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

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 $K_0(\mathcal{B})$.

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