The Stacks project

Lemma 68.7.2. With notation and assumptions as in Lemma 68.7.1. The category of $\mathcal{O}_ X$-modules of finite presentation is the colimit over $I$ of the categories $\mathcal{O}_{X_ i}$-modules of finite presentation.

Proof. Choose $0 \in I$. Choose an affine scheme $U_0$ and a surjective ├ętale morphism $U_0 \to X_0$. Set $U_ i = X_ i \times _{X_0} U_0$. Set $R_0 = U_0 \times _{X_0} U_0$ and $R_ i = R_0 \times _{X_0} X_ i$. Denote $s_ i, t_ i : R_ i \to U_ i$ and $s, t : R \to U$ the two projections. In the proof of Lemma 68.4.1 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). Moreover, it is also true that $R \times _{s, U, t} R = \mathop{\mathrm{colim}}\nolimits R_ i \times _{s_ i, U_ i, t_ i} R_ i$. Thus we know that $\mathit{QCoh}(\mathcal{O}_ U) = \mathop{\mathrm{colim}}\nolimits \mathit{QCoh}(\mathcal{O}_{U_ i})$, $\mathit{QCoh}(\mathcal{O}_ R) = \mathop{\mathrm{colim}}\nolimits \mathit{QCoh}(\mathcal{O}_{R_ i})$, and $\mathit{QCoh}(\mathcal{O}_{R \times _{s, U, t} R}) = \mathop{\mathrm{colim}}\nolimits \mathit{QCoh}(\mathcal{O}_{R_ i \times _{s_ i, U_ i, t_ i} R_ i})$ by Limits, Lemma 32.10.2. We have $\mathit{QCoh}(\mathcal{O}_ X) = \mathit{QCoh}(U, R, s, t, c)$ and $\mathit{QCoh}(\mathcal{O}_{X_ i}) = \mathit{QCoh}(U_ i, R_ i, s_ i, t_ i, c_ i)$, see Properties of Spaces, Proposition 64.32.1. Thus the result follows formally. $\square$


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