The derived category of the heart of a t-structure

Question

Suppose $\mathcal{D}$ is a triangulated category and that we are given a $t$-structure $(\mathcal{D}^{\leq 0},\mathcal{D}^{\geq 0})$ on $\mathcal{D}$. The heart of the $t$-structure, $\mathcal{A}=\mathcal{D}^{\leq 0} \cap \mathcal{D}^{\geq 0}$, is an abelian category. Is it true in general that $\mathcal{D}=D(\mathcal{A})$ is the derived category of the heart of the given $t$-structure on $\mathcal{D}$? If not, is there an easy example that shows why not?

Answer 1

Some examples from topology:

1) If D is the homotopy category Sp of spectra, then D has a canonical t-structure where the truncations correspond to Postnikov towers, so that the heart is the category Ab of abelian groups. The resulting functor D(Ab) --> Sp is the "generalized Eilenberg-Maclane" functor, usually denoted H. It is not fully faithful, since maps Z/p --> Z/p[n] are zero for n>1 in D(Ab) but there are plenty of maps HZ/p --> HZ/p[n] in Sp corresponding to Steenrod operations. It is also not essentially surjective: the cones of such nontrivial maps Z/p --> Z/p[n] cannot be in the image of H, since otherwise they'd have to be isomorphic to Z/p[1] \oplus Z/p[n].

2) For a Q-linear example, let X be a simply connected space, and let D be "local systems of complexes of Q-vector spaces on X up to quasi-isomoprhism" (you can realize this as a full subcategory of the derived category of sheaves of Q-vector spaces on X if you like). Then truncation on fibers defines a t-structure on D with heart the category of local systems of abelian groups on X, which, in view of our hypotheses, is just Q-vector spaces. But the functor F: D(Q-vect) --> D is again neither fully faithful nor essentially surjective -- e.g. maps F(Q) --> F(Q)[n] biject with the n^{th} rational cohomology of X.

There are a couple of ways to make 2) less topological:

a) Combinatorially: you can replace X by a small category (even a poset) which realizes X, e.g. make a category C out of the some triangulation of S^2, then take D to be the full subcategory of the derived category of C-diagrams of Q-vector spaces consisting of objects where each map in C gets sent to a quasi-isomoprhism of complexes of vector spaces.

b) Algebraically: you can realize local systems on $X$ as modules over chains on the based loop space $\Omega X$, and choose $X$ so that $C_\ast(\Omega X)$ has a very simple model as a DGA. For instance if $X = CP^\infty$ then $C_\ast(\Omega X) = C_\ast(S^1) = Q[e]/e^2, |e|=1$; so an algebraic example is given by the derived category of $Q[e]/e^2$ - modules, or more generally modules over (almost?) any nontrivial rational DGA in homologically non-negative degrees with just $Q$ in degree zero,

Answer 2

The most serious problem is that in general there is no natural functor from $D(A)$ to $D$. To construct one you need an additional structure on $D$. There are several approaches here. One was suggested by Beilinson and gives a notion of a filtered triangulated category. Another important approach uses derivators.

On the other hand, if you have enough structure to construct a functor, then the criterion for it to be an equivalence is rather simple. If I remember right, the necessary and sufficient condition is that each morphism $A \to A'[n]$ in $D$ (with both $A$ and $A'$ in the heart) should be decomposable into a sequence $A \to A_1[1] \to A_2[2] \to \dots \to A_{n-1}[n-1] \to A'[n]$ with all $A_i$ being objects in the heart (in other words, the graded algebra of $Ext$'s should be 1-generated).