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The Hirzebruch-Riemann-Roch theorem

Article in The Michigan Mathematical Journal · January 2000

DOI: 10.1307/mmj/1030132729 · Source: OAI

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Michigan Math. J. 48 (2000)

The Hirzebruch–Riemann–Roch Theorem

M a d h av V. Nor i

Dedicated to Professor William Fulton on his sixtieth birthday

It is indeed an honor to dedicate this essentially self-contained proof of HRR to

William Fulton, whose contributions to the study of Chow groups, intersection
theory, and the Riemann–Roch theorems have led to a deeper understanding of
these topics.
As is well known, Grothendieck formulated a relative version GRR of the
Riemann–Roch for proper morphisms f : X → Y, and HRR turned out to be
the special case when Y is a point. To prove GRR, Grothendieck showed that it
sufficed to prove GRR for the projection Y × P n → Y and for a closed immersion.
The former is easy, but the latter is much more subtle; in parallel, Grothendieck
also proved the Chern character induces an isomorphism
ch : Q ⊗ K(X) → Q ⊗ A(X).
Accounts of Grothendieck’s method are found in [SGA; BS; M]. Fulton proved
GRR without denominators for closed immersions quite directly by the famous
“degeneration to the normal cone”. This method has since been used in several
related contexts (see e.g. [Fa] and [GS]).
The aim of this note is to give a direct proof of HRR that does not rely of Grothen-
dieck’s method of factoring a morphism. What is crucially used here, however,
is the formalism introduced by Grothendieck, and in particular the isomorphism
K(X) → G(X) of the K-groups of vector bundles and coherent sheaves, respec-
tively, when X is regular (the hypothesis of quasi-projectivity was removed by
Kleiman; see [F]). Our method can be extended to deduce GRR itself directly, but
this has not been carried out here.
The HRR for compact complex manifolds was deduced by Atiyah and Singer
from their index theorem; it was also proved by methods of differential geometry
by Patodi [P] and Toledo–Tong [TT1]. What is more relevant to this paper is the
Atiyah–Bott version of the Lefschetz fixed-point formula (see [AB]) adapted to
cover the case where the set of fixed points is a submanifold. Such a version is due
to Toledo and Tong (see [TT2]). The fixed-point formula we obtain in Section 2
for periodic self-maps is a little stronger than the classical formula when the char-
acteristic of the ground field is positive: we get an identity in the Witt ring, which
reduced modulo the characteristic yields the classical formula.
The Adams–Riemann–Roch is deduced from the fixed-point formula in Sec-
tion 3. The Hirzebruch–Riemann–Roch is deduced from this in Section 4. The

Received March 13, 2000. Revision received March 17, 2000.

473
474 M a d h av V. Nor i

reductions in Section 4 involve use of the Adams operations and are, for the most
part, quite standard (see e.g. [GS]); thus, our exposition is brief.
If we take n = 2 in Sections 2 and 3 then the paper would be about one-third
in length, and this is enough to cover the Hirzebruch–Riemann–Roch whenever
char(k) 6= 2. Indeed, the paper rests on the following observation: If F is a co-
herent sheaf on a variety X, then the Lefschetz number of the natural involution σ
on the cohomologies of F  F on X × X is simply χ(X, F ). The Riemann–Roch
formalism is introduced in Section 1. The book by Fulton and Lang [FL] is a good
reference for this section.
The author would like to thank Dipendra Prasad and S. Ramanan for useful dis-
cussions, and also Kaj Gartz for doing an excellent job of putting the manuscript
into TEX.

1. The Riemann–Roch Formalism

We recall briefly the Adams power operations ψXn and the Bott classes θ n(X), the
Adams–Riemann–Roch, the Todd class, and the Hirzebruch–Riemann–Roch.
We use K(X) to denote the Grothendieck group of locally free coherent
sheaves of a scheme X, separated, and of finite type over Spec(k). For such a
P F, pits
sheaf ¡Vrepresentative
p
in K(X) is denoted by cl(F ). We often put δ(F ) =
p (−1) F .
The formalism of Hirzebruch (see [FL]) associates to a power series P ∈
A[[t − 1]] a homomorphism APX : K(X) → A ⊗ K(X) of Abelian groups, for
every such scheme X as just described, so that:
(a) f ∗ B APY = APX B f ∗ for every morphism f : X → Y ; and
(b) APX (cl(L)) = P(cl(L)) for every sheaf locally free of rank 1, L on X.
Furthermore, the APX are uniquely determined by these properties.
If P ∈ A[[t − 1]]× then there is a unique system of homomorphisms of Abelian
groups, MXP : K(X) → (A ⊗ K(X))× , so that (a) and (b) hold.

