Lemma 13.28.5. Let $\mathcal{B}$ be a weak Serre subcategory of the abelian category $\mathcal{A}$. There is a canonical isomorphism

The inverse sends the class $[X]$ of $X$ to the element $\sum (-1)^ i[H^ i(X)]$.

Lemma 13.28.5. Let $\mathcal{B}$ be a weak Serre subcategory of the abelian category $\mathcal{A}$. There is a canonical isomorphism

\[ K_0(\mathcal{B}) \longrightarrow K_0(D^ b_\mathcal {B}(\mathcal{A})),\quad [B] \longmapsto [B[0]] \]

The inverse sends the class $[X]$ of $X$ to the element $\sum (-1)^ i[H^ i(X)]$.

**Proof.**
We omit the verification that the rule for the inverse gives a well defined map $K_0(D^ b_\mathcal {B}(\mathcal{A})) \to K_0(\mathcal{B})$. It is immediate that the composition $K_0(\mathcal{B}) \to K_0(D^ b_\mathcal {B}(\mathcal{A})) \to K_0(\mathcal{B})$ is the identity. On the other hand, using the distinguished triangles of Remark 13.12.4 and an induction argument the reader may show that the displayed arrow in the statement of the lemma is surjective (details omitted). The lemma follows.
$\square$

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