The Stacks project

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.

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. $\square$

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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