Lemma 43.7.1. Let $f : X \to Y$ be a flat morphism of varieties. Set $r = \dim (X) - \dim (Y)$. Then $f^*[\mathcal{F}]_ k = [f^*\mathcal{F}]_{k + r}$ if $\mathcal{F}$ is a coherent sheaf on $Y$ and the dimension of the support of $\mathcal{F}$ is at most $k$.

**Proof.**
See Chow Homology, Lemma 42.14.4.
$\square$

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