10 PRAMOD N. ACHAR

from which we can see that part (3) holds as well.

3.2. Eﬀaceability. For the remainder of Section 3, we assume that the quasi-

exceptional set {∇s} has the following additional property.

Definition 3.5. An abelianesque quasi-exceptional set {∇s}s∈S is said to have

the eﬀaceability property if the following two conditions hold:

(1) For any morphism f : X[−d] → Δs where d 0 and X is s-quasistandard,

there is an object Y ∈ (≤s)A and an injective map g : Δs → Y such that

g ◦ f = 0.

(2) For any morphism f : ∇s → X[d] where d 0 and X is s-quasicostandard,

there is an object Y ∈

(≤s)A

and a surjective map h : Y

∇s

such that

f ◦ h = 0.

Lemma 3.6. For any morphism f : X[−d] → Δs where d 0 and X is

s-quasistandard, there is an s-quasistandard object Y and an injective map g :

Δs → Y such that g ◦ f = 0. Moreover, every standard subquotient of Y/g(Δs) is

isomorphic to some Δs m with m 0.

Proof. Let g : Δs → Y be an embedding as in Deﬁnition 3.5. We must show

how to replace this Y by a certain kind of s-quasistandard object. For now, we

know only that Y ∈ (≤s)A. This means that Y/g(Δs) has a ﬁltration with simple

subquotients lying in

(≤s)A.

We may write:

(3.1) Y ∈ Δs ∗ Σt1 p1 ∗ Σt2 p2 ∗ · · · ∗ Σtk pk ,

with ti ≤ s for all i. From the distinguished triangle Δs → Σs → Rs[1] →, we have

Σs m ∈ Δs m ∗ Rs[1] m .

For each factor Σti pi in (3.1) with ti = s, let us replace it by Δs pi ∗ Rs[1] pi .

We will then have

(3.2) Y ∈ Δs ∗ I1 ∗ · · · ∗ Il

where each Ii is one of:

⎧

⎪Δs

⎨

⎪

⎩

m for some m ∈ Z,

Rs[1] m for some m ∈ Z, or

Σt m for some t s and some m ∈ Z.

Note that each factor Ii that is not of the form Δs m belongs to

(s)D.

Now,

for I ∈ (s)D, Hom(Δs m , I[1]) = 0 by Deﬁnition 2.6(2), so any distinguished

triangle I → J → Δs m → splits. In other words, I ∗ Δs m contains only the

isomorphism class of the direct sum I ⊕ Δs m , and in particular, we have

I ∗ Δs m ⊂ Δs m ∗ I if I ∈

(s)D.

Using this fact, we can rearrange the expression (3.2) so that all factors of the form

Δs m occur to the left of all factors in

(s)D.

In other words, we may assume

without loss of generality that (3.1) reads as follows:

Y ∈ Δs ∗ Δs m1 ∗ · · · ∗ Δs mk ∗ Ik+1 ∗ · · · ∗ Il with Ii ∈

(s)D

for i ≥ k + 1.

This means that there is a distinguished triangle

(3.3) Y → Y → I →

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