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