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The Stacks project

See [Chapter V, Serre_algebre_locale].

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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