Lemma 63.11.2. 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

\[ \text{Tr}(f|_ P) = \sum \nolimits _ p \text{Tr}(f|_{\text{gr}^ p(P)}). \]

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