The Stacks project

Lemma 70.5.6. Notation and assumptions as in Situation 70.5.5. Suppose that $\mathcal{F}_0$ is a quasi-coherent sheaf on $X_0$. Set $\mathcal{F}_ i = f_{0i}^*\mathcal{F}_0$ for $i \geq 0$ and set $\mathcal{F} = f_0^*\mathcal{F}_0$. Then

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

Proof. Choose a surjective ├ętale morphism $U_0 \to X_0$ where $U_0$ is an affine scheme (Properties of Spaces, Lemma 66.6.3). Set $U_ i = X_ i \times _{X_0} U_0$. Set $R_0 = U_0 \times _{X_0} U_0$ and $R_ i = R_0 \times _{X_0} X_ i$. In the proof of Lemma 70.4.1 we have seen that there exists a presentation $X = U/R$ with $U = \mathop{\mathrm{lim}}\nolimits U_ i$ and $R = \mathop{\mathrm{lim}}\nolimits R_ i$. Note that $U_ i$ and $U$ are affine and that $R_ i$ and $R$ are quasi-compact and separated (as $X_ i$ is quasi-separated). Hence Limits, Lemma 32.4.7 implies that

\[ \mathcal{F}(U) = \mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i(U_ i) \quad \text{and}\quad \mathcal{F}(R) = \mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i(R_ i). \]

The lemma follows as $\Gamma (X, \mathcal{F}) = \mathop{\mathrm{Ker}}(\mathcal{F}(U) \to \mathcal{F}(R))$ and similarly $\Gamma (X_ i, \mathcal{F}_ i) = \mathop{\mathrm{Ker}}(\mathcal{F}_ i(U_ i) \to \mathcal{F}_ i(R_ i))$ $\square$

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