Lemma 43.6.1 (0AZD)—The Stacks project

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Part 3: Topics in Scheme Theory
Chapter 43: Intersection Theory
Section 43.6: Proper pushforward
Lemma 43.6.1 (cite)

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Lemma 43.6.1. Suppose that $f : X \to Y$ is a proper morphism of varieties. Let $\mathcal{F}$ be a coherent sheaf with $\dim (\text{Supp}(\mathcal{F})) \leq k$ , then $f_*[\mathcal{F}]_ k = [f_*\mathcal{F}]_ k$ . In particular, if $Z \subset X$ is a closed subscheme of dimension $\leq k$ , then $f_*[Z]_ k = [f_*\mathcal{O}_ Z]_ k$ .

Proof. See Chow Homology, Lemma 42.12.4. $\square$

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