Definition 1.1. A = Z , n ∈ Z , and P(t) = t n . Then APX = ψXn is the nth power
operation of Adams.

For example: if n = −1, then ψXn (cl(F )) = cl(F ∗ ); if n = 2, then ψXn (cl(F )) =
¡V

cl(Sym2 F ) − cl 2 F .
 
Definition 1.2. A = Z n1 , n is a nonzero integer, and P(t) = (t n − 1)/(t − 1).
Then θ n(X) = MXP (cl(X )) are Bott’s cannibalistic classes, where X is smooth.

With n = −1, for example, θ n(X) = (−1) d cl(Xd ) where d = dim(X); for n =
2, we have X p
θ n(X) = cl(X ).
p

Theorem (ARR). Assume that X is smooth and complete. Then, for all a ∈
K(X), we have
χ(X, a) = χ(X, θ n(X)−1 · ψXn (a)).
 The Hirzebruch–Riemann–Roch Theorem 475

Definition 1.3. e
R = Q and P(t) = log(t)/(t −1). Then td(X) = MXP (cl(X )).

Lemma 1.4. If X is smooth of dimension d, then

e
θ n(X) · ψXn (td(X)) e
= ndim(X) td(X).

Proof. Because ψXn is a ring homomorphism, we can see that ψXn B MXP = MXQ
where Q(t) = P(t n ) and P ∈ Q [[t − 1]]× . Thus, the lemma is a consequence of
the identity  
t n − 1 log(t n ) log(t)
· =n .
t −1 tn −1 t −1
In view of Lemma 1.4, ARR is equivalent to ARR0.

Theorem (ARR0 ). If X is smooth and complete of dimension d, and if a ∈

K(X), then
χ(X, a) = n−dim(X) χ(ted(X)−1 · ψXn (a · ted(X)).
L p
Definition–Notation 1.5. A(X) = p A (X) denotes the Chow ring, and
ch : Q ⊗ K(X) → Q ⊗ A(X) denotes the Chern character. P This is a ring ho-
momorphism, and chp B ψXn = np · ψXn , where ch(a) = p chp(a) and where
chp(a) ∈ Q ⊗ Ap(X). Note that the Todd class of X, denoted by td(X), equals
P
ch(ted(X)). For η ∈ Q ⊗ A(X), write η = p η p with η p ∈ Q ⊗ Ap(X). Let R d=
dim X. The degree of the zero cycle η d , denoted by deg(η d ), is written as X η.

Theorem (HRR). For all ξ ∈ Q ⊗ K(X), where X is smooth and complete, we

have Z
χ(X, ξ) = ch(ξ) · td(X).
X

That HRR implies ARR is obvious, simply because HRR gives a formula for
χ(X, ξ) and for χ(X, θ n(X)−1 · ψXn (ξ)).

2. The Atiyah–Bott–Lefschetz Formula

We fix a perfect field k and a natural number n so that k contains n distinct nth
roots of unity. The only schemes considered in this section are schemes Y of fi-
nite type over Spec(k), equipped with an action of a cyclic group G of order n. A
generator σ of G will be fixed once and for all.
We will consider sheaves of OY -modules on Y equipped with G-action. The
Grothendieck group of such sheaves that are locally free and finite rank (resp., co-
herent) will be denoted KG (Y ) (resp., G G (Y )). Every subgroup H of G also acts
on Y and thus KH (Y ) and GH (Y ) are also defined. For any H -sheaf F on Y we
have the induced representation IHG(F ), which is a G-sheaf on Y. This defines
IHG : KH (Y ) → KG (Y ) and IHG : GH (Y ) → G G (Y ). If H is generated by σ a with
L a−1 i ∗
a | n, and if F is an H -sheaf on Y, note that IHG(F ) = i=0 (σ ) F. If A is an
G G
H -sheaf on Y and B is a G-sheaf on Y, then B ⊗ IH (A) = IH (B ⊗ A). Thus we
see that IHG(KH (Y )) ⊂ KG (Y ) is an ideal. We define Kσ (Y ) to be the quotient of
476 M a d h av V. Nor i

