# Derived categories bleg

Is the following theorem true, and if so, where is the reference?

Theorem? Let $D$ be a triangulated category with finite homological dimension, and let $\{L_1,\ldots, L_n\}$ be a finite set of objects which generate $D$ and whose classes are a basis of the Grothendieck group, such that $\mathrm{Ext}^{-k}(L_i,L_j)=0$ for all $k>0$. Then there exists a $t$-structure on $D$ such that the $L_i$ 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 $t$-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 $L_i$‘s are the simple objects in the heart of a $t$-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.

## 6 thoughts on “Derived categories bleg”

1. The LaTeX paste didn’t work in your post!

2. Hi Ben –
I assume
1. you know this reference, and
2. someone will have a better answer,
but in the meanwhile I can say the closest things that comes to mind
is the paper “Quasi-exceptional sets and..” (arXiv:0102039)
by Bezrukavnikov (in particular Proposition 1) where he
characterizes t-structures generated by a quasi-exceptional
set, which can be constructed from your data under
some conditions.. I think there is a theory by Cline-Parshall-Scott
(or some subset) formalizing various properties of categories
like category O, that might be useful in this context..
(then again many of the world’s experts on just this kind
of problem are within 1/2 a mile of you right now, might be more
efficient to stand outside and shout out the problem?)

3. anonymous says:

I doubt it. Is the bounded derived category of modules over C[x] a counterexample? Take n =1, L_1 = C[x] itself. But I haven’t thought it through: it may be harder to show “there is no t-structure” than I think.

4. right, another obvious necessary condition is that Ext^0(L_i,L_i) = k
-otherwise you want to modify the question to talk about
constituents of the L_i rather than the L_i themselves..

5. b says:

Shouldn’t one require that the {L_i} be orthogonal in a stronger sense than just K-theory? If not, there could be non-zero maps L_i -> L_j for i != j, ruining simplicity. For example, consider the bounded derived category of coherent sheaves on P^1 with L_1 = O, and L_1 = O(-1).