ON THE COHOMOLOGY AND THE CHOW RING
OF THE CLASSIFYING SPACE OF PGL p
ANGELO VISTOLI
Abstract. We investigate the integral cohomology ring and the Chow
ring of the classifying space of the complex projective linear group PGL*
*p,
when p is an odd prime. In particular, we determine their additive
structures completely, and we reduce the problem of determining their
multiplicative structures to a problem in invariant theory.
Contents
1. Introduction 1
2. Notations and conventions 4
3. The main results 5
4. Preliminaries on equivariant intersection theory 7
5. On Cp x ~p 12
6. On Cp n TGL p 19
7. On Cp n TPGL p 26
8. On Sp n TPGL p 31
9. Some results on A *PGLp 32
10. Localization 34
11. The classes ae and fi 37
12. The splitting 39
13. The proofs of the main Theorems 43
14. On the ring A *TPGLpSp 45
References 49
1. Introduction
Let G be a complex linear group. One of the main invariants associated
with G is the cohomology H *Gof the classifying space BG. B. Totaro (see
[13]) has also introduced an algebraic version of the cohomology of the clas-
sifying space of an algebraic group G over a field k, the Chow ring A *Gof
the classifying space of G. When k = C there is a cycle ring homomorphism
____________
Date: April 24, 2005.
2000 Mathematics Subject Classification. 14C15, 14L30, 20G10, 55R35.
Partially supported by the University of Bologna, funds for selected researc*
*h topics.
1
2 ANGELO VISTOLI
A*G ! H *G. Chow rings are normally infinitely harder to study than coho-
mology; it remarkable that, in contrast, A*Gseems to be better behaved, and
easier to study, than H*G. For example, when G is a finite abelian group, A*G
is the symmetric algebra over Z of the dual group bG; while the cohomology
ring contains this symmetric algebra, but is much more complicated (for
example, will contain elements of odd degree), unless G is cyclic.
This ring A *Ghas also been computed for G = GL n, SLn , Spn by Totaro
and R. Pandharipande, and for G = SO n by R. Field. However, not much
is known for the PGL nseries. Even the cohomology of B PGL n seems to
be mysterious. Algebraic topologists tend to work with cohomology with
coefficients in a field, the case in which their very impressive toolkits work
the best. When p does not divide n, the comology ring H *(B PGL n, Z=pZ)
is a well understood polynomial ring. Also, since PGL 2= SO 3, the ring
H*(B PGL 2, Z=2Z) is also well understood. The other results that I am
aware of on H *(B PGL n, Z=pZ) are the following.
(1) In [9], the authors compute the cohomology ring H*(B PGL 3, Z=3Z).
(2) The ring H *(B PGL n, Z=2Z) is known when n 2 (mod 4) ([8] and
[12]).
(3) Some interesting facts on H *(B PGL p, Z=pZ) are proved in [14].
On the other hand, to my knowledge no one has studied the integral
cohomology ring H *PGLn.
In the algebraic case, the only known results about A *PGLn, apart from
the case of PGL 2 = SO 3, concern PGL 3 and were proved by Vezzosi in [15].
Here he determines almost completely the structure of A*PGL3by generators
and relations; the only ambiguity is about one of the generators, denoted by
O and living in A6PGL3, about which he knows that it is 3-torsion, but is not
able to determine whether it is 0. This O maps to 0 in the cohomology ring
H*PGL3; according to a conjecture of Totaro, the cycle map A *PGL3! H*PGL3
should be injective; so, if the conjecture is correct, O should be 0.
Despite this only partial success, the ideas of [15] are very important.
The main one is to make use of the stratification method to get generators.
This is how it works. Recall that Edidin and Graham ([2]) have generalized
Totaro's ideas to give a full-fledged equivariant intersection theory. Let V be
a representation of a group G; then we have A*G= A*G(V ). Suppose that we
have a stratification V0, . . . , Vt of V by locally closed invariant subvariet*
*ies,
such that each V i def=\j iVj is open in V , each Vi is closed in V i, and
Vt = V \ {0}. If we can determine generators for A*G(Vi) for each i, then one
can use the localization sequence
A *G(Vi) -! A*G(V i) -! A*G(V i-1) -! 0
and induction to get generators for A *G(V \ {0}); and since A *G(V \ {0}) =
A*G= cr(V ) , where cr(V ) 2 ArGis the rth Chern class of V , we obtain that
A*Gis generated by lifts to A*Gof the generators for A*G(V \ {0}), plus cr(V ).
ON THE CLASSIFYING SPACE OF PGLp 3
The stratification methods gives a unified approach for all the known
calculations of A *Gfor classical groups (see [10]).
Vezzosi applies the method to the adjoint representation space V = sl3
consisting of matrices with trace 0. The open subscheme V0 is the subscheme
of matrices with distict eigenvalues; its Chow ring is related with the Chow
ring of the normalizer N3 of a maximal torus TPGL p in PGL 3. In order to
get relations, Vezzosi uses an unpublished result of Totaro, implying that
the restriction homomorphism A *PGL3! A *N3is injective. The reason why
he is not able to determine whether O is 0 or not is that he does not have a
good description of the 3-torsion in A *N3.
In this paper we extend Vezzosi's approach to the case of PGL p, where p is
an odd prime; and we also show how this can give considerable information
on the cohomology of a classifying space.
Let TPGL p be the standard maximal torus in PGL p, consisting of classes
of diagonal matrices, Sp its Weyl group. Here are our main results (see
Section 3 for details).
Sp
(1) The natural homomorphism A*TPGLp! A*TPGLp is surjective, and
Sp
has a natural splitting A*TPGLp ! A*TPGLp, which is a ring homo-
morphism.
(2) The ring A *PGLpis generated as an algebra over A *TPGLpSp by a
single p-torsion element ae 2 Ap+1PGLp; we also describe the relations.
Sp
(3) The ring H *PGLpis generated as an algebra over A *TPGLp by two
elements: the image ae 2 H 2p+2PGLpof the class above and the Brauer
class fi 2 H3PGLp; we also describe the relations.
(4) Using (2) and (3) above, we describe completely the additive struc-
tures of A *PGLpand H *PGLp.
