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

12