PERVERSE COHERENT SHEAVES IN GOOD CHARACTERISTIC 5
Any object isomorphic to some Δs n is called a standard object, and any object
isomorphic to some
∇s
n is a costandard object.
2.4. Quasi-exceptional sets. We again let S be a set equipped with a partial
order satisfying (2.1). Let D be a triangulated category, either equipped with a
Tate twist (the “graded case”) or not (the “ungraded case”). As noted above, the
definitions and lemmas below are usually stated only for the graded case. However,
our first definition comes with a caveat; see the remark below.
Definition 2.4. A graded quasi-exceptional set in D is a collection of objects
{∇s}s∈S
such that the following conditions hold:
(1) If s t, then
Homi(∇s,
∇t) = 0 for all i Z.
(2) If i 0, then
Homi(∇s,
∇s)
= 0, and
Hom(∇s, ∇s)
k.
(3) If i 0 and n 0, then
Homi(∇s,
∇s n )) = 0.
(4) The objects {∇s n | s S, n Z} generate D as a triangulated category.
For s S, we denote by
(s)D
the full triangulated subcategory of D generated by
all
∇t
n with t s.
Remark 2.5. An ungraded quasi-exceptional set is defined with analogues of
conditions (1), (2), and (4) above, but without condition (3). The omission of
condition (3) makes the two cases substantially different. In particular, the results
of Section 3 apply only to the graded case.
Definition 2.6. A graded quasi-exceptional set
{∇s}
in D is said to be dual-
izable if for each s S, there is an object Δs and a morphism ιs : Δs
∇s
such
that:
(1) The cone of ιs lies in
(s)D.
(2) If s t,
Homi(Δs,
∇t) = 0 for all i Z.
The set {Δs} is known as the dual quasi-exceptional set.
It follows from the second condition that Hom(Δs,X) = 0 for all X
(s)D.
The proofs of the following two lemmas about a dual set are routine; we omit the
details.
Lemma 2.7. If
{∇s}
is a dualizable quasi-exceptional set, then the members of
the dual set {Δs} are uniquely determined up to isomorphism.
Lemma 2.8. Let
{∇s}
be a dualizable quasi-exceptional set, and let {Δs} be its
dual set. Then:
(1) If s t, then
Homi(Δs,
Δt) = 0 for all i Z.
(2) If i 0, then
Homi(Δs,
Δs) = 0, and Hom(Δs, Δs) k.
(3) If i 0 and n 0, then
Homi(Δs,
Δs n ) = 0.
(4) The objects
{∇s
n | s S, n Z} generate D as a triangulated category.
Furthermore, for all i Z, there are natural isomorphisms
Homi(Δs,
Δs)


Homi(Δs,
∇s)


Homi(∇s,
∇s).
Definition 2.9. A dualizable quasi-exceptional set
{∇s}s∈S
with dual set
{Δs}s∈S, is said to be abelianesque if we have
Homi(∇s, ∇t)
=
Homi(Δs,
Δt) = 0 for all i 0.
The main technical result we need about quasi-exceptional sets is the following.
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