S3
(5) For p = 3 we give a presentation of A *TPGL3 by generators and
relations (this is already in [15]); and this, together with (2) and
(3) above, gives presentations of A *PGL3and H *PGL3, completing the
work of [15].
(6) The cycle homomorphism A*PGLp! HevenPGLpinto the even-dimensional
cohomology is an isomorphism.
Sp
The ring A *TPGLp is complicated when p > 3; see the discussion in
Section 14.
The class ae in (2) seems interesting, and gives a new invariant for sheaves
of Azumaya algebras of prime rank (Remark 11.3). In [11], Elisa Targa shows
that ae is not a polynomial in Chern classes of representations of PGL p.
Many of the ideas in this paper come from [15]. The main new contri-
butions here are the contents of Sections 6 and 7 (the heart of these results
are Proposition 6.1, and the proof of Lemma 6.5), which substantially im-
prove our understanding of the cohomology and Chow ring of the classifying
4 ANGELO VISTOLI
space of Np, and Proposition 10.1, which gives a way of showing that in the
stratification method no new generators come from the strata correspond-
ing to non-zero matrices with multiple eigenvalues, thus avoiding the painful
case-by-case analysis that was necessary in [15].
Acknowledgments. I would like to thank Nitin Nitsure and Alejandro
Adem for pointing out references [8], [12] and [14] to me.
I am also in debt with Alberto Molina, who discovered a serious mistake
in the proof of Theorem 3.2 given in a preliminary version of the paper, and
with Marta Morigi, who helped me fix it.
2. Notations and conventions
All algebraic groups and schemes will be of finite type over a fixed field
k of charateristic 0. Furthermore, we will fix an odd prime p, and assume
that k contains a fixed pth root of 1, denoted by !. When k = C, we take
! = e2ssi=p.
The hypothesis that the characteristic be 0 is only used in the proof
Theorem 9.3, which should however hold over an arbitrary field. If so, it
would be enough to assume here that the characteristic of k be different
from p.
Our main tool is Edidin and Graham's equivariant intersection theory
(see [2]), which works over an arbitrary field; when we discuss cohomology,
instead, we will always assume that k = C. All finite groups will be con-
sidered as algebraic groups over k, in the usual fashion. We denote by Gm
the multiplicative group of non-zero scalars over k, ~ nthe algebraic group
of nth roots of 1 over k.
Whenever V is a vector space over k, we also consider it as a scheme over
k, as the spectrum of the symmetric algebra of the dual vector space V _. If
V is a representation of an algebraic group G, then there is an action of G
on V as a scheme over k.
We denote by TGL p, TSLp and TPGL p the standard maximal tori in the
respective groups, those consisting of diagonal matrices. We identify the
Weyl groups of these three groups with the symmetric group Sp. We also
denote the normalizer of TPGL p in PGL p by Sp n TPGL p.
If a1, . . . , ap are elements of k*, we will denote by [a1, . .,.ap] the dia*
*gonal
matrix in GL pwith entries a1, . . . , ap, and also its class in PGL p. In gene*
*ral,
we will ofter use the same symbol for a matrix in GL pand its class in PGL p;
this should not give rise to confusion.
It is well known that the arrows
A *GLp-! (A *TGLp)Sp, H*GLp-! (A *TGLp)Sp
and
A*SLp-! (A *TSLp)Sp, H*GLp-! (A *TSLp)Sp
induced by the embeddings TGL p ,! GL p and TSLp ,! SL p are isomor-
phisms. If we denote by xi 2 A *TGLp= H *TGLpthe first Chern class of
ON THE CLASSIFYING SPACE OF PGLp 5
the ith projection TGL p ! Gm , or its restriction to TSLp , then A *TGLp=
H*TGLp is the polynomial ring Z[x1, . .,.xp], while A *TSLp= H *TSLpequals
Z[x1, . .,.xp]=(x1 + . .+.xp). If we denote by oe1, . . . , oep the elementary
symmetric functions in the xi, then we conclude that
A *GLp= H*GLp= Z[oe1, . .,.oep]
while
A*SLp= H*SLp= Z[oe1, . .,.oep]=(oe1) = Z[oe2, . .,.oep].
The ring A *TPGLp= H *TPGLpis the subring of A *TGLpgenerated by the
Q
differences xi- xj. In particular it contains the element ffi = i6=j(xi- xj),
which we call the discriminant (up to sign, it is the classical discriminant);
it will play an important role in what follows.
We will use the following notation: if R is a ring, t1, . . . , tn are elemen*
*ts
of R, f1, . . . , fr are polynomials in Z[x1, . .,.xn], we write
R = Z[t1, . .,.tn]= f1(t1, . .,.tn), . .,.fr(t1, . .,.tn)
to indicate the the ring R is generated by t1, . . . , tn, and the kernel of the
evaluation map Z[x1, . .,.xn] ! R sending xito tiis generated by f1, . . . , fr.
When there are no fi this means that R is a polynomial ring in the ti.
3. The main results
Consider the embedding ~p ,! TPGL pdefined by i 7! [i, i2, . .,.ip-1, 1].
This induces a restriction homomorphism
A*TPGLp! A*~p= Z[j]=(pj),
where j is the first Chern class of the embedding ~p Gm .
2-p
The restriction of the discriminant ffi 2 (A pTPGLp)Sp to ~p is the element
Y `Y ' 2
(ij - jj) = (i - j) jp -p
i6=j i6=j
of Z[j]=(pj); this is non-zero multiple of jp2-p (in fact, it is easy to check
that it equals -jp2-p).
Proposition 3.1. The image of the restriction homomorphism
* Sp
ATPGLp -! Z[j]=(pj)
is the subring generated by jp2-p.
This is proved at the end of Section 7.
Sp
Theorem 3.2. There exists a canonical ring homomorphism A*TPGLp !
Sp
A*PGLpwhose composite with the restriction homomorphism A*PGLp! A*TPGLp
is the identity.
6 ANGELO VISTOLI
This is proved in Section 12.
As a consequence, A *PGLpand H *PGLpcan be regarded as A *TPGLpSp-
algebras.
