The Stacks project

Lemma 12.24.12. Let $\mathcal{A}$ be an abelian category. Let $(K^\bullet , F)$ be a filtered complex of $\mathcal{A}$. Assume that the filtration on each $K^ n$ is finite (see Definition 12.19.1) and that for some $r$ we have only a finite number of nonzero $E_ r^{p, q}$. Then only a finite number of $H^ n(K^\bullet )$ are nonzero and we have

\[ \sum (-1)^ n[H^ n(K^\bullet )] = \sum (-1)^{p + q} [E_ r^{p, q}] \]

in $K_0(\mathcal{A}')$ where $\mathcal{A}'$ is the smallest weak Serre subcategory of $\mathcal{A}$ containing the objects $E_ r^{p, q}$.

Proof. Denote $E_ r^{even}$ and $E_ r^{odd}$ the even and odd part of $E_ r$ defined as the direct sum of the $(p, q)$ components with $p + q$ even and odd. The differential $d_ r$ defines maps $\varphi : E_ r^{even} \to E_ r^{odd}$ and $\psi : E_ r^{odd} \to E_ r^{even}$ whose compositions either way give zero. Then we see that

\begin{align*} [E_ r^{even}] - [E_ r^{odd}] & = [\mathop{\mathrm{Ker}}(\varphi )] + [\mathop{\mathrm{Im}}(\varphi )] - [\mathop{\mathrm{Ker}}(\psi )] - [\mathop{\mathrm{Im}}(\psi )] \\ & = [\mathop{\mathrm{Ker}}(\varphi )/\mathop{\mathrm{Im}}(\psi )] - [\mathop{\mathrm{Ker}}(\psi )/\mathop{\mathrm{Im}}(\varphi )] \\ & = [E_{r + 1}^{even}] - [E_{r + 1}^{odd}] \end{align*}

Note that all the intervening objects are in the smallest Serre subcategory containing the objects $E_ r^{p, q}$. Continuing in this manner we see that we can increase $r$ at will. Since there are only a finite number of pairs $(p, q)$ for which $E_ r^{p, q}$ is nonzero, a property which is inherited by $E_{r + 1}, E_{r + 2}, \ldots $, we see that we may assume that $d_ r = 0$. At this stage we see that $H^ n(K^\bullet )$ has a finite filtration (Lemma 12.24.11) whose graded pieces are exactly the $E_ r^{p, n - p}$ and the result is clear. $\square$

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