The Stacks project

Remark 36.38.7. Let $f : X \to Y$ be a proper morphism of locally Noetherian schemes. There is a map

\[ f_* : K'_0(X) \longrightarrow K'_0(Y) \]

which sends $[\mathcal{F}]$ to

\[ [\bigoplus \nolimits _{i \geq 0} R^{2i}f_*\mathcal{F}] - [\bigoplus \nolimits _{i \geq 0} R^{2i + 1}f_*\mathcal{F}] \]

This is well defined because the sheaves $R^ if_*\mathcal{F}$ are coherent (Cohomology of Schemes, Proposition 30.19.1), because locally only a finite number are nonzero, and because a short exact sequence of coherent sheaves on $X$ produces a long exact sequence of $R^ if_*$ on $Y$. If $Y$ is quasi-compact (the only case most often used in practice), then we can rewrite the above as

\[ f_*[\mathcal{F}] = \sum (-1)^ i[R^ if_*\mathcal{F}] = [Rf_*\mathcal{F}] \]

where we have used the equality $K'_0(Y) = K_0(D^ b_{\textit{Coh}}(Y))$ from Lemma 36.38.1.

Comments (2)

Comment #7827 by Anonymous on

In this remark, maybe a reference to Proposition 30.19.1 is better than the currently referenced Lemma 30.16.3? (Proper vs locally projective.)

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Your email address will not be published.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 0FDL. Beware of the difference between the letter 'O' and the digit '0'.