Sp
Theorem 3.3. The A *TPGLp -algebra A *PGLpis generated by an element
ae 2 Ap+1pgl, and the ideal of relations is generated by the following:
(a)pae = 0, and
(b)aeu = 0 for all u in the kernel of the homomorphism A*TPGLp Sp! A*~p.
There is a similar description of the cohomology: besides the element
ae, now considered as living in H 2p+2PGLp, we need a single class fi in degree
3. This class is essentially the tautological Brauer class. That is, if we
call C the sheaf of complex valued continuous functions and C* the sheaf of
complex valued nowhere vanishing continuous functions on the classifying
space B PGL p, the tautological PGL p principal bundle on B PGL p has a
class in the topological Brauer group H 2(B PGL p, C*)tors(see [6]). On the
other hand, the exponential sequence
0 -! Z 2ssi--!C -! C* -! 1
induces a boundary homomorphism
H2(B PGL p, C*) -! H3(B PGL p, Z) = H3PGLp
which is an isomorphism, since B PGL pis paracompact, hence Hi(B PGL p, C) =
0 for all i > 0. Our class fi is, up to sign, the image under this boundary
homomorphism of the Brauer class of the tautological bundle.
Sp
Theorem 3.4. The ring H *PGLpis the commutative A*TPGLp -algebra gen-
erated by an element fi of degree 3 and the element ae of degree 2p + 2. The
ideal of relations is generated by the following:
(a)fi2 = 0,
(b)pae = pfi = 0, and
(c)aeu = fiu = 0 for all u in the kernel of the homomorphism A*TPGLp Sp!
A *~p.
Corollary 3.5. The cycle homomorphism induces an isomorphism of A*PGLp
with H evenPGLp.
From here it is not hard to get the additive structure of A*PGLpand H*PGLp.
For each integer m, denote by r(m, p) the number of partitions of m into
numbers between 2 and p. If we denote by ss(m, p) the number of partitions
of m with numbers at most equal to p (a more usual notation for this is
p(m, p), which does not look very good), then r(m, p) = ss(m, p)-ss(m-1, p).
We will also denote by s(m, p) the number of ways of writing m as a linear
combination (p2 - p)i + (p + 1)j, with i 0 and j > 0; and by s0(m, p) the
number of ways of writing m as a the same linear combination, with i 0
ON THE CLASSIFYING SPACE OF PGLp 7
and j 0. Obviously we have s0(m, p) = s(m, p), unless m is divisible by
p2 - p, in which case s0(m, p) = s(m, p) + 1.
Theorem 3.6.
(a)The groups A mPGLpis isomorphic to
Zr(m,p) (Z=pZ)s(m,p).
(b)The group H mPGLpis isomorphic to A m=2PGLpwhen m is even, and is iso-
morphic to
0(m-3_,p)
(Z=pZ)s 2
when m is odd.
When p = 3 we are able to get a description of A *PGLpand H *PGLpby
generator and relations, completing the work of [15].
Theorem 3.7.
(a)A *PGL3is the commutative Z-algebra generated by elements fl2, fl3, ffi, ae,
of degrees 2, 3, 6 and 4 respectively, with relations
27ffi - (4fl32+ fl23), 3ae, fl2ae, fl3ae.
(b)H *PGL3fl2, fl3, ffi, ae and fi of degrees 4, 6, 12, 8a and 3 respectively, *
*with
relations
27ffi - (4fl32+ fl23), 3ae, 3fi, fi2, fl2ae, fl3ae, fl2fi, *
*fl3fi.
The rest of the paper is dedicated to the proofs of these results. We start
by recalling some basic facts on equivariant intersection theory.
4. Preliminaries on equivariant intersection theory
In this section the base field k will be arbitrary.
We refer to [13], [2] and [15] for the definitions and the basic properites
of the Chow ring A *Gof the classifying space of an algebraic group G over
a field k, and of the Chow group A *G(X) when X is a scheme, or algebraic
space, over k on which G acts, and their main properties. Almost all X that
appear in this paper will be smooth, in which case A*G(X) is a commutative
ring; the single exception will be in the proof of Lemma 6.5.
The connection between these two notions is that A *G= A*G(Spec k).
Recall that A *G(X) is contravariant for equivariant morphism of smooth
varieties; that is, if f :X ! Y is a G-equivariant morphism of smooth G-
schemes, there in an induced ring homomorphism f* :A *G(X) ! A*G(Y ).
If k = C, and X is a smooth algebraic variety on which G acts, there is
a cycle ring homomorphism A *G(X) ! H *G(X) from the equivariant Chow
ring to the equivariant cohomology ring; this is compatible with pullbacks.
Furthermore, if f is proper there is a pushforward f*: A *G(Y ) ! A*G(X);
this is not a ring homomorphism, but it satisfies the projection formula
f*(, . f*j) = f*, . j
8 ANGELO VISTOLI
for any , 2 A*G(X) and j 2 A*G(Y ).
A *G(Y ) -'*!A*G(X) -! A*G(X \ Y ) -! 0.
The analogous statement for cohomology is different: here the restriction
homomorphism H*G(X) ! H*G(X \Y ) is not necessarily surjective. However,
when X and Y are smooth we have a long exact sequence
. ._.________//_Hi-1G(X \ Y )
fff
f@fffffffffff
ffffff
i-2r ssfffffff
HG (Y )___'*____//_HiG(X)________//_HiG(X \ Y )
ffff
f@ffffffffff
fffff
ssfffffff
H i-2r+1G(Y )_________//_. . .
where r is the codimension of Y in X.
Furthermore, if H ! G is a homomorphism of algebraic groups, and G
acts on a smooth scheme X, we can define an action of H on X by com-
posing with the given homomorphism H ! G. Then we have a restriction
homomorphism
resGH:A *G(X) -! A*H(X).
Here is another property that will be used often. Suppose that H is an
algebraic subgroup of G. We can define a ring homomorphism A*G(G=H) !
A*H by composing the restriction homomorphism A *G(G=H) ! A *H(G=H)
with the pullback A *H(G=H) ! A *H(Spec k) = A *Hobtained by the homo-
morphism Spec k ! G=H whose image is the image of the identity in G(k).
Then this ring homomorphism is an isomorphism.
