Lemma 64.11.1 (Additivity). Let $K\in DF_{\text{perf}}(\Lambda )$ and $f\in \text{End}_{DF}(K)$. Then

## 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

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

**Proof.**
Omitted.
$\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)