The Stacks project

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

In this tag and also 42.57.1, is it instead of or somehow we can prove that the Chern classes are finite polynomials?

Comment #7545 by on

Yes, because is quasi-compact will be zero for large enough.

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0FEY. Beware of the difference between the letter 'O' and the digit '0'.