The Stacks project

Lemma 89.4.4. The category $\widehat{\mathcal{C}}_\Lambda $ admits coproducts of pairs of objects.

Proof. Let $R$ and $S$ be objects of $\widehat{\mathcal{C}}_\Lambda $. Consider the ring $C = R \otimes _\Lambda S$. There is a canonical surjective map $C \to R \otimes _\Lambda S \to k \otimes _\Lambda k \to k$ where the last map is the multiplication map. The kernel of $C \to k$ is a maximal ideal $\mathfrak m$. Note that $\mathfrak m$ is generated by $\mathfrak m_ R C$, $\mathfrak m_ S C$ and finitely many elements of $C$ which map to generators of the kernel of $k \otimes _\Lambda k \to k$. Hence $\mathfrak m$ is a finitely generated ideal. Set $C^\wedge $ equal to the completion of $C$ with respect to $\mathfrak m$. Then $C^\wedge $ is a Noetherian ring complete with respect to the maximal ideal $\mathfrak m^\wedge = \mathfrak mC^\wedge $ with residue field $k$, see Algebra, Lemma 10.97.5. Hence $C^\wedge $ is an object of $\widehat{\mathcal{C}}_\Lambda $. Then $R \to C^\wedge $ and $S \to C^\wedge $ turn $C^\wedge $ into a coproduct in $\widehat{\mathcal{C}}_\Lambda $ (details omitted). $\square$

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