The Stacks project

66.11 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 66.11.1. 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 65.18.4. 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

66.11.1.1
\begin{equation} \label{spaces-morphisms-equation-representable-pushforward} f'_*(\mathcal{F}|_ U) = (f_*\mathcal{F})|_ V, \end{equation}

see Properties of Spaces, Lemma 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 $f'_*(\mathcal{F}|_ U)$ 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 66.11.2. 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 65.29.6. 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 66.8.4 and 66.4.4. By (66.11.1.1) 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 65.6.3. By Lemma 66.11.1 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 65.29.7. $\square$

Higher direct images are discussed in Cohomology of Spaces, Section 68.3.


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