Lemma 63.11.1 (Additivity). Let $K\in DF_{\text{perf}}(\Lambda )$ and $f\in \text{End}_{DF}(K)$. Then

**Proof.**
By Proposition 63.10.2, 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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