The Stacks project

84.10 Comparison with the case of schemes

We should add a lot more in this section.

Lemma 84.10.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of quasi-compact and quasi-separated algebraic spaces over $S$. Assume $X$ and $Y$ are representable and let $f_0 : X_0 \to Y_0$ be a morphism of schemes representing $f$ (awkward but temporary notation). Let $a : D_\mathit{QCoh}(\mathcal{O}_ Y) \to D_\mathit{QCoh}(\mathcal{O}_ X)$ be the right adjoint of $Rf_*$ from Lemma 84.3.1. Let $a_0 : D_\mathit{QCoh}(\mathcal{O}_{Y_0}) \to D_\mathit{QCoh}(\mathcal{O}_{X_0})$ be the right adjoint of $Rf_*$ from Duality for Schemes, Lemma 48.3.1. Then

\[ \xymatrix{ D_\mathit{QCoh}(\mathcal{O}_{X_0}) \ar@{=}[rrrrrr]_{\text{Derived Categories of Spaces, Lemma 071Q}} & & & & & & D_\mathit{QCoh}(\mathcal{O}_ X) \\ D_\mathit{QCoh}(\mathcal{O}_{Y_0}) \ar[u]^{a_0} \ar@{=}[rrrrrr]^{\text{Derived Categories of Spaces, Lemma 071Q}} & & & & & & D_\mathit{QCoh}(\mathcal{O}_ Y) \ar[u]_ a } \]

is commutative.

Proof. Follows from uniqueness of adjoints and the compatibilities of Derived Categories of Spaces, Remark 73.6.3. $\square$


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