PERVERSE COHERENT SHEAVES IN GOOD CHARACTERISTIC 21

that the kernel of the map Fn → Fn−1 is free as a graded k[u]-module, i.e. weakly

free. Thus, every module admits a ﬁnite resolution whose terms are either free or

weakly free. It follows that

DbC

is generated by the weakly free modules.

The lemma then follows from the following claim: Every weakly free module

admits a ﬁnite ﬁltration whose subquotients are free modules. Let M be a weakly

free module, and let m1,...,mn

be a set of homogeneous elements that constitute

a basis for it as a free k[u]-module. Let N = max{deg mi},

and assume without

loss of generality that m1,...,mk have degree N and that mk+1,...,mn have

degree N. Then m1,...,mk must constitute a k-basis for the vector space MN .

The k[u]-submodule M generated by m1,...,mk is a free k[u]-module and a direct

summand of R(M). It is also stable under B and isomorphic to k[u] ⊗ MN as

an object of C. In other words, M is a subobject of M in C; it is free, and the

quotient M/M is weakly free. The claim then follows by induction on the rank of

R(M).

Via the equivalences C

∼

=

CohB×Gm

(u)

∼

=

CohG×Gm

(

˜

N ), we obtain the following

result.

Corollary 5.8.

DbCohG×Gm

(

˜

N ) is generated as a triangulated category by

the objects of the form p∗S (V ) n , where V ranges over all ﬁnite-dimensional B-

representations.

Lemma 5.9.

DbCohG×Gm

(N ) is generated as a triangulated category by objects

of the form Rπ∗F, where F ∈

DbCohG×Gm

(

˜

N ).

Proof. Let D ⊂

DbCohG×Gm

(N ) be the subcategory generated by objects

Rπ∗F for F ∈

DbCohG×Gm

(

˜

N ). Because

PCohG×Gm

(N ) is a ﬁnite-length category

that is the heart of a bounded t-structure, we have that the simple perverse coherent

sheaves generate

DbCohG×Gm

(N ) as a triangulated category, so it suﬃces to show

that the simple perverse coherent sheaves lie in D.

Consider a simple perverse coherent sheaf IC(C, V), where C ⊂ N is a nilpotent

orbit, and V is an irreducible G-equivariant vector bundle on C. Let Z = C C.

We proceed by induction on C with respect to the closure partial order on nilpotent

orbits. That is, we assume that IC(C , V ) ∈ D for all C ⊂ Z. The latter objects

generate the full triangulated subcategory DZ b

CohG×Gm

(N ) ⊂

DbCohG×Gm

(N )

consisting of objects whose support is contained in Z. Thus, our assumption implies

that DZ b

CohG×Gm

(N ) ⊂ D.

By[Ja, Proposition 5.9 and 8.8(II)], there is a parabolic subgroup P ⊃ B and

a P -stable subspace v ⊂ u ∩ C such that the natural map q : G

×P

v → C is a

resolution of singularities of C. Consider the variety X = G

×B

v. We have an

inclusion ˜ ı : X →

˜

N , as well as an obvious smooth map h : X → G

×P

v whose

ﬁbers are isomorphic to P/B. Let i : C →

˜

N be the inclusion map.

Let G ∈

DbCohG×Gm

(C) be an object such that i∗G

∼

= IC(C, V). (Because

coherent pullback is not exact, some care must be taken to distinguish between

these two objects.) Let

˜

G = (q

◦h)∗G,

and let F = ˜∗ ı

˜.

G Since RΓ(P/B, OP/B)

∼

= k,

it follows from the projection formula that the canonical adjunction morphism

q∗G

∼

→

Rh∗h∗(q∗G)

is an isomorphism. Applying Rq∗, we obtain an isomorphism

Rq∗q∗G → R(q ◦ h)∗

˜.

G Then, composing with G → Rq∗q∗G, we get a morphism

G → R(q ◦ h)∗

˜.

G

21