The Stacks project

Remark 48.11.5. If $f : Y \to X$ is a finite morphism of Noetherian schemes, then the diagram

\[ \xymatrix{ Rf_*a(K) \ar[r]_-{\text{Tr}_{f, K}} \ar@{=}[d] & K \ar@{=}[d] \\ R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(f_*\mathcal{O}_ Y, K) \ar[r] & K } \]

is commutative for $K \in D_\mathit{QCoh}^+(\mathcal{O}_ X)$. This follows from Lemma 48.11.4. The lower horizontal arrow is induced by the map $\mathcal{O}_ X \to f_*\mathcal{O}_ Y$ and the upper horizontal arrow is the trace map discussed in Section 48.7.

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  • 2 comment(s) on Section 48.11: Right adjoint of pushforward for finite morphisms

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