Lemma 36.38.1. Let $X$ be a Noetherian scheme. Then

$K_0(\textit{Coh}(\mathcal{O}_ X)) = K_0(D^ b(\textit{Coh}(\mathcal{O}_ X)) = K_0(D^ b_{\textit{Coh}}(\mathcal{O}_ X))$

Proof. The first equality is Derived Categories, Lemma 13.28.2. The second equality holds because $D^ b(\textit{Coh}(\mathcal{O}_ X)) = D^ b_{\textit{Coh}}(\mathcal{O}_ X)$ by Proposition 36.11.2. $\square$

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).