The Stacks project

Remark 73.6.3. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of representable algebraic spaces $X$ and $Y$ over $S$. Let $f_0 : X_0 \to Y_0$ be a morphism of schemes representing $f$ (awkward but temporary notation). Then the diagram

\[ \xymatrix{ D_\mathit{QCoh}(\mathcal{O}_{X_0}) \ar@{=}[rrrrrr]_{\text{Lemma 071Q}} & & & & & & D_\mathit{QCoh}(\mathcal{O}_ X) \\ D_\mathit{QCoh}(\mathcal{O}_{Y_0}) \ar[u]^{Lf^*_0} \ar@{=}[rrrrrr]^{\text{Lemma 071Q}} & & & & & & D_\mathit{QCoh}(\mathcal{O}_ Y) \ar[u]_{Lf^*} } \]

(Lemma 73.5.5 and Derived Categories of Schemes, Lemma 36.3.8) is commutative. This follows as the equivalences $D_\mathit{QCoh}(\mathcal{O}_{X_0}) \to D_\mathit{QCoh}(\mathcal{O}_ X)$ and $D_\mathit{QCoh}(\mathcal{O}_{Y_0}) \to D_\mathit{QCoh}(\mathcal{O}_ Y)$ of Lemma 73.4.2 come from pulling back by the (flat) morphisms of ringed sites $\epsilon : X_{\acute{e}tale}\to X_{0, Zar}$ and $\epsilon : Y_{\acute{e}tale}\to Y_{0, Zar}$ and the diagram of ringed sites

\[ \xymatrix{ X_{0, Zar} \ar[d]_{f_0} & X_{\acute{e}tale}\ar[l]^\epsilon \ar[d]^ f \\ Y_{0, Zar} & Y_{\acute{e}tale}\ar[l]_\epsilon } \]

is commutative (details omitted). If $f$ is quasi-compact and quasi-separated, equivalently if $f_0$ is quasi-compact and quasi-separated, then we claim

\[ \xymatrix{ D_\mathit{QCoh}(\mathcal{O}_{X_0}) \ar[d]_{Rf_{0, *}} \ar@{=}[rrrrrr]_{\text{Lemma 071Q}} & & & & & & D_\mathit{QCoh}(\mathcal{O}_ X) \ar[d]^{Rf_*} \\ D_\mathit{QCoh}(\mathcal{O}_{Y_0}) \ar@{=}[rrrrrr]^{\text{Lemma 071Q}} & & & & & & D_\mathit{QCoh}(\mathcal{O}_ Y) } \]

(Lemma 73.6.1 and Derived Categories of Schemes, Lemma 36.4.1) is commutative as well. This also follows from the commutative diagram of sites displayed above as the proof of Lemma 73.4.2 shows that the functor $R\epsilon _*$ gives the equivalences $D_\mathit{QCoh}(\mathcal{O}_ X) \to D_\mathit{QCoh}(\mathcal{O}_{X_0})$ and $D_\mathit{QCoh}(\mathcal{O}_ Y) \to D_\mathit{QCoh}(\mathcal{O}_{Y_0})$.


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