivalence of categories between the category (85.11.0.1) for $A$ with the category (85.11.0.1) for the completion $A^\wedge $.

Proof. Set $S = \mathop{\mathrm{Spec}}(A)$ as in (85.11.0.1) and $T = V(I)$. Similarly, write $S' = \mathop{\mathrm{Spec}}(A^\wedge )$ and $T' = V(IA^\wedge )$. The morphism $S' \to S$ defines an isomorphism $S'_{/T'} \to S_{/T}$ of formal completions. Let $\mathcal{C}_{S, T}$, $\mathcal{C}_{S_{/T}}$, $\mathcal{C}_{S'_{/T'}}$, and $\mathcal{C}_{S', T'}$ be the corresponding categories as used in (85.10.3.1). By Theorem 85.10.9 (in fact we only need the affine case treated in Lemma 85.10.8) we see that

\[ \mathcal{C}_{S, T} = \mathcal{C}_{S_{/T}} = \mathcal{C}_{S_{/T'}'} = \mathcal{C}_{S', T'} \]

Since $\mathcal{C}_{S, T}$ is the category (85.11.0.1) for $A$ and $\mathcal{C}_{S', T'}$ the category (85.11.0.1) for $A^\wedge $ this proves the lemma. $\square$


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