The Stacks project

Lemma 84.4.4. Let $S$ be a scheme. Let $f : X \to Y$ and $g : Y' \to Y$ be a morphisms of algebraic spaces over $S$. Assume $f$ is proper. Set $X' = Y' \times _ Y X$ with projections $f' : X' \to Y'$ and $g' : X' \to X$. Let $\mathcal{F}$ be any sheaf on $X_{\acute{e}tale}$. Then $g^{-1}f_*\mathcal{F} = f'_*(g')^{-1}\mathcal{F}$.

Proof. The question is étale local on $Y'$. Choose a scheme $V$ and a surjective étale morphism $V \to Y$. Choose a scheme $V'$ and a surjective étale morphism $V' \to V \times _ Y Y'$. Then we may replace $Y'$ by $V'$ and $Y$ by $V$. Hence we may assume $Y$ and $Y'$ are schemes. Then we may work Zariski locally on $Y$ and $Y'$ and hence we may assume $Y$ and $Y'$ are affine schemes.

Assume $Y$ and $Y'$ are affine schemes. Choose a surjective proper morphism $h_1 : X_1 \to X$ where $X_1$ is a scheme, see Cohomology of Spaces, Lemma 69.18.1. Set $X_2 = X_1 \times _ X X_1$ and denote $h_2 : X_2 \to X$ the structure morphism. Observe this is a scheme. By the case of schemes (Étale Cohomology, Lemma 59.91.5) we know the lemma is true for the cartesian diagrams

\[ \vcenter { \xymatrix{ X'_1 \ar[r] \ar[d] & X_1 \ar[d] \\ Y' \ar[r] & Y } } \quad \text{and}\quad \vcenter { \xymatrix{ X'_2 \ar[r] \ar[d] & X_2 \ar[d] \\ Y' \ar[r] & Y } } \]

and the sheaves $\mathcal{F}_ i = (X_ i \to X)^{-1}\mathcal{F}$. By Lemma 84.4.1 we have an exact sequence $0 \to \mathcal{F} \to h_{1, *}\mathcal{F}_1 \to h_{2, *}\mathcal{F}_2$ and similarly for $(g')^{-1}\mathcal{F}$ because $X'_2 = X'_1 \times _{X'} X'_1$. Hence we conlude that the lemma is true (some details omitted). $\square$

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