The Stacks project

Lemma 98.21.6. 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 98.18.3 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 98.21.1.

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 90.11.5.

The rest of the proof is exactly the same as the proof of Formal Deformation Theory, Lemma 90.20.1. $\square$

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