## 68.3 Higher direct images

Let $S$ be a scheme. Let $X$ be a representable algebraic space over $S$. Let $\mathcal{F}$ be a quasi-coherent module on $X$ (see Properties of Spaces, Section 65.29). By Descent, Proposition 35.9.3 the cohomology groups $H^ i(X, \mathcal{F})$ agree with the usual cohomology group computed in the Zariski topology of the corresponding quasi-coherent module on the scheme representing $X$.

More generally, let $f : X \to Y$ be a quasi-compact and quasi-separated morphism of representable algebraic spaces $X$ and $Y$. Let $\mathcal{F}$ be a quasi-coherent module on $X$. By Descent, Lemma 35.9.5 the sheaf $R^ if_*\mathcal{F}$ agrees with the usual higher direct image computed for the Zariski topology of the quasi-coherent module on the scheme representing $X$ mapping to the scheme representing $Y$.

More generally still, suppose $f : X \to Y$ is a representable, quasi-compact, and quasi-separated morphism of algebraic spaces over $S$. Let $V$ be a scheme and let $V \to Y$ be an étale surjective morphism. Let $U = V \times _ Y X$ and let $f' : U \to V$ be the base change of $f$. Then for any quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$ we have

68.3.0.1
\begin{equation} \label{spaces-cohomology-equation-representable-higher-direct-image} R^ if'_*(\mathcal{F}|_ U) = (R^ if_*\mathcal{F})|_ V, \end{equation}

see Properties of Spaces, Lemma 65.26.2. And because $f' : U \to V$ is a quasi-compact and quasi-separated morphism of schemes, by the remark of the preceding paragraph we may compute $R^ if'_*(\mathcal{F}|_ U)$ by thinking of $\mathcal{F}|_ U$ as a quasi-coherent sheaf on the scheme $U$, and $f'$ as a morphism of schemes. We will frequently use this without further mention.

Next, we prove that higher direct images of quasi-coherent sheaves are quasi-coherent for any quasi-compact and quasi-separated morphism of algebraic spaces. In the proof we use a trick; a “better” proof would use a relative Čech complex, as discussed in Sheaves on Stacks, Sections 95.18 and 95.19 ff.

Lemma 68.3.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. If $f$ is quasi-compact and quasi-separated, then $R^ if_*$ transforms quasi-coherent $\mathcal{O}_ X$-modules into quasi-coherent $\mathcal{O}_ Y$-modules.

Proof. Let $V \to Y$ be an étale morphism where $V$ is an affine scheme. Set $U = V \times _ Y X$ and denote $f' : U \to V$ the induced morphism. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. By Properties of Spaces, Lemma 65.26.2 we have $R^ if'_*(\mathcal{F}|_ U) = (R^ if_*\mathcal{F})|_ V$. Since the property of being a quasi-coherent module is local in the étale topology on $Y$ (see Properties of Spaces, Lemma 65.29.6) we may replace $Y$ by $V$, i.e., we may assume $Y$ is an affine scheme.

Assume $Y$ is affine. Since $f$ is quasi-compact we see that $X$ is quasi-compact. Thus we may choose an affine scheme $U$ and a surjective étale morphism $g : U \to X$, see Properties of Spaces, Lemma 65.6.3. Picture

$\xymatrix{ U \ar[r]_ g \ar[rd]_{f \circ g} & X \ar[d]^ f \\ & Y }$

The morphism $g : U \to X$ is representable, separated and quasi-compact because $X$ is quasi-separated. Hence the lemma holds for $g$ (by the discussion above the lemma). It also holds for $f \circ g : U \to Y$ (as this is a morphism of affine schemes).

In the situation described in the previous paragraph we will show by induction on $n$ that $IH_ n$: for any quasi-coherent sheaf $\mathcal{F}$ on $X$ the sheaves $R^ if\mathcal{F}$ are quasi-coherent for $i \leq n$. The case $n = 0$ follows from Morphisms of Spaces, Lemma 66.11.2. Assume $IH_ n$. In the rest of the proof we show that $IH_{n + 1}$ holds.

