Lemma 36.38.8. Let $f : X \to Y$ be a proper morphism of locally Noetherian schemes. Then we have $f_*(\alpha \cdot f^*\beta ) = f_*\alpha \cdot \beta $ for $\alpha \in K'_0(X)$ and $\beta \in K_0(Y)$.

**Proof.**
Follows from Lemma 36.22.1, the discussion in Remark 36.38.7, and the definition of the product $K'_0(X) \times K_0(X) \to K'_0(X)$ in Remark 36.38.6.
$\square$

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