The Stacks project

Lemma 89.4.3. 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). $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 06SB. Beware of the difference between the letter 'O' and the digit '0'.