8 PRAMOD N. ACHAR
Moreover, from (2.7), we see that
Ext1(P,
X) = 0 for all X B. To prove that
P is a projective object of A, it remains only to show that
Ext1(P,
L) = 0. Using
Lemma 2.1, we may form the long exact sequence
· · ·
Ext1(P,
M)
Ext1(P,
L) HomT(P,
2
R) · · · .
We saw in (2.5) that the first term vanishes, and the assumption (2.2) implies that
the last term does as well. Thus,
Ext1(P,
L) = 0, as desired.
3. Derived equivalences from quasi-exceptional sets
In this section, D will be a triangulated category equipped with a Tate twist
and an abelianesque dualizable graded quasi-exceptional set {∇s}s∈S with dual
set {Δs}, with S satisfying (2.1). Let A denote the heart of t-structure on D as
in Theorem 2.10. Under mild assumptions, there is a natural t-exact functor of
triangulated categories
real :
DbA
D,
called a realization functor. For a construction of real in various settings, see [AR,
Be, BBD]. The goal of this section is to prove that under an additional assumption
(the “effaceability property” of Section 3.2), this is an equivalence of categories.
Below, Sections 3.1–3.3 contain a number preparatory results. The derived
equivalence result, Theorem 3.15, is proved in Section 3.4.
3.1. Standard filtrations and quasistandard objects. We begin with a
number of technical lemmas on the existence and properties of certain objects
which are filtered by standard objects. Most of the results of this section are trivial
in the case where the quasi-exceptional set is actually exceptional, meaning that
Homi(∇s, ∇s)
= 0 for i 0.
Definition 3.1. Let X A. A filtration
0 = X0 X1 · · · Xk = X
is called a standard filtration if there are elements s1,...,sk S and integers
n1,...,nk Z such that Xi/Xi−1 Δsi ni for each i. If X has such a filtration
with s1 = · · · = sk = s, X is said to be s-quasistandard. The notions of costandard
filtration and s-quasicostandard are defined similarly.
Definition 3.2. The standard order is the partial order
Δ
on S ×Z given by
(s, n)
Δ
(t, m) if s t, or else if s = t and n m.
Similarly, the costandard order

is given by
(s, n)

(t, m) if s t, or else if s = t and n m.
A member of a subset Ξ S × Z is said to be standard-maximal (resp. costandard-
maximal) if it is a maximal element of Ξ with respect to
Δ
(resp. ∇).
A number of statements in this section, starting with the following lemma,
contain both a “standard” part and a “costandard” part. In each instance, we will
only prove the part pertaining to standard objects. It is, of course, a routine matter
to adapt these arguments to the costandard case.
For the maps φs : Δs Σs and ψs : Σs ∇s as in Definition 2.3, we introduce
the notation
Rs = ker φs, Qs = cok
ψs.
8
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