Lemma 75.12.4. Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. If $\mathcal{J}$ is an injective object of $\mathit{QCoh}(\mathcal{O}_ X)$, then
$H^ p(U, \mathcal{J}|_ U) = 0$ for $p > 0$ and for every quasi-compact and quasi-separated algebraic space $U$ étale over $X$,
for any morphism $f : X \to Y$ of algebraic spaces over $S$ with $Y$ quasi-separated we have $R^ pf_*\mathcal{J} = 0$ for $p > 0$.
Proof. Proof of (1). Write $\mathcal{J}$ as a direct summand of $\mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i$ with $\mathcal{F}_ i = (Z_ i \to X)_*\mathcal{G}_ i$ as in Lemma 75.12.2. It is clear that it suffices to prove the vanishing for $\mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i$. Since pullback commutes with colimits and since $U$ is quasi-compact and quasi-separated, it suffices to prove $H^ p(U, \mathcal{F}_ i|_ U) = 0$ for $p > 0$, see Cohomology of Spaces, Lemma 69.5.1. Observe that $Z_ i \to X$ is an affine morphism, see Morphisms of Spaces, Lemma 67.20.12. Thus
where $\mathcal{G}'_ i$ is the pullback of $\mathcal{G}_ i$ to $Z_ i \times _ X U$, see Cohomology of Spaces, Lemma 69.11.1. Since $Z_ i \times _ X U$ is affine we conclude that $\mathcal{G}'_ i$ has no higher cohomology on $Z_ i \times _ X U$. By the Leray spectral sequence we conclude the same thing is true for $\mathcal{F}_ i|_ U$ (Cohomology on Sites, Lemma 21.14.6).
Proof of (2). Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $V \to Y$ be an étale morphism with $V$ affine. Then $V \times _ Y X \to X$ is an étale morphism and $V \times _ Y X$ is a quasi-compact and quasi-separated algebraic space étale over $X$ (details omitted). Hence $H^ p(V \times _ Y X, \mathcal{J})$ is zero by part (1). Since $R^ pf_*\mathcal{J}$ is the sheaf associated to the presheaf $V \mapsto H^ p(V \times _ Y X, \mathcal{J})$ the result is proved. $\square$
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