KG (Y ) by the sum of the IHG(KH (Y )) taken over all proper subgroups H of G; we

define Gσ (Y ) in a like manner. Thus Kσ (Y ) is a ring, Gσ (Y ) is a Kσ (Y )-module,
and the natural arrow Kσ (Y ) → Gσ (Y ) is a module homomorphism.
Lemma 2.1. If Y is a regular scheme, then
(a) KG (Y ) → G G (Y ) and
(b) Kσ (Y ) → Gσ (Y )
are both isomorphisms.
Proof. The standard method of proof (see e.g. [F, Apx. B.8.3]) shows that
KG (Y ) → G G (Y ) is an isomorphism, once it is checked that every coherent
G-sheaf F on Y is a quotient of a locally free G-sheaf on Y of finite rank. An epi-
morphism A → F of ordinary sheaves induces a G-epimorphism IHG(A) → F
where H = {e}. Taking A to be locally free of finite rank, the result follows.
Part (b) is a consequence of part (a) because G can be replaced by any proper
subgroup H in part (a).
Lemma 2.2. Let F be a G-stable closed subset of Y and let U be its comple-
ment. Let i : F → Y and j : U → Y denote the inclusions. Then i∗ and j ∗ induce
the exact sequences
(a) G G (F ) → G G (Y ) → G G (U ) → 0 and
(b) Gσ (F ) → Gσ (Y ) → Gσ (U ) → 0.
Proof. For part (a), again the standard proof applies. Exactness at G G (Y ) is the
issue, and for this we need φ : G G (U ) → T = coker(i∗ : G G (F ) → G G (Y )).
For a coherent G-sheaf F on U, choose a coherent G-subsheaf F 0 of j∗ F so that
F 0 | U = F. We define φ(cl(F )) = cl(F 0 ) in T .
For part (b), one need only note that (a) is valid for all proper subgroups H of
G as well.
Lemma 2.3. If σ acts without fixed points on Y (i.e., if {y ∈ Y(k̄) | σy = y} =
∅), then  
1
Z ⊗ Gσ (Y ) = 0.
n
Proof. We may choose a G-stable, nonempty Zariski-open affine subset U of Y
such that the G-action of U comes from a fixed-point–free action of G/H where
H is a proper subgroup. We attempt to prove the lemma for U first. If F is a
G-sheaf on U, then IHG(F ) = F ⊗ IHG(OU ). Thus, if IHG(OU ) ∼= OUa as a G-sheaf,
where a = #(G/H ), it would follow that a · G G (U ) = 0. Let π : U → V be
the quotient by the (G/H )-action. By descent, the (G/H )-sheaf IHG(OU ) = π ∗A
for some locally free rank-a sheaf A on V ; actually A = π∗ OU , but this does
not concern us. Replacing V by a suitable nonempty open V 0 and replacing U by
π −1(V 0 ), we may assume that A ∼= OVa . Induction on dimension and Lemma 2.2
now show that Gσ (Y ) is annihilated by ne , where e = 1 + dim Y.
Proposition 2.4. Let F Y denote the closed subscheme
   of σ in Y.
of fixed points
Let i : F Y → Y denote the inclusion. Then i∗ : Z n1 ⊗ Gσ (F Y ) → Z n1 ⊗ Gσ (Y )
is surjective.
 The Hirzebruch–Riemann–Roch Theorem 477

Proof. This is an immediate consequence of Lemmas 2.3 and 2.2.

Remark 2.5. In Proposition 2.4, i∗ is an isomorphism, as we can see from

Quillen’s exact sequence (see [Q, Thm. 5.4, p. 131] adapted to G̃ ). We prove
this here only when Y is smooth (see Theorem 2.8).

