The Stacks project

Lemma 98.6.1. Let $S$ be a locally Noetherian scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$ satisfying (RS). For any field $k$ of finite type over $S$ and any object $x_0$ of $\mathcal{X}$ lying over $k$ the predeformation category $p : \mathcal{F}_{\mathcal{X}, k, x_0} \to \mathcal{C}_\Lambda $ ( is a deformation category, see Formal Deformation Theory, Definition 90.16.8.

Proof. Set $\mathcal{F} = \mathcal{F}_{\mathcal{X}, k, x_0}$. Let $f_1 : A_1 \to A$ and $f_2 : A_2 \to A$ be ring maps in $\mathcal{C}_\Lambda $ with $f_2$ surjective. We have to show that the functor

\[ \mathcal{F}(A_1 \times _ A A_2) \longrightarrow \mathcal{F}(A_1) \times _{\mathcal{F}(A)} \mathcal{F}(A_2) \]

is an equivalence, see Formal Deformation Theory, Lemma 90.16.4. Set $X = \mathop{\mathrm{Spec}}(A)$, $X' = \mathop{\mathrm{Spec}}(A_2)$, $Y = \mathop{\mathrm{Spec}}(A_1)$ and $Y' = \mathop{\mathrm{Spec}}(A_1 \times _ A A_2)$. Note that $Y' = Y \amalg _ X X'$ in the category of schemes, see More on Morphisms, Lemma 37.14.3. We know that in the diagram of functors of fibre categories

\[ \xymatrix{ \mathcal{X}_{Y'} \ar[r] \ar[d] & \mathcal{X}_ Y \times _{\mathcal{X}_ X} \mathcal{X}_{X'} \ar[d] \\ \mathcal{X}_{\mathop{\mathrm{Spec}}(k)} \ar@{=}[r] & \mathcal{X}_{\mathop{\mathrm{Spec}}(k)} } \]

the top horizontal arrow is an equivalence by Definition 98.5.1. Since $\mathcal{F}(B)$ is the category of objects of $\mathcal{X}_{\mathop{\mathrm{Spec}}(B)}$ with an identification with $x_0$ over $k$ we win. $\square$

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Your email address will not be published.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 07WU. Beware of the difference between the letter 'O' and the digit '0'.