More generally, suppose that H acts on a scheme X. We define the
induced space GxH X as usual, as the quotient (GxX)=H by the free right
action given by the formula (g, x)h = (gh, h-1x). This carries a natural
left action of G defined by the formula g0(g, x) = (g0g, x). There is also
an embedding X ' H xH X ,! G xH X that is H-equivariant: and the
composite of the restriction homomorphism A *G(G xH X) ! A*H(G xH X)
with the pullback A *H(G xH X) ! A*H(X) is an isomorphism.
Furthermore, if V is a representation of G, then there are Chern classes
ci(V ) 2 AiG, satisfying the usual properties. More generally, if X is a smooth
scheme over k with an action of G, every G-equivariant vector bundles E !
X has Chern classes ci(E) 2 AiG(X).
The following fact will be used often.
Lemma 4.1. Let E ! X be an equivariant vector bundle of constant rank
r, s: X ! E the 0-section, E0 E the complement of the 0-section. Then
the sequence
cr(E) * *
A*G(X) ---! AG (X) -! AG (E0) -! 0,
ON THE CLASSIFYING SPACE OF PGLp 9
where the second arrow is the pullback along E0 ! Speck, is exact.
Furthermore, when k = C we also have a long exact sequence
. . ._________//_Hi-1G(E0)
ggggg
g@gggggggggg
ggggg
ssggggggg
H i-2rG(X)_c_______//HiG(X)_________//_HiG(E0)g
r(E) ggggggg
@gggggggg
ggggg
ssggggggg
H i-2r+1G(X)__________//. . .
Proof.This follows from the following facts:
(1) the pullbacks A *G(X) ! A *G(E) and H *G(X) ! H *G(E) are isomor-
phisms,
(2) the self-intersection formula, that says that the homomorphisms
s*s*: A *G(X) ! A *G(X) and s*s*: H *G(X) ! H *G(X) are multi-
plication by cr(E), and
(3) the localization sequences for Chow rings and cohomology.
Let us recall the following results from [13].
(1) If T = Gnmis a torus, and we denote by xi2 A1Tthe first Chern class
of the ith projection T ! Gm , considered as a representation, then
A*T= Z[x1, . .,.xn].
(2) If TGL n is the standard maximal torus in GL n consisting of diag-
onal matrices, then the restriction homomorphism A *GLn! A *TGLn
induces an isomorphism
A*GLn' Z[x1, . .,.xn]Sn
= Z[oe1, . .,.oen]
where the oei are the elementary symmetric functions of the xi.
(3) If TSLn is the standard maximal torus in SLn consisting of diagonal
matrices, and we denote by xi the restriction to A *SLnof xi2 A*GLn,
then we have
A*TSLn= Z[x1, . .,.xn]=(oe1);
furthermore the restriction homomorphism A *SLn! A *TSLninduces
an isomorphism
Sn
A*SLn' Z[x1, . .,.xn]=(oe1)
= Z[oe1, oe2, . .,.oen]=(oe1).
(4) If t 2 A *~nis the first Chern class of the embedding ~ n ,! Gm ,
considered as a 1-dimensional representation, then we have
A *~n= Z[t](nt).
10 ANGELO VISTOLI
Furthemore, if G is any of the groups above and k = C, then the cycle
homomorphism A *G! H*Gis an isomorphism.
The following result is implicit in [13]. Let G be a finite algebraic group
that is a product of copies of ~n , for various n. This is equivalent to saying
that G is a finite diagonalizable group scheme, or that G is the Cartier dual
of a finite abelian group , considered as a group scheme over k. By Cartier
duality, we have that is the character group bGdef=Hom(G, Gm ).
Proposition 4.2. Consider the group homomorphism bG! A 1Gthat sends
each character O: G ! Gm into c1(O). The induced ring homomorphism
Sym ZGb ! A*G is an isomorphism.
A more concrete way of stating this is the following. Set
G = ~n1 x . .x.~nr.
For each i = 1, . . . , n call Oi the character obtained by composing the ith
projection G ! ~ni with the embedding ~ ni,! Gm , and set ,i = c1(Oi) 2
A1G. Then
A *G= Z[,1, . .,.,r]=(n1,1, . .,.nr,r).
Proof.When G = ~ n, this follows from Totaro's calculation cited above.
The general case follows by induction on r from the following Lemma.
Lemma 4.3. If H is an algebraic group over k, the ring homomorphism
A*H Z A*~n-! A*Hx~n
induced by the pullbacks A *H! A *Hx~n and A *~n! A *Hx~n along the two
projections H x ~n ! H and H x ~n ! ~n is an isomorphism.
Proof.This follows easily, for example, from the Chow-K"unneth formula in
[13, Section 6], because ~ phas a representation V = kn on which it acts
by multiplication, with an open subscheme U def=V \ {0} on which it acts
trivally; and the quotient U=~ pis the total space of a Gm -torsor on Pn-1,
and, as such, it is a union of open subschemes of affine spaces.
It is also not hard to prove directly.
There is also a very important transfer operation (sometimes called in-
duction). Suppose that H is an algebraic subgroup of G of finite index. The
transfer homomorphism
tsfHG:A *H-! A*G
(see [15]) is the proper pushforward from A*H ' A*G(G=H) to A*G(Spec k) =
A*G.
This is not a ring homomorphism; however, the projection formula holds,
that is, if , 2 A*G(X) and j 2 A*H(X), we have
G H
tsfHG, . resHj = , . tsfGj
(in other words, tsfHGis a homomorphism of A *G(X)-modules).
ON THE CLASSIFYING SPACE OF PGLp 11
We are going to need the analogue of Mackey's formula in this context.
Let H and K be algebraic subgroups of G, and assume that H has finite
index in G. We will also assume that the quotient G=H is reduced, and
a disjoint union of copies of Spec k (this is automatically verified when k
is algebraically closed of characteristic 0). Then it is easy to see that the
double quotient K\G=H is also the disjoint union of copies of Spec k. Fur-
thermore, we assume that every element of (K\G=H)(k) is in the image of
some element of G(k). Of course this will always happen if k is algebraically
closed; with some work, this hypothesis can be removed, but it is going to
be verified in all the cases to which we will apply the formula).
Denote by C a set of representatives in G(k) for classes in (K\G=H)(k).
