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, Lemma 30.16.3), 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.


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