The Stacks project

Example 89.15.8. In Lemma 89.9.5 we constructed objects $R \in \widehat{\mathcal{C}}_\Lambda $ such that $\underline{R}|_{\mathcal{C}_\Lambda }$ is smooth and such that

\[ H_1(L_{k/\Lambda }) = \mathfrak m_ R/\mathfrak m_ R^2 \quad \text{and}\quad \Omega _{R/\Lambda } \otimes _ R k = \Omega _{k/\Lambda } \]

Let us reinterpret this using the theorem above. Namely, consider $\mathcal{F} = \mathcal{C}_\Lambda $ as a category cofibred in groupoids over itself (using the identity functor). Then $\mathcal{F}$ is a predeformation category, satisfies (S1) and (S2), and we have $T\mathcal{F} = 0$. Thus $\mathcal{F}$ satisfies condition (3) of Theorem 89.15.5. The theorem implies that (2) holds, i.e., we can find a minimal versal formal object $\xi \in \widehat{\mathcal{F}}(S)$ over some $S \in \widehat{\mathcal{C}}_\Lambda $ satisfying (89.15.0.2). Lemma 89.9.3 shows that $\Lambda \to S$ is formally smooth in the $\mathfrak m_ S$-adic topology (because $\underline{\xi } : \underline{R}|_{\mathcal{C}_\Lambda } \to \mathcal{F} = \mathcal{C}_\Lambda $ is smooth). Now condition (89.15.0.2) tells us that $\text{Der}_\Lambda (S, k) \to 0$ is bijective on $\text{Der}_\Lambda (k, k)$-orbits. This means the injection $\text{Der}_\Lambda (k, k) \to \text{Der}_\Lambda (S, k)$ is also surjective. In other words, we have $\Omega _{S/\Lambda } \otimes _ S k = \Omega _{k/\Lambda }$. Since $\Lambda \to S$ is formally smooth in the $\mathfrak m_ S$-adic topology, we can apply More on Algebra, Lemma 15.40.4 to conclude the exact sequence (89.3.10.2) turns into a pair of identifications

\[ H_1(L_{k/\Lambda }) = \mathfrak m_ S/\mathfrak m_ S^2 \quad \text{and}\quad \Omega _{S/\Lambda } \otimes _ S k = \Omega _{k/\Lambda } \]

Reading the argument backwards, we find that the $R$ constructed in Lemma 89.9.5 carries a minimal versal object. By the uniqueness of minimal versal objects (Lemma 89.14.5) we also conclude $R \cong S$, i.e., the two constructions give the same answer.


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Your email address will not be published.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 06T9. Beware of the difference between the letter 'O' and the digit '0'.