The Stacks project

Lemma 97.21.6. Let $S$ be a scheme. Let $p : \mathcal{X} \to \mathcal{Y}$ and $q : \mathcal{Z} \to \mathcal{Y}$ be $1$-morphisms of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Assume $\mathcal{X}$, $\mathcal{Y}$, $\mathcal{Z}$ satisfy (RS*). Let $A$ be an $S$-algebra and let $w$ be an object of $\mathcal{W} = \mathcal{X} \times _\mathcal {Y} \mathcal{Z}$ over $A$. Denote $x, y, z$ the objects of $\mathcal{X}, \mathcal{Y}, \mathcal{Z}$ you get from $w$. For any $A$-module $M$ there is a $6$-term exact sequence

\[ \xymatrix{ 0 \ar[r] & \text{Inf}_ w(M) \ar[r] & \text{Inf}_ x(M) \oplus \text{Inf}_ z(M) \ar[r] & \text{Inf}_ y(M) \ar[lld] \\ & T_ w(M) \ar[r] & T_ x(M) \oplus T_ z(M) \ar[r] & T_ y(M) } \]

of $A$-modules.

Proof. By Lemma 97.18.3 we see that $\mathcal{W}$ satisfies (RS*) and hence $T_ w(M)$ and $\text{Inf}_ w(M)$ are defined. The horizontal arrows are defined using the functoriality of Lemma 97.21.1.

Definition of the “boundary” map $\delta : \text{Inf}_ y(M) \to T_ w(M)$. Choose isomorphisms $p(x) \to y$ and $y \to q(z)$ such that $w = (x, z, p(x) \to y \to q(z))$ in the description of the $2$-fibre product of Categories, Lemma 4.35.7 and more precisely Categories, Lemma 4.32.3. Let $x', y', z', w'$ denote the trivial deformation of $x, y, z, w$ over $A[M]$. By pullback we get isomorphisms $y' \to p(x')$ and $q(z') \to y'$. An element $\alpha \in \text{Inf}_ y(M)$ is the same thing as an automorphism $\alpha : y' \to y'$ over $A[M]$ which restricts to the identity on $y$ over $A$. Thus setting

\[ \delta (\alpha ) = (x', z', p(x') \to y' \xrightarrow {\alpha } y' \to q(z')) \]

we obtain an object of $T_ w(M)$. This is a map of $A$-modules by Formal Deformation Theory, Lemma 89.11.5.

The rest of the proof is exactly the same as the proof of Formal Deformation Theory, Lemma 89.20.1. $\square$


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Your email address will not be published.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