For each s 2 C, set
Ks def=K \ sHs-1 G.
Obviously Ks is a subgroup of finite index of K; there is also an embedding
Ks ,! H defined by k 7! s-1ks.
Proposition 4.4 (Mackey's formula).
X
resGKtsfHG= tsfKsKresHKs:A*H-! A*K.
s2C
Proof.This is standard. We have that the equivariant cohomology rings
A*G(G=H) and A *G(G=K) are canonically isomorphic to A *Hand A *K, re-
spectively. The retriction homomorphism A *G! A *Kcorresponds to the
pullback A *G(Spec k) ! A *G(G=K), and the tranfer homomorphism corre-
sponds to the proper pushforward A *G(G=H) ! A*G(Spec k).
Since proper pushforwards and flat pullbacks commute, from the cartesian
diagram
pr2
G=K x G=H _____//_G=H
pr1|| |ss|
fflffl|ae fflffl|
G=K ________//_Speck
we get the equality
resGKtsfHG= ae*ss* = pr1*pr*2:A*H- ! A*K.
Now we need to express G=K x G=H as a disjoint union of orbits by the
diagonal action of G. There is a G-invariant morphism G x G ! G, defined
by the rule (a, b) 7! a-1b, that induces a morphism G=K xG=H ! K\G=H.
For each s 2 C, call s the inverse`image of s 2 (K\G=H)(k), so that
G=K x G=H is a disjoint union s2C s. It is easy to verify that s is
the orbit of the class [1, s] 2 (G=K x G=H)(k) of the element (1, s) 2
(G x G)(k), and that the stabilizer of [1, s] is precisely Ks. From this we get
an isomorphism a
G=K x G=H ' G=Ks
s2C
from which the statement follows easily.
12 ANGELO VISTOLI
Proposition 4.5. Assume that G is smooth. Let f :X ! Y a proper G-
equivariant morphism of G-schemes. Assume that for every G-invariant
closed subvariety W Y there exists a G-invariant closed subvariety of X
mapping birationally onto W . Then the pushforward f*: A *GX ! A*GY is
surjective.
Here by G-invariant closed subvariety of X we mean a closed subscheme V
of X that is reduced, and such that G permutes the irreducible components
of V transitively (one sometimes says that V is primitive).
This property can be expressed by saying that X is an equivariant Chow
envelope of Y (see [3, Definition 18.3]).
Proof.In the non-equivariant setting the result follows from the definition
of proper pushforward.
In our setting, let us notice first of all that if Y 0! Y is a G-equivariant
morphism and X0 def=Y 0xY X, the projection X0 ! Y 0is also a Chow
envelope (this is easy, and left to the reader). Therefore, if U is an open
subscheme of a representation of G on which G acts freely, the morphism
f x idU:X x U ! Y x U is an equivariant Chow envelope. But since G is
smooth, it is easily seen that pullback from (X x U)=G to X x U defines a
bijective correspondence between closed subvarieties of (XxU)=G and closed
invariant subvarieties of X x U; hence the (X x U)=G is a Chow envelope of
(Y x U)=G. So the proper pushforward A* (X x U)=G ! A* (Y x U)=G
is surjective, and this completes the proof.
5.On Cp x ~p
A key role in our proof is played by a finite subgroups Cp x ~p PGL p.
We denote by Cp Spthe subgroup generated by the cycle oe def=(1 2 . .p.).
We embed Sp into PGL p as usual by identifying a permutation ff 2 Sp with
the corresponding permutation matrix, obtained by applying ff to the indices
of the canonical basis e1, . . . , ep of V (so that oeei= ei+1, where addition *
*is
modulo p).
If we denote by o the generator
[!, . .,.!p-1, 1]
of ~p PGL p, we have that
ooe = !oeo inGL p;
so oe and o commute in PGL p, and they generate a subgroup
Cp x ~p PGL p.
We denote by ff and fi the characters Cp x ~p ! Gm defined as
ff(oe) = ! and ff(o) = 1
and
fi(oe) = 1 and fi(o) = !.
ON THE CLASSIFYING SPACE OF PGLp 13
The following fact will be useful later.
Lemma 5.1. If i and j are integers between 1 and p, consider the matrix
oeioj in the algebra glpof p x p matrices. Then if (i, j) 6= (p, p), the matrix
oeioj has trace 0, and its eigenvalues are precisely the p-roots of 1.
Each oeioj is a semi-invariant for the action of Cp x ~p , with character
ff-jfii. Furthermore the oeioj form a basis of glp, and those with (i, j) 6=
(p, p) form a bases of slp.
Proof.The fact that Cp x ~p acts on oeioj via the character ff-jfii is an
elementary calculation, using the relation ooe = !oeo. From this it follows
that the oeioj are linearly independent, and therefore form a basis of glp.
The statement about the trace is also easy.
Let us check that the oeioj with (i, j) 6= (p, p) have the elements of ~ p
as eigenvalues. When i = p we get a diagonal matrix with eigenvalues are
!j, . . . , !pj, which are all the elements of ~p, because p is a prime and j is
not divisible by p. Assume that i 6= p. The numbers i, 2i, . . . , pi, reduced
modulo p, coincide with 1, . . . , p. If ~ is a pth root of 1, and e1, . . . , *
*ep is
the canonical basis of kn, then the vector
Xp t
~-t!ij(2)eti
t=1
is easily seen to be an eigenvector of oeioj with eigenvalue ~ (using the fact
that ` ' ` '
t1 t2
(mod p)
2 2
when t1 t2 (mod p), which holds because p is odd, and the relations
oeei= ei+1 and oei= !iei). This concludes the proof of the Lemma.
Corollary 5.2. Any two elements in Cp x ~p different from the identity are
conjugate in PGL p.
Remark 5.3. It is interesting to observe that the Proposition, andpits_Corol-
lary, are false for p = 2; then the matrix oeo has eigenvalues -1 , which
are not square roots of 1.
We will denote by , and j the first Chern classes in A 1Cpx~pof the char-
acters ff and fi. Then we have
A*Cpx~p= Z[,, j]=(p,, pj).
We will identify Cp x ~p with Fp x Fp, by sending oe to (1, 0) and o to
(0, 1); this identifies the automorphism group of Cp x ~p with GL 2(Fp).
