67.11 Pushforward of quasi-coherent sheaves
We first prove a simple lemma that relates pushforward of sheaves of modules for a morphism of algebraic spaces to pushforward of sheaves of modules for a morphism of schemes.
Lemma 67.11.1. Let S be a scheme. Let f : X \to Y be a morphism of algebraic spaces over S. Let U \to X be a surjective étale morphism from a scheme to X. Set R = U \times _ X U and denote t, s : R \to U the projection morphisms as usual. Denote a : U \to Y and b : R \to Y the induced morphisms. For any object \mathcal{F} of \textit{Mod}(\mathcal{O}_ X) there exists an exact sequence
0 \to f_*\mathcal{F} \to a_*(\mathcal{F}|_ U) \to b_*(\mathcal{F}|_ R)
where the second arrow is the difference t^* - s^*.
Proof.
We denote \mathcal{F} also its extension to a sheaf of modules on X_{spaces, {\acute{e}tale}}, see Properties of Spaces, Remark 66.18.4. Let V \to Y be an object of Y_{\acute{e}tale}. Then V \times _ Y X is an object of X_{spaces, {\acute{e}tale}}, and by definition f_*\mathcal{F}(V) = \mathcal{F}(V \times _ Y X). Since U \to X is surjective étale, we see that \{ V \times _ Y U \to V \times _ Y X\} is a covering. Also, we have (V \times _ Y U) \times _ X (V \times _ Y U) = V \times _ Y R. Hence, by the sheaf condition of \mathcal{F} on X_{spaces, {\acute{e}tale}} we have a short exact sequence
0 \to \mathcal{F}(V \times _ Y X) \to \mathcal{F}(V \times _ Y U) \to \mathcal{F}(V \times _ Y R)
where the second arrow is the difference of restricting via t or s. This exact sequence is functorial in V and hence we obtain the lemma.
\square
Let S be a scheme. Let f : X \to Y be a quasi-compact and quasi-separated morphism of representable algebraic spaces X and Y over S. By Descent, Proposition 35.9.4 the functor f_* : \mathit{QCoh}(\mathcal{O}_ X) \to \mathit{QCoh}(\mathcal{O}_ Y) agrees with the usual functor if we think of X and Y as schemes.
More generally, 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
67.11.1.1
\begin{equation} \label{spaces-morphisms-equation-representable-pushforward} f'_*(\mathcal{F}|_ U) = (f_*\mathcal{F})|_ V, \end{equation}
see Properties of Spaces, Lemma 66.26.2. 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 f'_*(\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.
The next level of generality is to consider an arbitrary quasi-compact and quasi-separated morphism of algebraic spaces.
Lemma 67.11.2. 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 f_* transforms quasi-coherent \mathcal{O}_ X-modules into quasi-coherent \mathcal{O}_ Y-modules.
Proof.
Let \mathcal{F} be a quasi-coherent sheaf on X. We have to show that f_*\mathcal{F} is a quasi-coherent sheaf on Y. For this it suffices to show that for any affine scheme V and étale morphism V \to Y the restriction of f_*\mathcal{F} to V is quasi-coherent, see Properties of Spaces, Lemma 66.29.6. Let f' : V \times _ Y X \to V be the base change of f by V \to Y. Note that f' is also quasi-compact and quasi-separated, see Lemmas 67.8.4 and 67.4.4. By (67.11.1.1) we know that the restriction of f_*\mathcal{F} to V is f'_* of the restriction of \mathcal{F} to V \times _ Y X. Hence we may replace f by f', and assume that Y is an affine scheme.
Assume Y is an affine scheme. 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 U \to X, see Properties of Spaces, Lemma 66.6.3. By Lemma 67.11.1 we get an exact sequence
0 \to f_*\mathcal{F} \to a_*(\mathcal{F}|_ U) \to b_*(\mathcal{F}|_ R).
where R = U \times _ X U. As X \to Y is quasi-separated we see that R \to U \times _ Y U is a quasi-compact monomorphism. This implies that R is a quasi-compact separated scheme (as U and Y are affine at this point). Hence a : U \to Y and b : R \to Y are quasi-compact and quasi-separated morphisms of schemes. Thus by Descent, Proposition 35.9.4 the sheaves a_*(\mathcal{F}|_ U) and b_*(\mathcal{F}|_ R) are quasi-coherent (see also the discussion preceding this lemma). This implies that f_*\mathcal{F} is a kernel of quasi-coherent modules, and hence itself quasi-coherent, see Properties of Spaces, Lemma 66.29.7.
\square
Higher direct images are discussed in Cohomology of Spaces, Section 69.3.
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