The Stacks project

63.11 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 63.11.1 (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 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. $\square$

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

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

Proof. Omitted. $\square$


Comments (0)


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Your email address will not be published.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.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 03TJ. Beware of the difference between the letter 'O' and the digit '0'.