Lemma 13.28.5. Let $\mathcal{B}$ be a weak Serre subcategory of the abelian category $\mathcal{A}$. Then there are canonical maps

$K_0(\mathcal{B}) \longrightarrow K_0(D^ b_\mathcal {B}(\mathcal{A})) \longrightarrow K_0(\mathcal{B})$

whose composition is zero. The second arrow sends the class $[X]$ of the object $X$ to the element $\sum (-1)^ i[H^ i(X)]$ of $K_0(\mathcal{B})$.

Proof. Omitted. $\square$

Comment #7815 by Anonymous on

Should the lemma say instead that the composition is the identity map? Or is the first map $K_0(\mathcal{B}) \rightarrow K_0(D^b_{\mathcal{B}}(\mathcal{A}))$ not the map $[B] \mapsto [B[0]]$?

Comment #7833 by Anonymous on

Unless I have misunderstood something, it also seems to me that the natural map $K_0(\mathcal{B}) \rightarrow K_0(D^b_{\mathcal{B}}(\mathcal{A}))$ given by $[B] \mapsto [B[0]]$ is a group isomorphism, with inverse as described in this lemma.

To see that the composite $K_0(D^b_{\mathcal{B}}(\mathcal{A})) \rightarrow K_0(\mathcal{B}) \rightarrow K_0(D^b_{\mathcal{B}}(\mathcal{A}))$ is the identity map, we can use the distinguished triangle of Remark 13.12.4 and repeatedly take canonical truncations for any bounded complex $X$ in $D^b_{\mathcal{B}}(\mathcal{A})$.

This would generalize Lemma 13.28.2.

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