Notation 2.6. Let R(k) be the subring generated by n1 and all the nth roots on
unity in k (resp., the Witt ring W(k) of k) if char(k) = 0 (resp., > 0). We have a
ring homomorphism R(k) → k, and every nth root of unity λ in k lifts uniquely
to an nth root of unity hλi in R(k).
Let Ĝ = Hom(G, k × ). Let Y be a scheme on which G acts trivially (this will
be applied to Spec(k) and F Y in the sequel). Every G-sheaf F on Y is the direct
sum of its eigensheaves Fg for g ∈ Ĝ. Thus G G (Y ) = Z [Ĝ] ⊗ G(Y ). For τ ∈ G
we define X
tr(τ | F ) = hg(τ )i cl(Fg ).
g∈ Ĝ

This extends to an R(k)-module homomorphism

tr(τ | ·) : R(k) ⊗ G G (Y ) → R(k),
which is an R(k)-algebra homomorphism if Y is smooth.
Put gen = {τ ∈ G | τ generates G}. Taking all τ ∈ G and then all τ ∈ gen, we
obtain the isomorphisms
tr : R(k) ⊗ G G (Y ) → G(Y )G ,
tr : R(k) ⊗ Gσ (Y ) → G(Y ) gen .
These arrows are R(k)-algebra homomorphisms if Y is smooth.
P ¡V

Notation 2.7. We put δ(F ) = p (−1) p cl p F for a locally free sheaf F.

Theorem 2.8. If Y is smooth, then so is F Y. Let I be the sheaf of ideals in OY

that vanish on F Y. Then:
 
(a) δ(I/I 2 ) is a unit in Z n1 ⊗ Gσ (F Y );
1  
(b) i∗ : Z n ⊗ Gσ (F Y ) → Z n1 ⊗ Gσ (Y ) is an isomorphism; and
 
(c) i ∗ i∗ a = aδ(I/I 2 ) for all a ∈ Z n1 ⊗ Gσ (F Y ).
Vp
Proof. If F is locally free on F Y, then torpOY (F, OF Y ) ∼ =F⊗ (I/I 2 ), so this
proves part (c). FromL Proposition 2.4, we see that (a) impliesQ (b).
Because I/I 2 = g∈ Ĝ (I/I 2 )g we see that δ(I/I 2 ) = g∈ Ĝ δ(I/I 2 )g . Because
 
R(k) is a nonzero free Z n1 -module, it suffices to check that δ(I/I 2 )g is a unit in
R(k) ⊗ Gσ (F Y ); in view of 2.6, this is the same as checking that tr(τ | δ(I/I 2 )g )
is a unit in R(k) ⊗ G(F Y ) for every τ ∈ gen and every g ∈ G. Let r(g) be the rank
of (I/I 2 )g . Because {a ∈ K(F Y ) | rank(a) = 0} is a nilpotent ideal, it suffices to
check that rank tr(τ | δ(I/I 2 )g ) = (1 − hg(τ )i) r(g) is a unit in R(k).
Because F Y is also the fixed points of τ, no eigenvalue of τ on I/I 2 can equal 1.
Thus, if r(g) is positive at some point of Y, then g(τ ) 6= 1. Thus hg(τ )i 6= 1, and
that 1 − hg(τ )i is a unit in R(k) is standard.
478 M a d h av V. Nor i

Notation 2.9. We will assume that Y is complete. P Then we have χG (Y, ·):
G G (Y ) → KG (Spec(k)) given by χG (Y, cl F ) = p (−1) p cl(H p(Y, F )). Be-
cause IHGH p(Y, F ) = H p(Y, IHG(F )) for an H -sheaf F on Y, the χG (Y, ·) factors
through as
χ σ (Y, ·) : Gσ (Y ) → Kσ (Spec(k)).
We put Lef(σ, Y, a) = tr(σ | χ σ (Y, a)) with tr(σ | ·) as in 2.6. Clearly, if G acts
trivially on Y then
Lef(σ, Y, a) = χ(Y, tr(σ | a)).
 
