The Stacks project

Lemma 96.19.5. Let

\[ \xymatrix{ \mathcal{V} \ar[d]_ g \ar[r]_ h & \mathcal{U} \ar[d]^ f \\ \mathcal{Y} \ar[r]^ e & \mathcal{X} } \]

be a $2$-fibre product of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$ and let $\mathcal{F}$ be an abelian presheaf on $\mathcal{X}$. Then the map $e^{-1}\mathcal{K}^\bullet (f, \mathcal{F}) \to \mathcal{K}^\bullet (g, e^{-1}\mathcal{F})$ of Lemma 96.19.2 is an isomorphism of complexes of abelian presheaves.

Proof. Let $y$ be an object of $\mathcal{Y}$ lying over the scheme $T$. Set $x = e(y)$. We are going to show that the map induces an isomorphism on sections over $y$. Note that

\[ \Gamma (y, e^{-1}\mathcal{K}^\bullet (f, \mathcal{F})) = \Gamma (x, \mathcal{K}^\bullet (f, \mathcal{F})) = \check{\mathcal{C}}^\bullet ( (\mathit{Sch}/T)_{fppf} \times _{x, \mathcal{X}} \mathcal{U} \to (\mathit{Sch}/T)_{fppf}, x^{-1}\mathcal{F}) \]

by Remark 96.19.4. On the other hand,

\[ \Gamma (y, \mathcal{K}^\bullet (g, e^{-1}\mathcal{F})) = \check{\mathcal{C}}^\bullet ( (\mathit{Sch}/T)_{fppf} \times _{y, \mathcal{Y}} \mathcal{V} \to (\mathit{Sch}/T)_{fppf}, y^{-1}e^{-1}\mathcal{F}) \]

also by Remark 96.19.4. Note that $y^{-1}e^{-1}\mathcal{F} = x^{-1}\mathcal{F}$ and since the diagram is $2$-cartesian the $1$-morphism

\[ (\mathit{Sch}/T)_{fppf} \times _{y, \mathcal{Y}} \mathcal{V} \to (\mathit{Sch}/T)_{fppf} \times _{x, \mathcal{X}} \mathcal{U} \]

is an equivalence. Hence the map on sections over $y$ is an isomorphism by Lemma 96.18.1. $\square$

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