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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