Theorem 2.10. Let Y be smooth and complete. Let a ∈ Z n1 ⊗ Gσ (Y ) and put
 
b = δ(I/I 2 )−1 · i ∗a, with the notation of 2.7; thus b ∈ Z n1 ⊗ Gσ (F Y ). Then
Lef(σ, Y, a) = χ(F Y, tr(σ | b)).

Proof. From Theorem 2.8, we see that i∗ b = a. Because χ σ (Y, i∗ b) = χ σ (F Y, b),

the result follows from the setup in 2.9.

3. The Adams–Riemann–Roch Theorem

We now apply the results of the previous section to the following special situation.
Let X be a smooth variety defined over k, and let Y = X n be equipped with the
natural action of the permutation group. We choose an n-cycle σ once and for all
and denote by G the subgroup generated by σ. We denote by i : X → Y the diag-
onal embedding; in the notation of the previous section this is F Y, the fixed points
of σ in Y.

Lemma 3.1. For a coherent sheaf F on X, we put SF = F  F  · · ·  F on Y.

Thus SF is a G-sheaf. There is an additive homomorphism S : G(X) → Gσ (Y ),
so that S(cl(F )) = cl(SF ) for all coherent F on X.

Proof. Let 0 → F 0 → F → F 00 → 0 be a short exact sequence of coherent

sheaves on X. We put
F = F 0 F ⊃ F 1 F = F 0 ⊃ F 2 F = 0.
The decreasing filtration F • F on F induces a decreasing filtration F • SF of SF
for which we see that F 0SF = SF, F n+1SF = 0, and:
(1) F 0SF/F 1SF ∼ = SF 00 and F nSF/F n+1SF ∼ = SF 0 ;
p p+1
(2) for 0 < p < n, F SF/F SF is a direct sum of sheaves induced from
proper subgroups of G.
Thus, cl(SF ) = cl(SF 0 ) + cl(SF 00 ) in Gσ (Y ), and the lemma follows.

Naturally, we also have S : K(X) → Kσ (Y ).

Lemma 3.2. For all a in K(X), we have

 
1
i ∗Sa = ψXn a in Z ⊗ Kσ (X).
n
 The Hirzebruch–Riemann–Roch Theorem 479

Lemma 3.3. If X is smooth, then

   
I 1
δ 2 = θ n(X) in Z ⊗ Kσ (X).
I n
The right-hand sides of the equations in Lemmas 3.2 and 3.3 belong to K(X),
and we have a natural arrow K(X) → Kσ (X) because G acts trivially on X. The
Adams operations ψXn and the Bott classes θ n(X) have been defined in 1.1 and 1.2,
respectively, and δ(I/I 2 ) occurs in 2.7 and 2.8.
Proof of Lemma 3.2. For every generator τ of G, the operation a 7→ tr(τ | i ∗Sa)
is additive in a by Lemma 3.1, and it is obviously functorial in X. If a =
cl(L) where L is locally free of rank 1, then the G-action on SL is trivial and
cl(i ∗SL) = a n . Since these properties characterize the Adams operations, we see
that tr(τ | i ∗Sa) = ψXn a in R(k) ⊗ K(X). From 2.6, it follows that i ∗Sa = ψXn a
in R(k) ⊗ Kσ (X), and this implies the result.
V P ¡V

Proof of Lemma 3.3. We set (A) = p (−T ) p cl p A in Z [T ] ⊗ Kσ (X)
for every locally free G-sheaf A on X. For every locally free sheaf F with trivial
G-action, we have an exact sequence of G-sheaves on X:
0 → sF → F n → F → 0,
where F n = IHG(F ) and H = {e}. V If F = X , then sF = I/I and thus
2

δ(I/I ) is the
2
V at T
= 1, of (sX ). Let
¡ value, ¡ τ be
V a generator

of G and put
L(F ) = tr τ | (sF ) . We first compute tr τ | (L n ) where L is locally
free of rank 1 on X. Because the action of τ on {S ⊂ {1, ¡2,V. . . , n}
| #S = p},
where 0 < p < n, is fixed-point–free, ¡it follows that cl p L n = 0 in this
V n

range. Putting a = cl(L), we see that tr τ | (L ) = 1 + hε(τ¡)ia n(−T ) n
=
V
1 − (aT ) n , where
¡ Vε denotes

¡ theVsign of
the ¡permutation.

