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