We are interested in the action of the normalizer NCpx~p PGL p of Cp x ~p
in PGL p on Cp x ~p and on the Chow ring A *Cpx~p.
Proposition 5.4. Consider the homomorphism
NCpx~p PGL p -! GL 2(Fp)
14 ANGELO VISTOLI
defined by the action of NCpx~p PGL p on Cp x ~p . Its kernel is Cp x ~p ,
while its image is SL 2(Fp).
Furthermore, the ring of invariants
* SL2(Fp)
A Cpx~p
is the subring of A *Cpx~pgenerated by the two homogeneous polynomials
2-p p-1 p-1 p-1 p-1
q def=jp + , (, - j )
2-p p-1 p-1 p-1 p-1
= ,p + j (, - j )
and
r def=,j(,p-1 - jp-1)
The equality of the two polynomials that appear in the definition of q
is not immediately obvious, but is easy to prove, by subtracting them and
using the identity
2-p p2-p
(,p-1 - jp-1)p = ,p - j .
Proof.First of all, let us show that the image of the homomorphism above
is contained in SL2(Fp). There is canonical symplectic form
^ 2
Cp x ~p -! ~p
defined as follows: if a and b are in Cp x ~p PGL p, lift them to matrices
__aand _bin GL ____-1_-1
p. Then the commutator aba b is a scalar multiple of the
identity matrix Ip; it is easy to see that the scalar factor, which we denote by
, is in ~p, and that it only depends on a and b, that is, it is independe*
*nt
of the liftings. The resulting function
<-, ->: (Cp x ~p ) x (Cp x ~p ) -! ~p
is the desired symplectic form.
Now, SL 2(Fp) has p(p2 - 1) elements. According to Corollary 5.2, the
action of NCpx~p PGL p is transitive on the non-zero vectors in F2p; so the
order of the image of NCpx~p PGL p in SL2(Fp) has order divisible by p2 - 1.
It is easy to check that the diagonal matrix
i
[1, !, !3, . .,. !(2)_|-z|", . .,.!, 1]
ithplace
is also in NCpx~p PGL p, acts non-trivially on Cp x ~p, and has order p. So
the order of the image of NCpx~p PGL p is divisible by p; it follows that it is
equal to all of SL2(Fp).
It is not hard to check that the centralizer of Cp x ~p equals Cp x ~p ;
and this completes the proof of the first part of the statement.
ON THE CLASSIFYING SPACE OF PGLp 15
SL2(Fp)
To study the invariant subring A *Cpx~p , we use the natural sur-
jective homomorphism
A *Cpx~p= Z[,, j]=(p,, pj) -! Fp[,, j],
which is an isomorphism in all degrees except 1; it is enough to show that
the ring of invariants Fp[,, j]SL2(Fp)is the polynomial subring Fp[q, r].
To look for invariants in Fp[,, j], we compute the symmetric functions
of the vectors in the dual vector space F2p_; these are the homogeneous
components of the polynomial
Y
(1 + i, + jj),
i,j2Fp
which are evidently invariant under GL 2(Fp).
Lemma 5.5. Y
(1 + i, + jj) = 1 - q + rp-1.
0 i,j p-1
Proof.Using the formula
Y
(a + ib) = ap - abp-1,
i2Fp
which holds for any two elements a and b of a commmutative Fp-algebra,
we obtain
Y Y
(1 + i, + jj)= (1 + i,)p - (1 + i,)jp-1
i,j2Fp i2Fp
Y
= (1 - jp-1) + i(,p - ,jp-1)
i2Fp
= (1 - jp-1)p - (1 - jp-1)(,p - ,jp-1)p-1
2-p p p-1 p-1
= 1 - (jp + (, - ,j ) )
+ ,p-1jp-1(,p-1 - jp-1)p-1
= 1 - q + rp-1.
This shows that q and rp-1 are invariant under GL 2(Fp). The polynomial
r is not invariant under GL 2(Fp), but it is invariant under SL 2(Fp). The
simplest way to verify this is to observe that r must be a semi-invariant of
GL 2(Fp) (if g 2 GL 2(Fp), then (gr)p-1 = rp-1, and this means that gr and
r differ by a constant in F*p). But the commutator subgroup of GL 2(Fp)
is well known to be SL 2(Fp); so any character GL 2(Fp) ! F*pis trivial on
SL2(Fp), and r is invariant under SL2(Fp).
We have left to check that q and r generate the ring of invariants Fp[,, j]SL*
*2(Fp).
The equalities
2-1 p-1 p-1 p2-1 p-1 p-1
,p - q, + r = j - qj + r = 0,
16 ANGELO VISTOLI
which are easily checked by homogenizing the equality of Lemma 5.5, that
is, by adding an indeterminate t and obtaining
Y 2
(t + i, + jj) = tp -1 - qtp-1 + rp-1,
0 i,j p-1
(i,j)6=(0,0)
ensure that the extension Fp[q, r] Fp[,, j] is finite. Hence it is flat, and *
*its
degree equals
dim FpFp[,, j]=(q, r)= dimFp F[,, j]=(q, ,)
+ dimFpF[,, j]=(q, j)
p-1 p-1
+ dimFpF[,, j]= q, , - j
2-p
= dimFp Fp[,, j]=(jp , ,)
2-p
+ dimFpFp[,, j]=(,p , j)
p2-p p-1 p-1
+ dimFpFp[,, j]= , , , - j
= (p2 - p) + (p2 - p) + (p2 - p)(p - 1)
= p(p2 - 1),
which is the order of SL2(Fp). So the degrees of the field extensions
Fp(q, r) Fp(,, j) and Fp(,, j)SL2(Fp) Fp(,, j)
both equal p(p2 - 1), so Fp(q, r) = Fp(,, j)SL2(Fp); and the result follows,
because Fp[q, r] is integrally closed.
For later use, let us record the following fact. The image the restriction
homomorphism A *PGLp! A *Cpx~pis contained in A *Cpx~pSL2(Fp). We are
going to need formulas for the restriction of the Chern classes ci(slp) to
A*Cpx~p.
