The Stacks project

Remark 36.39.2. The construction of Lemma 36.39.1 is compatible with pullbacks. More precisely, given a morphism $f : X \to Y$ of schemes and a perfect object $K$ of $D(\mathcal{O}_ Y)$ of tor-amplitude in $[-1, 0]$ then $Lf^*K$ is a perfect object $K$ of $D(\mathcal{O}_ X)$ of tor-amplitude in $[-1, 0]$ and we have a canonical identification

\[ f^*\det (K) \longrightarrow \det (Lf^*K) \]

Moreover, if $K$ has rank $0$, then $\delta (K)$ pulls back to $\delta (Lf^*K)$ via this map. This is clear from the affine local construction of the determinant.

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