Also, tr τ | (L) =
V
1− aT and tr τ | (L) · tr τ | (sL) = tr τ | (L n ) . Furthermore, 1− aT is
not a zero divisor in R(k)[T ] ⊗ K(X) because its rank, 1 − T, is¡ notVa zero
di-
visor in R(k)[T ]. Setting P(t) = (t n − 1)/(t − 1), we see that tr τ | (sL) =
P(aT ) and, evaluating at T = 1, we have tr(τ | δ(sL)) = P(a). It follows that
tr(τ | δ(sF )) = MXP (cl(F )) with MXP as in Section 1; this holds for all generators
τ of G, so δ(sF ) = MXP (cl(F )) in R(k) ⊗ Kσ (X). Putting F = X then yields
the result.

Theorem 3.4. Assume that X is smooth. For every a ∈ K(X),

Sa = i∗ (θ n(X)−1 · ψXn (a))
1
holds in Z n
⊗ Kσ (X).
1
As in 3.2 and 3.3, the right-hand side belongs to Z n
⊗ K(X). Theorem 3.4 is
immediate from 2.8, 3.2, and 3.3.
Proposition 3.5. Assume that X is complete. For all a ∈ G(X),
χ σ (Y, Sa) = ψ nχ(X, a)
holds in Kσ (Spec(k)).
480 M a d h av V. Nor i

Once again, χ(X, a) ∈ K(Spec(k)) = Z and so ψ nχ(X, a) = χ(X, a).

Proof. It suffices to check this for a = cl(F ) where F is a coherent sheaf on

X. The cohomologies of SF on Y are computed by the Kunneth formula, and
the off-diagonal terms can be dropped because they are induced from proper sub-
groups. The G-representation on H p(X, F ) ⊗ H p(X, F ) ⊗ · · · ⊗ H p(X, F ) ⊂
H np(Y, SF ) is the permutation representation tensored with εp , where ε : G →
{±1} is the sign of the permutation. By Lemma 3.2 applied to Spec(k), the
image of the former in Kσ (Spec(k)) is ψ n(cl H p(X, F )) = rk H p(X, F ) ∈ Z ⊂
Kσ (Spec(k)). Because the image of ε ∈ KG (Spec(k)) → Kσ (Spec(k)) is (−1) n+1,
it follows that
X
χ σ (Y, Sa) = (−1) pn · (−1) p(n+1) · rk H p(X, F )
p
X
= (−1) p rk H p(X, F )
p
= χ(X, F ).

Theorem 3.6 (ARR). If X is complete and smooth and if a ∈ K(X), then

χ(X, a) = χ(X, θ n(X)−1 · ψXn (a)).

Proof. By Theorem 3.4,

χ σ (Y, Sa) = χ σ (X, θ n(X)−1 · ψXn (a)) = χ(X, θ n(X)−1 · ψXn (a)).
But χ σ (Y, Sa) = χ(X, F ) from Proposition 3.5. This completes the proof ofARR.

4. The Hirzebruch–Riemann–Roch Theorem

The notation is as in Section 3; in addition, we assume that k is algebraically
closed. For HRR, we may assume this without any loss of generality.

Notation 4.1. X is smooth of dimension d. For every coherent sheaf F on X, we

have cl(F ) ∈ G(X) = K(X). We use G p(X) to denote the subgroup of G(X) gen-
erated by cl(F ) with dim supp(F ) ≤ d − p. We put K p(X) = {a ∈ Q ⊗ K(X) :
ψXn (a) = npa}.

Proposition 4.2 is proved in [GS]; a proof is included here for completeness.

L
Proposition 4.2. Q ⊗ G p(X) = q≥p K q(X).

