6 PRAMOD N. ACHAR

Theorem 2.10. Let D be a triangulated category with a Tate twist, and let

{∇s}s∈S

be an abelianesque dualizable quasi-exceptional set with dual set {Δs}s∈S.

The categories

D≤0

= {X ∈ D | Hom(X,

∇s

n [d] = 0 for all n ∈ Z and all d 0},

D≥0

= {X ∈ D | Hom(Δs n [d],X) = 0 for all n ∈ Z and all d 0}

constitute a bounded t-structure on D. In addition, its heart A = D≤0 ∩ D≥0 has

the following properties:

(1) A contains all Δs n and ∇s n .

(2) There is a natural bijection Irr(A)

∼

→ S × Z; the simple object Σs corre-

sponding to (s, 0) ∈ S × Z is the image of the map ιs : Δs →

∇s.

(3) A is a ﬁnite-length, graded quasi-hereditary category; the Δs n are the

standard objects, and the

∇s

n are the costandard objects.

Proof sketch. A similar statement in the ungraded case, with the “abelian-

esque” condition omitted, is proved in [B2, Propositions 1 and 2]. In loc. cit., the

standard and costandard objects in the heart are

tH0(Δs)

and

tH0(∇s),

where

tH0(−)

denotes cohomology with respect to the t-structure in the statement of

the theorem. But the abelianesque condition clearly implies that the Δs and

∇s

already lie in the heart of the t-structure, so the ungraded version of the theorem

follows from the aforementioned results. The same arguments work in the graded

case as well; cf. [B3, Proposition 4].

2.5. Projective covers. We end this section with a result that lets us con-

struct projectives in an abelian category starting from projectives in a Serre sub-

catgory. Its proof is similar to that of [BGS, Theorem 3.2.1].

Proposition 2.11. Let A be a ﬁnite-length abelian category. Let L ∈ A be a

simple object with a projective cover M, and let R denote the kernel of M L.

Let B ⊂ A be the Serre subcategory of objects that do not have L as a subquotient.

Let L ∈ B be a simple object. Assume that the following conditions hold:

(1) We have Hom(M, M) k.

(2) Inside B, L admits a projective cover P .

(3) A and B are admissible subcategories of a triangulated category T, and

(2.2) HomT(M,

2

R) = HomT(P

2

, R) = 0.

Then L admits a projective cover P in A, arising in a short exact sequence

(2.3) 0 → ExtA(P

1

,

M)∗

⊗ M → P → P → 0.

Proof. Let E =

Ext1(P

, M), and consider the identity map id : E → E as an

element of Hom(E, E). Following this element through the chain of isomorphisms

Hom(E, E)

E∗

⊗ E

E∗

⊗

Ext1(P

, M)

Ext1(P

,

E∗

⊗ M),

we obtain a canonical element ν ∈

Ext1(P

,

E∗⊗M).

Form the short exact sequence

corresponding to ν, and deﬁne P to be its middle term. We have thus constructed

the sequence (2.3). We must now show that P is a projective cover of L .

Because Hom(M, M) k, we have natural isomorphisms

Hom(E∗

⊗ M, M) E ⊗ Hom(M, M) E.

6