Lemma 48.17.2 (0G4J)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 48: Duality for Schemes
Section 48.17: Properties of upper shriek functors
Lemma 48.17.2 (cite)

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$\xymatrix{ U \ar[r]_ j \ar[d]_ g & X \ar[d]^ f \\ V \ar[r]^{j'} & Y }$

be a commutative diagram of $\textit{FTS}_ S$ where $j$ and $j'$ are open immersions. Then $j^* \circ f^! = g^! \circ (j')^*$ as functors $D^+_\mathit{QCoh}(\mathcal{O}_ Y) \to D^+(\mathcal{O}_ U)$ .

Proof. Let $h = f \circ j = j' \circ g$ . By Lemma 48.16.3 we have $h^! = j^! \circ f^! = g^! \circ (j')^!$ . By Lemma 48.17.1 we have $j^! = j^*$ and $(j')^! = (j')^*$ . $\square$

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