Let $\mathcal{H}$ be a quasi-coherent $\mathcal{O}_ U$-module. Consider the Leray spectral sequence

$E_2^{p, q} = R^ pf_* R^ qg_* \mathcal{H} \Rightarrow R^{p + q}(f \circ g)_*\mathcal{H}$

Cohomology on Sites, Lemma 21.14.7. As $R^ qg_*\mathcal{H}$ is quasi-coherent by $IH_ n$ all the sheaves $R^ pf_*R^ qg_*\mathcal{H}$ are quasi-coherent for $p \leq n$. The sheaves $R^{p + q}(f \circ g)_*\mathcal{H}$ are all quasi-coherent (in fact zero for $p + q > 0$ but we do not need this). Looking in degrees $\leq n + 1$ the only module which we do not yet know is quasi-coherent is $E_2^{n + 1, 0} = R^{n + 1}f_*g_*\mathcal{H}$. Moreover, the differentials $d_ r^{n + 1, 0} : E_ r^{n + 1, 0} \to E_ r^{n + 1 + r, 1 - r}$ are zero as the target is zero. Using that $\mathit{QCoh}(\mathcal{O}_ X)$ is a weak Serre subcategory of $\textit{Mod}(\mathcal{O}_ X)$ (Properties of Spaces, Lemma 65.29.7) it follows that $R^{n + 1}f_*g_*\mathcal{H}$ is quasi-coherent (details omitted).

Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Set $\mathcal{H} = g^*\mathcal{F}$. The adjunction mapping $\mathcal{F} \to g_*g^*\mathcal{F} = g_*\mathcal{H}$ is injective as $U \to X$ is surjective étale. Consider the exact sequence

$0 \to \mathcal{F} \to g_*\mathcal{H} \to \mathcal{G} \to 0$

where $\mathcal{G}$ is the cokernel of the first map and in particular quasi-coherent. Applying the long exact cohomology sequence we obtain

$R^ nf_*g_*\mathcal{H} \to R^ nf_*\mathcal{G} \to R^{n + 1}f_*\mathcal{F} \to R^{n + 1}f_*g_*\mathcal{H} \to R^{n + 1}f_*\mathcal{G}$

The cokernel of the first arrow is quasi-coherent and we have seen above that $R^{n + 1}f_*g_*\mathcal{H}$ is quasi-coherent. Thus $R^{n + 1}f_*\mathcal{F}$ has a $2$-step filtration where the first step is quasi-coherent and the second a submodule of a quasi-coherent sheaf. Since $\mathcal{F}$ is an arbitrary quasi-coherent $\mathcal{O}_ X$-module, this result also holds for $\mathcal{G}$. Thus we can choose an exact sequence $0 \to \mathcal{A} \to R^{n + 1}f_*\mathcal{G} \to \mathcal{B}$ with $\mathcal{A}$, $\mathcal{B}$ quasi-coherent $\mathcal{O}_ Y$-modules. Then the kernel $\mathcal{K}$ of $R^{n + 1}f_*g_*\mathcal{H} \to R^{n + 1}f_*\mathcal{G} \to \mathcal{B}$ is quasi-coherent, whereupon we obtain a map $\mathcal{K} \to \mathcal{A}$ whose kernel $\mathcal{K}'$ is quasi-coherent too. Hence $R^{n + 1}f_*\mathcal{F}$ sits in an exact sequence

$R^ nf_*g_*\mathcal{H} \to R^ nf_*\mathcal{G} \to R^{n + 1}f_*\mathcal{F} \to \mathcal{K}' \to 0$

with all modules quasi-coherent except for possibly $R^{n + 1}f_*\mathcal{F}$. We conclude that $R^{n + 1}f_*\mathcal{F}$ is quasi-coherent, i.e., $IH_{n + 1}$ holds as desired. $\square$

Lemma 68.3.2. Let $S$ be a scheme. Let $f : X \to Y$ be a quasi-separated and quasi-compact morphism of algebraic spaces over $S$. For any quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$ and any affine object $V$ of $Y_{\acute{e}tale}$ we have

$H^ q(V \times _ Y X, \mathcal{F}) = H^0(V, R^ qf_*\mathcal{F})$

for all $q \in \mathbf{Z}$.

Proof. Since formation of $Rf_*$ commutes with étale localization (Properties of Spaces, Lemma 65.26.2) we may replace $Y$ by $V$ and assume $Y = V$ is affine. Consider the Leray spectral sequence $E_2^{p, q} = H^ p(Y, R^ qf_*\mathcal{F})$ converging to $H^{p + q}(X, \mathcal{F})$, see Cohomology on Sites, Lemma 21.14.5. By Lemma 68.3.1 we see that the sheaves $R^ qf_*\mathcal{F}$ are quasi-coherent. By Cohomology of Schemes, Lemma 30.2.2 we see that $E_2^{p, q} = 0$ when $p > 0$. Hence the spectral sequence degenerates at $E_2$ and we win. $\square$

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