PERVERSE COHERENT SHEAVES IN GOOD CHARACTERISTIC 9
Lemma 3.3. If (s, n) is standard-maximal in Ξ, then Δs n is a projective cover
of Σs n in
ΞA.
If (s, n) is costandard-maximal, then
∇s
n is an injective hull of
Σs n .
Proof. We already know that Δs n has Σs n as its unique simple quotient,
and that
Ext1(Δs
n , Σt m ) = 0 whenever t s. To prove that Δs n is projective
in
ΞA,
it remains to show that
Ext1(Δs
n , Σs m ) = 0 if m n.
Consider the short exact sequence
0 Rs m Δs m Σs m 0.
Since Rs m
(s)A,
we have
Homi(Δs
n , Rs m ) = 0 for all i 0. It follows
that there is an isomorphism
Ext1(Δs
n , Δs m )


Ext1(Δs
n , Σs m ).
When m n, Lemma 2.8(3) tells us that
Ext1(Δs
n , Δs m ) = 0, as desired.
Proposition 3.4. For each k 0, there is an s-quasistandard object
˜
Δ
(k)
s
with
the following properties:
(1) Hom(
˜
Δ
(k)
s
, Σs) k, and Hom(
˜
Δ
(k)
s
, Σt m ) = 0 if (t, m) = (s, 0).
(2)
Ext1(
˜
Δ
(k)
s
, Σt m ) = 0 if (t, m)
Δ
(s, −k).
(3)
˜
Δ
(k)
s
has a standard filtration whose subquotients are various Δs m with
−k m 0.
Similarly, there is an s-quasicostandard object
˜

s
(k)
such that
(1) Hom(Σs,
˜

s
(k)
) k, and Hom(Σt m ,
˜

s
(k)
) = 0 if (t, m) = (s, 0).
(2)
Ext1(Σt
m ,
˜

s
(k)
) = 0 if (t, m)

(s, k).
(3)
˜
s
(k)
has a costandard filtration whose subquotients are various ∇s m
with 0 m k.
Proof. We proceed by induction on k. When k = 0, we set
˜
Δ
(0)
s
= Δs. For
this object, parts (1) and (2) are contained in Definition 2.3, and part (3) is trivial.
Suppose now that k 0, and that we have already defined
˜
Δ
(k−1)
s
with the desired
properties. Let
Ξ = {(t, m) | (t, m)
Δ
(s, −k)},
and let Ψ = Ξ (s, −k).
We will define
˜
Δ
(k)
s
by invoking Proposition 2.11, but we must first check that
the hypotheses of that proposition are satisfied. Parts (1) and (2) say that
˜
Δ
(k−1)
s
is a projective cover of Σs in
ΨA.
By Lemma 3.3, we have that Δs−k is a
projective cover of Σs−k in
ΞA.
Moreover, because Rs−k
(s)D,
we know
from Definition 2.6(2) that
Hom2(Δs−k
, Rs−k ) =
Hom2(
˜
Δ
(k−1),Rs−k
s
) = 0.
(The latter vanishing holds because
˜
Δ
(k−1)
s
is s-quasistandard.)
Let
˜
Δ
(k)
s
be the projective cover of Σs in
ΞA
obtained from Proposition 2.11.
It is clear then that parts (1) and (2) of the proposition hold for
˜
Δ
(k)
s
. Moreover,
we have an exact sequence
0
Ext1(
˜
Δ
(k−1),
s
Δs−k
)∗
Δs−k
˜
Δ
(k)
s

˜
Δ
(k−1)
s
0
9
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