Lemma 32.4.7 (01Z0)—The Stacks project

Lemma 32.4.7. In Situation 32.4.5. Suppose that $\mathcal{F}_0$ is a quasi-coherent sheaf on $S_0$. Set $\mathcal{F}_ i = f_{i0}^*\mathcal{F}_0$ for $i \geq 0$ and set $\mathcal{F} = f_0^*\mathcal{F}_0$. Then

\[ \Gamma (S, \mathcal{F}) = \mathop{\mathrm{colim}}\nolimits _{i \geq 0} \Gamma (S_ i, \mathcal{F}_ i) \]

Proof. Write $\mathcal{A}_ j = f_{i0, *} \mathcal{O}_{S_ i}$. This is a quasi-coherent sheaf of $\mathcal{O}_{S_0}$-algebras (see Morphisms, Lemma 29.11.5) and $S_ i$ is the relative spectrum of $\mathcal{A}_ i$ over $S_0$. In the proof of Lemma 32.2.2 we constructed $S$ as the relative spectrum of $\mathcal{A} = \mathop{\mathrm{colim}}\nolimits _{i \geq 0} \mathcal{A}_ i$ over $S_0$. Set

\[ \mathcal{M}_ i = \mathcal{F}_0 \otimes _{\mathcal{O}_{S_0}} \mathcal{A}_ i \]

and

\[ \mathcal{M} = \mathcal{F}_0 \otimes _{\mathcal{O}_{S_0}} \mathcal{A}. \]

Then we have $f_{i0, *} \mathcal{F}_ i = \mathcal{M}_ i$ and $f_{0, *}\mathcal{F} = \mathcal{M}$. Since $\mathcal{A}$ is the colimit of the sheaves $\mathcal{A}_ i$ and since tensor product commutes with directed colimits, we conclude that $\mathcal{M} = \mathop{\mathrm{colim}}\nolimits _{i \geq 0} \mathcal{M}_ i$. Since $S_0$ is quasi-compact and quasi-separated we see that

\begin{eqnarray*} \Gamma (S, \mathcal{F}) & = & \Gamma (S_0, \mathcal{M}) \\ & = & \Gamma (S_0, \mathop{\mathrm{colim}}\nolimits _{i \geq 0} \mathcal{M}_ i) \\ & = & \mathop{\mathrm{colim}}\nolimits _{i \geq 0} \Gamma (S_0, \mathcal{M}_ i) \\ & = & \mathop{\mathrm{colim}}\nolimits _{i \geq 0} \Gamma (S_ i, \mathcal{F}_ i) \end{eqnarray*}

see Sheaves, Lemma 6.29.1 and Topology, Lemma 5.27.1 for the middle equality. $\square$

The Stacks project

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