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}).
64.11 Filtrations and perfect complexes
We now present a filtered version of the category of perfect complexes. An object (M, F) of \text{Fil}^ f(\text{Mod}_\Lambda ) is called filtered finite projective if for all p, \text{gr}^ p_ F (M) is finite and projective. We then consider the homotopy category KF_{\text{perf}}(\Lambda ) of bounded complexes of filtered finite projective objects of \text{Fil}^ f(\text{Mod}_\Lambda ). We have a diagram of categories
\begin{matrix} KF(\Lambda )
& \supset
& KF_{\text{perf}}(\Lambda )
\\ \downarrow
& & \downarrow
\\ DF(\Lambda )
& \supset
& DF_{\text{perf}}(\Lambda )
\end{matrix}
where the vertical functor on the right is fully faithful and the category DF_{\text{perf}}(\Lambda ) is its essential image, as before.
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
Lemma 64.11.2. Let P \in \text{Fil}^ f(\text{Mod}_\Lambda ) be filtered finite projective, and f : P \to P an endomorphism in \text{Fil}^ f(\text{Mod}_\Lambda ). Then
\text{Tr}(f|_ P) = \sum \nolimits _ p \text{Tr}(f|_{\text{gr}^ p(P)}).
Proof. Omitted. \square
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