Lemma 48.17.3. In Situation 48.16.1 let $Y$ be an object of $\textit{FTS}_ S$ and let $f : X = \mathbf{A}^1_ Y \to Y$ be the projection. Then there is a (noncanonical) isomorphism $f^!(-) \cong Lf^*(-) [1]$ of functors.

**Proof.**
Since $X = \mathbf{A}^1_ Y \subset \mathbf{P}^1_ Y$ and since $\mathcal{O}_{\mathbf{P}^1_ Y}(-2)|_ X \cong \mathcal{O}_ X$ this follows from Lemmas 48.15.1 and 48.13.3.
$\square$

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