PERVERSE COHERENT SHEAVES IN GOOD CHARACTERISTIC 17
where D is the Serre–Grothendieck duality functor given by
D = RHom(−, ON ).
In fact, the functor D can be defined using any equivariant dualizing complex [B1].
The fact that ON is a dualizing complex is equivalent to the fact that it is Goren-
stein, cf. [BK, Theorem 5.3.2]. There is a choice of shifts and Tate twists here; our
normalization agrees with the convention of [B3] but not with that of [B2].
The category
PCohG(N
) of perverse coherent sheaves has a number of features
in common with the more familiar perverse constructible sheaves. Key among
these are that every object has finite length, and that the simple objects admit
a characterization resembling that of intersection cohomology complexes. Simple
objects are classified by pairs (C, V), where V is an irreducible G-equivariant vector
bundle on C. The corresponding simple object will be denoted IC(C, V).
The category
PCohG×Gm
(N ) is defined in the same way as above. (Recall
that the orbits of G × Gm coincide with those of G.) The forgetful functor U :
DbCohG×Gm
(N )
DbCohG(N
) restricts to an exact functor
U :
PCohG×Gm
(N )
PCohG(N
)
that takes simple objects of
PCohG×Gm
(N ) to simple objects of
PCohG(N
).
5. Andersen–Jantzen sheaves and perverse coherent sheaves
In this section, we prove a number of lemmas on Andersen–Jantzen sheaves.
We work with (G × Gm)-equivariant sheaves throughout. For the most part, the
Gm-action will play no essential role; nearly every statement in this section has an
obvious G-equivariant analogue, with the same proof. The only exception to this is
part (3) of Proposition 5.6, whose statement and proof involve imposing conditions
on Tate twists.
Many proofs in this section are closely modeled on those in [B2, Section 3],
suitably modified to handle the difficulties that arise in positive characteristic.
Lemma 5.1. For all λ Λ, we have DA(λ)

= A(−λ).
Proof. Recall that proper pushforward Rπ∗ commutes with Serre–Grothen-
dieck duality, where the duality functor on
˜
N is given by D
˜
N
= RHom(−,
π!ON
).
It is a consequence of [BK, Lemma 3.4.2 and Lemma 5.1.1] that
π!ON

=
O
˜
N
, so
D
˜
N
(p∗S
(kλ))

=
RHom(p∗S
(kλ), O
˜
N
)

=
p∗S
(k−λ),
and the lemma follows.
Lemma 5.2. For all λ Λ, we have A(λ)
PCohG×Gm
(N ). In particular, for
all μ, λ Λ, we have
Homi(A(μ),A(λ))
= 0 if i 0.
Proof. Recall that the Springer resolution is semismall [Ja, Theorem 10.11].
This means that for any closed point x in an orbit C N , we have dim
π−1(x)

1
2
codim C. Let UC be the union of the nilpotent orbits whose closure contains C.
Thus, UC is an open G-stable subset of N , and C is the unique closed orbit therein.
Let
˜
N
C
=
π−1(UC
). Every fiber of the proper map π :
˜
N
C
UC has dimension
1
2
codim C, so by [H, Corollary III.11.2], it follows that
Riπ∗(p∗S
(kλ))|
˜
N
U
= 0
for i
1
2
codim C. Since iC

is an exact functor (where iC : {ηC } N is as in
Section 4.5), it follows that
Hi(iCA(λ))
= 0 for i
1
2
codim C. The same reasoning
applies to A(−λ)

= DA(λ), so A(λ)
PCohG×Gm
(N ).
17
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