The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Lemma 25.24.1. 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 25.21.7. 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) \\ & = & \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}})\right)(V) \\ & = & \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 $\mathcal{O}_ X$-modules then $\mathcal{F}_ i$, $\mathcal{F}_{ijk}$ is a quasi-coherent $\mathcal{O}_{U_ i}$, $\mathcal{O}_{U_{ijk}}$-module. Hence by Lemma 25.7.3 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. $\square$


Comments (1)

Comment #3619 by Dario WeiƟmann on

Typo: the bracket in the third line should be .

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  • 2 comment(s) on Section 25.24: Functoriality for quasi-coherent modules

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