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68.8 Vanishing for higher direct images

We apply the results of Section 68.7 to obtain vanishing of higher direct images of quasi-coherent sheaves for quasi-compact and quasi-separated morphisms. This is useful because it allows one to argue by descending induction on the cohomological degree in certain situations.

Lemma 68.8.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume that

  1. $f$ is quasi-compact and quasi-separated, and

  2. $Y$ is quasi-compact.

Then there exists an integer $n(X \to Y)$ such that for any algebraic space $Y'$, any morphism $Y' \to Y$ and any quasi-coherent sheaf $\mathcal{F}'$ on $X' = Y' \times _ Y X$ the higher direct images $R^ if'_*\mathcal{F}'$ are zero for $i \geq n(X \to Y)$.

Proof. Let $V \to Y$ be a surjective ├ętale morphism where $V$ is an affine scheme, see Properties of Spaces, Lemma 65.6.3. Suppose we prove the result for the base change $f_ V : V \times _ Y X \to V$. Then the result holds for $f$ with $n(X \to Y) = n(X_ V \to V)$. Namely, if $Y' \to Y$ and $\mathcal{F}'$ are as in the lemma, then $R^ if'_*\mathcal{F}'|_{V \times _ Y Y'}$ is equal to $R^ if'_{V, *}\mathcal{F}'|_{X'_ V}$ where $f'_ V : X'_ V = V \times _ Y Y' \times _ Y X \to V \times _ Y Y' = Y'_ V$, see Properties of Spaces, Lemma 65.26.2. Thus we may assume that $Y$ is an affine scheme.

Moreover, to prove the vanishing for all $Y' \to Y$ and $\mathcal{F}'$ it suffices to do so when $Y'$ is an affine scheme. In this case, $R^ if'_*\mathcal{F}'$ is quasi-coherent by Lemma 68.3.1. Hence it suffices to prove that $H^ i(X', \mathcal{F}') = 0$, because $H^ i(X', \mathcal{F}') = H^0(Y', R^ if'_*\mathcal{F}')$ by Cohomology on Sites, Lemma 21.14.6 and the vanishing of higher cohomology of quasi-coherent sheaves on affine algebraic spaces (Proposition 68.7.2).

Choose $U \to X$, $d$, $V_ p \to U_ p$ and $d_ p$ as in Lemma 68.7.3. For any affine scheme $Y'$ and morphism $Y' \to Y$ denote $X' = Y' \times _ Y X$, $U' = Y' \times _ Y U$, $V'_ p = Y' \times _ Y V_ p$. Then $U' \to X'$, $d' = d$, $V'_ p \to U'_ p$ and $d'_ p = d$ is a collection of choices as in Lemma 68.7.3 for the algebraic space $X'$ (details omitted). Hence we see that $H^ i(X', \mathcal{F}') = 0$ for $i \geq \max (p + d_ p)$ and we win. $\square$

Lemma 68.8.2. Let $S$ be a scheme. Let $f : X \to Y$ be an affine morphism of algebraic spaces over $S$. Then $R^ if_*\mathcal{F} = 0$ for $i > 0$ and any quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$.

Proof. Recall that an affine morphism of algebraic spaces is representable. Hence this follows from ( and Cohomology of Schemes, Lemma 30.2.3. $\square$

Lemma 68.8.3. Let $S$ be a scheme. Let $f : X \to Y$ be an affine morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Then $H^ i(X, \mathcal{F}) = H^ i(Y, f_*\mathcal{F})$ for all $i \geq 0$.

Proof. Follows from Lemma 68.8.2 and the Leray spectral sequence. See Cohomology on Sites, Lemma 21.14.6. $\square$

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