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
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