Example 48.9.8 (0B6M)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 48: Duality for Schemes
Section 48.9: Right adjoint of pushforward for closed immersions
Example 48.9.8 (cite)

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Example 48.9.8. If $i : Z \to X$ is closed immersion of Noetherian schemes, then the diagram

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

is commutative for $K \in D_\mathit{QCoh}^+(\mathcal{O}_ X)$. Here the horizontal equality sign is Lemma 48.9.3 and the lower horizontal arrow is induced by the map $\mathcal{O}_ X \to i_*\mathcal{O}_ Z$. The commutativity of the diagram is a consequence of Lemma 48.9.7.

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