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
Previous Page Next Page