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