The Stacks project

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. $\square$


Comments (2)

Comment #7815 by Anonymous on

Should the lemma say instead that the composition is the identity map? Or is the first map not the map ?

Comment #7833 by Anonymous on

Unless I have misunderstood something, it also seems to me that the natural map given by is a group isomorphism, with inverse as described in this lemma.

To see that the composite is the identity map, we can use the distinguished triangle of Remark 13.12.4 and repeatedly take canonical truncations for any bounded complex in .

This would generalize Lemma 13.28.2.


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