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
10
Previous Page Next Page