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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 étale 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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