Lemma 64.11.1 (Additivity). Let K\in DF_{\text{perf}}(\Lambda ) and f\in \text{End}_{DF}(K). Then
\text{Tr}(f|_ K) = \sum \nolimits _{p\in \mathbf{Z}} \text{Tr}(f|_{\text{gr}^ p K}).
Proof. By Proposition 64.10.2, we may assume we have a bounded complex P^\bullet of filtered finite projectives of \text{Fil}^ f(\text{Mod}_\Lambda ) and a map f^\bullet : P^\bullet \to P^\bullet in \text{Comp}(\text{Fil}^ f(\text{Mod}_\Lambda )). So the lemma follows from the following result, which proof is left to the reader. \square
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