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## Tag: 01LC

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Lemma 22.24.1. Let $f : X \to S$ be a morphism of schemes. If $f$ is quasi-compact and quasi-separated then $f_*$ transforms quasi-coherent $\mathcal{O}_X$-modules into quasi-coherent $\mathcal{O}_S$-modules.

Proof. The question is local on $S$ and hence we may assume that $S$ is affine. Because $X$ is quasi-compact we may write $X = \bigcup_{i = 1}^n U_i$ with each $U_i$ open affine. Because $f$ is quasi-separated we may write $U_i \cap U_j = \bigcup_{k = 1}^{n_{ij}} U_{ijk}$ for some affine open $U_{ijk}$, see Lemma 22.21.7. Denote $f_i : U_i \to S$ and $f_{ijk} : U_{ijk} \to S$ the restrictions of $f$. For any open $V$ of $S$ and any sheaf $\mathcal{F}$ on $X$ we have \begin{eqnarray*} f_*\mathcal{F}(V) & = & \mathcal{F}(f^{-1}V) \\ & = & \text{Ker}\left( \bigoplus\nolimits_i \mathcal{F}(f^{-1}V \cap U_i) \to \bigoplus\nolimits_{i, j, k} \mathcal{F}(f^{-1}V \cap U_{ijk})\right) \\ & = & \text{Ker}\left( \bigoplus\nolimits_i f_{i, *}(\mathcal{F}|_{U_i})(V) \to \bigoplus\nolimits_{i, j, k} f_{ijk, *}(\mathcal{F}|_{U_{ijk}})\right)(V) \\ & = & \text{Ker}\left( \bigoplus\nolimits_i f_{i, *}(\mathcal{F}|_{U_i}) \to \bigoplus\nolimits_{i, j, k} f_{ijk, *}(\mathcal{F}|_{U_{ijk}})\right)(V) \end{eqnarray*} In other words there is a short exact sequence of sheaves $$0 \to f_*\mathcal{F} \to \bigoplus f_{i, *}\mathcal{F}_i \to \bigoplus f_{ijk, *}\mathcal{F}_{ijk}$$ where $\mathcal{F}_i, \mathcal{F}_{ijk}$ denotes the restriction of $\mathcal{F}$ to the corresponding open. If $\mathcal{F}$ is a quasi-coherent $\mathcal{O}_X$-modules then $\mathcal{F}_i$, $\mathcal{F}_{ijk}$ is a quasi-coherent $\mathcal{O}_{U_i}$, $\mathcal{O}_{U_{ijk}}$-module. Hence by Lemma 22.7.3 we see that the second and third term of the exact sequence are quasi-coherent $\mathcal{O}_S$-modules. Thus we conclude that $f_*\mathcal{F}$ is a quasi-coherent $\mathcal{O}_S$-module. $\square$

\begin{lemma}
\label{lemma-push-forward-quasi-coherent}
Let $f : X \to S$ be a morphism of schemes.
If $f$ is quasi-compact and quasi-separated then
$f_*$ transforms quasi-coherent $\mathcal{O}_X$-modules
into quasi-coherent $\mathcal{O}_S$-modules.
\end{lemma}

\begin{proof}
The question is local on $S$ and hence we may assume that
$S$ is affine. Because $X$ is quasi-compact we may write
$X = \bigcup_{i = 1}^n U_i$ with each $U_i$ open affine.
Because $f$ is quasi-separated we may write
$U_i \cap U_j = \bigcup_{k = 1}^{n_{ij}} U_{ijk}$ for some
affine open $U_{ijk}$, see Lemma \ref{lemma-characterize-quasi-separated}.
Denote $f_i : U_i \to S$ and $f_{ijk} : U_{ijk} \to S$ the
restrictions of $f$. For any open $V$ of $S$ and any sheaf
$\mathcal{F}$ on $X$ we have
\begin{eqnarray*}
f_*\mathcal{F}(V) & = & \mathcal{F}(f^{-1}V) \\
& = &
\text{Ker}\left(
\bigoplus\nolimits_i \mathcal{F}(f^{-1}V \cap U_i)
\to
\bigoplus\nolimits_{i, j, k} \mathcal{F}(f^{-1}V \cap U_{ijk})\right) \\
& = &
\text{Ker}\left(
\bigoplus\nolimits_i f_{i, *}(\mathcal{F}|_{U_i})(V)
\to
\bigoplus\nolimits_{i, j, k} f_{ijk, *}(\mathcal{F}|_{U_{ijk}})\right)(V) \\
& = &
\text{Ker}\left(
\bigoplus\nolimits_i f_{i, *}(\mathcal{F}|_{U_i})
\to
\bigoplus\nolimits_{i, j, k} f_{ijk, *}(\mathcal{F}|_{U_{ijk}})\right)(V)
\end{eqnarray*}
In other words there is a short exact sequence of sheaves
$$0 \to f_*\mathcal{F} \to \bigoplus f_{i, *}\mathcal{F}_i \to \bigoplus f_{ijk, *}\mathcal{F}_{ijk}$$
where $\mathcal{F}_i, \mathcal{F}_{ijk}$ denotes the
restriction of $\mathcal{F}$ to the corresponding open.
If $\mathcal{F}$ is a quasi-coherent $\mathcal{O}_X$-modules
then $\mathcal{F}_i$, $\mathcal{F}_{ijk}$ is a quasi-coherent
$\mathcal{O}_{U_i}$, $\mathcal{O}_{U_{ijk}}$-module.
Hence by Lemma \ref{lemma-widetilde-pullback} we see that the second and
third term of the exact sequence are quasi-coherent
$\mathcal{O}_S$-modules. Thus we conclude that
$f_*\mathcal{F}$ is a quasi-coherent $\mathcal{O}_S$-module.
\end{proof}


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