Lemma 42.23.3 (0FDR)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 42: Chow Homology and Chern Classes
Section 42.23: Chow groups and K-groups
Lemma 42.23.3 (cite)

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Lemma 42.23.3. Let $\pi : X \to Y$ be a finite morphism of schemes locally of finite type over $(S, \delta )$ as in Situation 42.7.1. Then $\pi _* : \textit{Coh}(X) \to \textit{Coh}(Y)$ is an exact functor which sends $\textit{Coh}_{\leq k}(X)$ into $\textit{Coh}_{\leq k}(Y)$ and induces homomorphisms on $K_0$ of these categories and their quotients. The maps of Lemma 42.23.2 fit into a commutative diagram

$\xymatrix{ Z_ k(X) \ar[d]^{\pi _*} \ar[r] & K_0(\textit{Coh}_{\leq k}(X)/\textit{Coh}_{\leq k - 1}(X)) \ar[d]^{\pi _*} \ar[r] & Z_ k(X) \ar[d]^{\pi _*} \\ Z_ k(Y) \ar[r] & K_0(\textit{Coh}_{\leq k}(Y)/\textit{Coh}_{\leq k - 1}(Y)) \ar[r] & Z_ k(Y) }$

Proof. A finite morphism is affine, hence pushforward of quasi-coherent modules along $\pi$ is an exact functor by Cohomology of Schemes, Lemma 30.2.3. A finite morphism is proper, hence $\pi _*$ sends coherent sheaves to coherent sheaves, see Cohomology of Schemes, Proposition 30.19.1. The statement on dimensions of supports is clear. Commutativity on the right follows immediately from Lemma 42.12.4. Since the horizontal arrows are bijections, we find that we have commutativity on the left as well. $\square$

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