Lemma 26.24.1. 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 26.21.6. 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) \\ & = & \mathop{\mathrm{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) \\ & = & \mathop{\mathrm{Ker}}\left( \bigoplus \nolimits _ i f_{i, *}(\mathcal{F}|_{U_ i})(V) \to \bigoplus \nolimits _{i, j, k} f_{ijk, *}(\mathcal{F}|_{U_{ijk}})(V)\right) \\ & = & \mathop{\mathrm{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 an 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 $\mathcal{O}_ X$-module then $\mathcal{F}_ i$ is a quasi-coherent $\mathcal{O}_{U_ i}$-module and $\mathcal{F}_{ijk}$ is a quasi-coherent $\mathcal{O}_{U_{ijk}}$-module. Hence by Lemma 26.7.3 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. $\square$

Comment #3619 by Dario Weißmann on

Typo: the bracket $)(V)$ in the third line should be $(V))$.

Comment #3722 by on

Thanks very much. This is a very confusing typo. Fixed here.

Comment #4729 by Simon on

additional s in last pragraph: it should say "If $\mathcal{F}$ is a quasi-coherent $\mathcal{O}_X$-module"

Comment #6664 by WhatJiaranEatsTonight on

Why does the lemma not hold for general $f$? I cannot figure out which equality not holds if we replace finite directed sum by general infinite product of quasi-coherent sheaves.

Comment #6671 by on

@#6664. The product of quasi-coherent modules isn't usually quasi-coherent.

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