Lemma 42.58.1 (0FEY)—The Stacks project

Table of contents
Part 3: Topics in Scheme Theory
Chapter 42: Chow Homology and Chern Classes
Section 42.58: Rational intersection products on regular schemes
Lemma 42.58.1 (cite)

Lemma 42.58.1. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be a quasi-compact regular scheme of finite type over $S$ with affine diagonal and $\delta _{X/S} : X \to \mathbf{Z}$ bounded. Then the composition

\[ K_0(\textit{Vect}(X)) \otimes \mathbf{Q} \longrightarrow A^*(X) \otimes \mathbf{Q} \longrightarrow \mathop{\mathrm{CH}}\nolimits _*(X) \otimes \mathbf{Q} \]

of the map $ch$ from Remark 42.56.5 and the map $c \mapsto c \cap [X]$ is an isomorphism.

Proof. We have $K'_0(X) = K_0(X) = K_0(\textit{Vect}(X))$ by Derived Categories of Schemes, Lemmas 36.38.4, 36.36.8, and 36.38.5. By Remark 42.56.12 the composition given agrees with the map of Proposition 42.57.1 for $X = Y$. Thus the result follows from the proposition. $\square$

Comments (3)

Comment #7542 by Hao Peng on July 31, 2022 at 03:53

In this tag and also 42.57.1, is it $\prod_{p\ge0}A^p(X)\otimes \mathbb Q$ instead of $A^*(X)\otimes \mathbb Q$ or somehow we can prove that the Chern classes are finite polynomials?

Comment #7545 by Johan on July 31, 2022 at 09:21

Yes, because $X$ is quasi-compact $ch_i(E)$ will be zero for $i$ large enough.

Comment #7667 by Stacks Project on August 15, 2022 at 11:45

Going to leave as is.

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