Let $D$ be a triangulated category, and let $\sigma = (Z, P)$ be a Bridgeland stability condition on $D$. Let $I \subset \mathbb{R}$ be any interval (open, closed, or half-open).

The category $P(I)$ is defined to be the extension closed subcategory generated by $P(\phi)$ for all $\phi \in I$. Bridgeland claims that if $I = (a,b)$, then $P(I)$ consists of all objects $E$ with $$a < \phi^-(E) \leq \phi^+(E) < b. \tag{$*$}$$

Q: Why does $(*)$ define an extension closed subcategory? For example if $$a < \phi < \psi < \phi + 1 < b,$$ one could have a triangle $A \to B \to C \to A[1]$ where $A \in P(\phi)$ and $C \in P(\psi)$. Then I don't see how the Harder-Narasimhan-filtration of $B$ looks like.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.