Notation 4.3. Let π : Q ⊗ K(X) → K d (X) denote the projection obtained

from p = 0 and q = d in Proposition 4.2.
R
Remark 4.4. Assume, in addition, that X is complete. Then a 7→ X ch(a) is
a homomorphism from G(X) to (n!)−1 Z. Restrict this homomorphism to G d (X).
The kernel of χ(X, ·) : G d (X) → Z is a divisible group because Jacobians of
curves are divisible. It follows that there is e(X) ∈ (n!)−1 Z so that
 The Hirzebruch–Riemann–Roch Theorem 481
Z
ch(a) = e(X)χ(X, a)
X

for all a ∈ G d (X). If f : X1 → X2 is a surjective morphism (both smooth and

complete of dimension d ) then, by considering f ∗a for a ∈ G d (X2 ), we see that
e(X1 ) = e(X2 ). Checking that e( P d ) = 1, by Chow’s lemma we see that e(X) =
1 for all smooth complete X.

Proof of HRR. We proved ARR in Section 3; thus we may assume ARR0 (see
e
Section 1). For a ∈ Q ⊗ K(X), we put χ 0(a) = χ(X, td(X) −1
a). By ARR0 , we
have
χ 0(a) = n−d · χ 0(ψXn a) for all a ∈ Q ⊗ K(X).
It follows that χ 0(a) = 0 for all a ∈ K p(X) and for all p 6= d. In other words,
χ 0(a) = χ 0(πa) with π as in 4.3. Also, for a ∈ Q ⊗ G d (X), a = ted(X) −1
R a be-
0
cause P(1) = 1, where P(t) = log t/(t − 1). Thus χ (a) = χ(X, a) = X ch(a)
fromR4.4 if a ∈ QR ⊗ G d (X). Finally, because ch(K p(X)) ⊂ Q ⊗ Ap(X), we see
that X ch(a) = X ch(πa) for all a ∈ Q ⊗ K(X). It follows that
Z Z
e
χ(X, td(X) −1
a) = χ 0(a) = χ 0(πa) = ch(πa) = ch(a),
X X

and replacing a by ted(X)a yields the statement of HRR.

Proof of Proposition 4.2. For any closed subscheme Z of X, we have [Z] =

cl(OZ ) ∈ G(X) = K(X). It is standard that G p(X)/G p+1(X) is generated by [Z]
for Z closed and irreducible of codimension p in X. To prove 4.2 (by decreasing
induction on p), it suffices to prove that ψXn [Z] − np [Z] is in G p+1(X) for such
Z ⊂ X.
n
Let F be a finite locally free resolution of OZ . We put D = F ⊗ ; thus, D is a
complex of G-sheaves on X. We see that
X 
n n q
ψX [Z] = ψX (−1) cl(Fq )
q
X  
q n 1
= (−1) cl(Fq⊗ ) in Z ⊗ Gσ (X) (by Lemma 3.2)
q
n
X
= (−1) q cl(Dq ) (ignoring the off-diagonal terms as with 3.5)
q
X
= (−1) q cl(H q(D)).
q
 
This is an equality in Z n1 ⊗Gσ (X). Let I be the sheaf of ideals in OX that anni-
hilates OZ . Then
P I annihilates the G-sheaf H q (D) and tr(τ | H q (D)) for any τ ∈
gen. If m = q (−1) q l(tr(τ | H q (D)), where l denotes the length of the sheaf at
the generic point of Z, then ψXn [Z] − m[Z] ∈ G p+1(X), so we have to prove m =
np . For this, we may replace X by any Zariski-open subset whose intersection
 482 M a d h av V. Nor i

with Z is nonempty. Thus we may assume that Z is a local complete intersec-

tion in X. Let j : Z → X denote the inclusion and define N ∈ K(Z)V by j∗ N ∗ =
I/I . With s as in the proof of Lemma 3.3, we see that H q (D) = j∗ q(sN ∗ ) as
2

a G-sheaf, just by checking the actionPof transpositions in the permutation group.

It follows, as in the proof of 3.3, that q (−1) q cl(H q (D)) = j∗ (MZP (N ∗ )) where
 
P(t) = (t n −1)/(t −1), and the equality holds in Z n1 ⊗ Gσ (X). Because P(1) =
n and rk N ∗ = p, it follows that m = np as desired; this completes the proof of
the proposition. We remark that this method of proof also yields ARR for closed
immersions directly.

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