Lemma 48.17.1 (0AU0)—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 48.17.1 (cite)

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Lemma 48.17.1. In Situation 48.16.1 let $Y$ be an object of $\textit{FTS}_ S$ and let $j : X \to Y$ be an open immersion. Then there is a canonical isomorphism $j^! = j^*$ of functors.

Proof. In this case we may choose $\overline{X} = Y$ as our compactification. Then the right adjoint of Lemma 48.3.1 for $\text{id} : Y \to Y$ is the identity functor and hence $j^! = j^*$ by definition. $\square$

Comments (2)

Comment #4632 by Noah Olander on October 28, 2019 at 17:01

I think it's useful to point out either here or later that by the end of the chapter you know that $f^! = f^*$ for $f$ etale in $FTS_S$ .

Comment #4785 by Johan on December 09, 2019 at 20:05

OK, I have added a lemma saying this. See this commit.

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