The Stacks project

Example 48.3.2. Let $A \to B$ be a ring map. Let $Y = \mathop{\mathrm{Spec}}(A)$ and $X = \mathop{\mathrm{Spec}}(B)$ and $f : X \to Y$ the morphism corresponding to $A \to B$. Then $Rf_* : D_\mathit{QCoh}(\mathcal{O}_ X) \to D_\mathit{QCoh}(\mathcal{O}_ Y)$ corresponds to restriction $D(B) \to D(A)$ via the equivalences $D(B) \to D_\mathit{QCoh}(\mathcal{O}_ X)$ and $D(A) \to D_\mathit{QCoh}(\mathcal{O}_ Y)$. Hence the right adjoint corresponds to the functor $K \longmapsto R\mathop{\mathrm{Hom}}\nolimits (B, K)$ of Dualizing Complexes, Section 47.13.

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