Lemma 5.6. Let i be a positive integer. Then the restriction of ci(slp) to
A*Cpx~p is -q if i = p2 - p, is rp-1 if i = p2 - 1, and is 0 in all other cases.
Proof.The total Chern class of glpcoincides with the total Chern class of
slp, because glpis the direct some of slpand a trivial representation. From
Lemma 5.1 we see that this total Chern class, when restricted to A *Cpx~p,
equals
Xp
(1 + i, + jj);
i,j=1
and then the result follows from Lemma 5.5.
We will also need to know about the cohomology ring H *Cpx~p. For any
cyclic group Cn ' ~n, the homomorphism A *Cn! H *Cnis an isomorphism.
ON THE CLASSIFYING SPACE OF PGLp 17
This does not extend to Cp x ~p ; however, from the universal coefficients
theorem for cohomology, for each index k we have a split exact sequence
M M
0 -! H iCp Hj~p-! HkCpx~p- ! TorZ1H iCp, Hj~p -! 0;
i+j=k i+j=k+1
furthermore, since the exterior product homomorphism A*Cp A*~p! A*Cpx~p
L i j *
is an isomorphism, the image of the term i+j=k HCp H~p into H Cpx~p
is the image of the cycle homomorphism A *Cpx~p! H *Cpx~p. From this it
is easy to deduce that the cycle homomorphism induces an isomorphism of
A*Cpx~p with the even dimensional part H evenCpx~pof the cohomology.
We have isomorphisms
2 2
H 3Cpx~p' TorZ1H Cp, H~p ' Z=pZ;
chose a generator s of H 3Cpx~p(later we will make a canonical choice). We
have that s2 = 0, because p is odd, and s has odd degree.
The odd-dimensional part H oddCpx~pof the cohomology is isomorphic to
L Z i j
the direct sum i,jTor1H Cp, H~p , with a shift by 1 in degree. Both
L Z i j
HoddCpx~pand i,jTor1 H Cp, H~p have natural structures of modules over
H*Cp H*~p= H evenCpx~p, and the isomorphism above is an isomorphism of
L Z i j even
modules. But i,jTor1H Cp, H~p is easily seen to be a cyclic H Cpx~p-
module generated by s. From this we obtain the following result.
Proposition 5.7.
H*Cpx~p= Z[,, j, s]=(p,, pj, ps, s2).
We are also interested in the action of SL 2(Fp) on H *Cpx~p. I claim that
the class s is invariant: this is equivalent to the following.
Lemma 5.8. The action of SL2(Fp) on H 3Cpx~pis trivial.
This follows, for example, from the construction of Section 11, where we
construct a class fi 2 H3PGLpthat maps to a non-zero element of H 3Cpx~p. It
would be logically correct to postpone the proof to Section 11, as this fact
is not used before then; but this would not be very satisfactory, so we prove
it now directly.
Proof.Consider the exact sequence
H 2(Cp x ~p , Z=pZ) -fi!H3(Cp x ~p , Z) -p! H3(Cp x ~p , Z)
coming from the short exact sequence
0 -! Z -p! Z -! Z=pZ -! 0;
since H 3(Cp x ~p , Z) = H 3Cpx~pis Z=pZ, we see that the Bockstein homo-
morphism fi :H 2(Cp x ~p , Z=pZ) ! H 3(Cp x ~p , Z) is surjective. It is also
18 ANGELO VISTOLI
SL2(Fp)-equivariant. By K"unneth's formula, the exterior product induces
an isomorphism of the direct sum
1 1 2
H2(Cp , Z=pZ) H (Cp , Z=pZ) H (~ p, Z=pZ) H (~ p, Z=pZ)
with H 2(Cp x ~p , Z=pZ). Now, from the commutativity of the diagram
H 2(Cp , Z=pZ)_________//H3(Cp , Z) = 0
| |
| |
fflffl| fi fflffl|
H2(Cp x ~p , Z=pZ)_____//H3(Cp x ~p , Z)
where the rows are Bockstein homomorphisms and the columns are induced
by projection Cp x ~p ! Cp, we see that the Bockstein homomorphism fi
sends H 2(Cp , Z=pZ), and H 2(~ p, Z=pZ) for analogous reasons, to 0. Hence
the composite of the exterior product map
H1(Cp , Z=pZ) H1(~ p, Z=pZ) -! H2(Cp x ~p , Z=pZ)
with fi is surjective. But we have an isomorphism
H1(Cp x ~p , Z=pZ) ' H1(Cp , Z=pZ) H1(~ p, Z=pZ)
which induces an isomorphism
^ 2
H 1(Cp x ~p , Z=pZ) ' H1(Cp , Z=pZ) H1(~ p, Z=pZ).
This shows that the composite of the map
^ 2
H1(Cp x ~p , Z=pZ) -! H2(Cp x ~p , Z=pZ)
with the Bockstein homomorphism fi is surjective, hence an isomorphism,
because both groups are isomorphic to Z=pZ. It is also evidentlyVGL 2(F2)-
equivariant. The action of GL 2(Fp) on the exterior power 2 H1(Cp x ~p , Z=pZ)
is by multiplication by the inverse of the determinant; hence SL 2(Fp) acts
trivially, and this completes the proof.
From this we deduce the following fact.
SL2(Fp)
Proposition 5.9. The ring of invariants H *Cpx~p is generated by
q, r and s.
Remark 5.10. The group Cp x ~p is important in the theory of division
algebras. Suppose that K is a field containing k, and E ! SpecK is a non-
trivial PGL p principal bundle. This corresponds to a central division algebra
D over K of degree p. Recall that D is cyclic when there are elements a and
b of K*, such that D is generated by two elements x and y, satisfying the
relations xp = a, yp = b, yx = !xy. It is not hard to show that D is cyclic
if and only if E has a reduction of structure group to Cp x ~p.
One of the main open problems in the theory of division algebra is whether
all division algebras of prime degree is cyclic. Let V be a representation of
PGL p over k with a non-empty open invariant subset U on which PGL p acts
ON THE CLASSIFYING SPACE OF PGLp 19
freely. Let K be the fraction field of U=PGL p, E the pullback to Spec K of
the PGL p-torsor U ! U=G and D the corresponding division algebra; it is
well known that D cyclic if and only if every division algebra of degree p
over a field containing k is cyclic.
