Remark 48.11.5 (0AX3)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 48: Duality for Schemes
Section 48.11: Right adjoint of pushforward for finite morphisms
Remark 48.11.5 (cite)

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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.

Comments (1)

Comment #1586 by Pieter Belmans on July 14, 2015 at 22:46

The word morphism is missing in the first sentence.

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