12 TOM BRIDGELAND
Let D be a triangulated category and suppose A D is the heart of a bounded
t-structure on D. We write U(A) for the subset of Stab(D) consisting of stability
conditions with heart A. In general this subset could be empty. Suppose though
that A is a finite length category with finitely many isomorphism classes of simple
objects S1, ··· , Sn. The Grothendieck group K(D) = K(A) is then a free abelian
group on the generators [Si]. Set
H = {r exp(iπφ) : r R
0
and 0 φ 1} C.
According to Proposition 5.1 we can define a stability condition on D with heart A
by choosing a central charge Z(Si) H for each i and extending linearly to give a
map Z : K(D) C. This argument gives
Lemma 5.2. Let A D be the heart of a bounded t-structure on D and suppose
A is finite length with n simple objects S1, ··· , Sn. Then the subset U(A) Stab(D)
consisting of stability conditions with heart A is isomorphic to
Hn.
The next step is to understand stability conditions on the boundary of the
region U(A) described above. To do this we need the method of tilting.
5.2. Tilting. In the level of generality we shall need, tilting was introduced
by Happel, Reiten and Smalø [30], although the name and the basic idea go back
to a paper of Brenner and Butler [11].
Definition 5.3. A torsion pair in an abelian category A is a pair of full sub-
categories (T , F) of A which satisfy HomA(T, F ) = 0 for T T and F F, and
such that every object E A fits into a short exact sequence
0 −→ T −→ E −→ F −→ 0
for some pair of objects T T and F F.
The objects of T and F are called torsion and torsion-free. The following result
[30, Proposition 2.1] is easy to check.
Lemma 5.4. (Happel, Reiten, Smalø) Suppose A D is the heart of a bounded
t-structure on a triangulated category D. Given an object E D let H
i(E)
A
denote the ith cohomology object of E with respect to this t-structure. Suppose
(T , F) is a torsion pair in A. Then the full subcategory
A = E D :
Hi(E)
= 0 for i / {−1, 0},
H−1(E)
F and
H0(E)
T
is the heart of a bounded t-structure on D.
Recall that a bounded t-structure on D determines and is determined by its
heart A D. In the situation of the Lemma one says that the the subcategory
A is obtained from the subcategory A by tilting with respect to the torsion pair
(T , F). In fact one could equally well consider A [−1] to be the tilted subcategory;
we shall be more precise about this where necessary. Note that the pair (F[1], T ) is
a torsion pair in A and that tilting with respect to this pair gives back the original
subcategory A with a shift.
Suppose A D is the heart of a bounded t-structure and is a finite length
abelian category. Given a simple object S A define S A to be the full
subcategory consisting of objects E A all of whose simple factors are isomorphic
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