Lemma 98.8.2. Let $S$ be a locally Noetherian 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 $k$ be a field of finite type over $S$ and let $w_0$ be an object of $\mathcal{W} = \mathcal{X} \times _\mathcal {Y} \mathcal{Z}$ over $k$. Denote $x_0, y_0, z_0$ the objects of $\mathcal{X}, \mathcal{Y}, \mathcal{Z}$ you get from $w_0$. Then there is a $6$-term exact sequence
of $k$-vector spaces.
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