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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