The Stacks project

Lemma 88.4.3. The category $\widehat{\mathcal{C}}_\Lambda $ admits pushouts.

Proof. Let $R \to S_1$ and $R \to S_2$ be morphisms of $\widehat{\mathcal{C}}_\Lambda $. Consider the ring $C = S_1 \otimes _ R S_2$. This ring has a finitely generated maximal ideal $\mathfrak m = \mathfrak m_{S_1} \otimes S_2 + S_1 \otimes \mathfrak m_{S_2}$ with residue field $k$. 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 $ whose residue field is identified with $k$, see Algebra, Lemma 10.97.5. Hence $C^\wedge $ is an object of $\widehat{\mathcal{C}}_\Lambda $. Then $S_1 \to C^\wedge $ and $S_2 \to C^\wedge $ turn $C^\wedge $ into a pushout over $R$ in $\widehat{\mathcal{C}}_\Lambda $ (details omitted). $\square$


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