Derived categories bleg March 24, 2008Posted by Ben Webster in Category Theory, homological algebra.
Is the following theorem true, and if so, where is the reference?
Theorem? Let be a triangulated category with finite homological dimension, and let be a finite set of objects which generate and whose classes are a basis of the Grothendieck group, such that for all . Then there exists a -structure on such that the are the set of simple objects in the heart.
These conditions are obviously necessary, but I’m a bit less confident that they are sufficient.
Hmmm, I guess this means we’re going to have to explain what a -structure is now, doesn’t it?
EDIT: I found the theorem I wanted in the paper David mentioned, by Bezrukavnikov. Roughly, the right theorem is that the ‘s are the simple objects in the heart of a -structure if there is a semi-simple abelian subcategory of the triangulated category in which they are the simple objects, and they satisfy the conditions above.