Lemma 86.10.1. 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 86.3.1. 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.
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