The obvious way to show that D is not cyclic is to show that there is an
invariant for division algebras that is 0 for cyclic algebras, but not 0 for D.
However, the result proved here implies that there is no such invariant in the
cohomology ring H *PGLp. In fact, consider a non-zero invariant , 2 H *PGLp.
Then either , has even degree, so it comes from A*PGLp, hence it restricts to
0 in V=PGL pfor some open invariant subset V U, or it has odd degree,
and then it maps to 0 in A *TPGLp, and it does not map to 0 in A *Cpx~p.
This is related with the fact that one can not find such an invariant in
'etale cohomology with Z=pZ coefficients (see [4, x22.10]).
6. On Cp n TGL p
Proposition 6.1. Assume that k = C. Then the cycle homomorphism
A*CpnTGLp! H*CpnTGLp is an isomorphism.
Proof.This the first illustration of the stratification method: we take a
geometrically meaningful representation of Cp n TGL pand we stratify it.
Denote by V def=Ap the standard representation of GL p, restricted to
Cp n TGL p. We denote by V i the Zariski open Cp n TGL p-invariant sub-
set consisting of p-uples of complex numbers such that at most i of them
are 0, and by Vi def=V i \ V i-1 the smooth locally closed subvariety of p-
uples consisting of vectors with exactly i coordinates that are 0. Obviously
V p-1 = V \ {0} and Vp = 0.
Lemma 6.2. For each 0 i p-1, the cycle homomorphism A*CpnTGLpVi !
H*CpnTGLpVi is an isomorphism.
Proof.First of all, assume that i = 0. Then the action of Cp n TGL pon V0
is transitive, and the stabilizer of (1, . .,.1) 2 V0(k) is Cp; hence we have a
commutative diagram
A*CpnTGLp(V0)_____//H*CpnTGLp(V0)
| |
| |
fflffl| fflffl|
A *Cp_____________//_H*Cp
where the rows are cycle homomorphisms and the columns are isomorphisms.
Since the bottom row is also an isomorphism, the thesis follows.
When i > 0 the argument is similar. The action of Cp n TGL p on Vi
p
expresses Vi as a disjoint union of open orbits 1, . . . , r, where r def=1_p*
*i,
and the stabilizer of a point of each j is an i-dimensional torus Tj; hence
20 ANGELO VISTOLI
we get a commutative diagram
A *CpnTGLp(Vi)____//H*CpnTGLp(Vi)
| |
| |
L r fflffl| L fflffl|
* ________// r *
h=1 ATj h=1H Tj
where the columns and the bottom row are isomorphisms.
Lemma 6.3. For each 0 i p-1, the cycle homomorphism A*CpnTGLpV i !
H*CpnTGLpV i is an isomorphism.
Proof.We proceed by induction on i. When i = 0 we have V 0 = V0, and
the thesis follows from the previous lemma. For the inductive step, we have
a commutative diagram with exact rows
A *CpnTGLp(Vi)____//A*CpnTGLp(V i)____//A*CpnTGLp(V i-1)___//_0
| | |/.-,()*+
| | |1
fflffl| /.-,()*+3 fflffl| /.-,()*+2 fflffl|
H *CpnTGLp(Vi)____//H*CpnTGLp(V i)____//H*CpnTGLp(V i-1);
by inductive hypothesis, the arrow marked with /.-,()*+1is an isomorphism, hence
the arrow marked with /.-,()*+2is surjective. However, the bottom row of the
diagram extends to a Gysin exact sequence
H*CpnTGLp(Vi) ! H*CpnTGLp(V i) ! H*CpnTGLp(V i-1) ! H*CpnTGLp(Vi) ! . . .
showing that the arrow marked with /.-,()*+3is injective. From this, and the
fact that the left hand column is an isomorphism, it follows that the middle
column is also an isomorphism, as desidered.
Let us proceed with the proof of the Theorem. For each i we have an
commutative diagram with exact rows
A *CpnTGLpcp(V_)////_A*CpnTGLp//_//_A*CpnTGLp(V \_{0})//_//_0
| | |
| | |
fflffl|c(V ) fflffl| fflffl|
H *CpnTGLp_p___////_H*CpnTGLp_////_H*CpnTGLp(V \ {0})
Now, by Lemma 6.3 the right hand column is an isomorphism, hence, arguing
as in the proof of Lemma 6.3, we conclude that the bottom row of the
diagram is a short exact sequence.
If i is odd, we have H iCpnTGLp(V \ {0}) = 0, hence the multiplication
homomorphism
cp(V ) i
H i-2pCpnTGLp---!HCpnTGLp
ON THE CLASSIFYING SPACE OF PGLp 21
is an isomorphism. From this we deduce, by induction on i, that HiCpnTGLp=
0 for all odd i.
When i is even, one proceeds similarly by induction on i, with a straight-
forward diagram chasing in the diagram above.
Let us compute the Chow ring of the classifying space of Cp n TGL p.
The Weyl group Sp acts on A *TGLp= Z[x1, . .,.xp] by permuting the xi's.
Consider the action of Cp on A*TGLp: the group permutes the monomials, and
the only monomials that are left fixed are the ones of the form oerp= xr1. .x.r*
*p,
while on the others the action of Cp is free. We will call the monomials that
are not powers of oep free monomials. Then A *TGLpsplits as a direct sum
Z[oep] M, where M is the free ZCp -module generated by the free monomials.
Hence the ring of invariants (A *TGLp)Cp is a direct sum Z[oep] MCp , and
P
MCp is a free abelian group on the generators s2Cpsm, where m is a free
monomial.
We will denote by , 2 A1Cpthe first Chern class of the character Cp ! Gm
obtained by sending the generator (1, . .,.p) of Cp to the fixed generator ! of
~p, and also its pullback to Cp n TGL pthrough the projection Cp n TGL p!
Cp.
We will also use the subgroup ~ p TGL p of matrices of the form iIp,
where i 2 ~p. The Chow ring A *~pis of the form Z[j]=(pj), where j is the
first Chern class of the 1-dimensional representation given by the embedding
~p ,! Gm . The action of Cp on ~p is trivial, so there a copy of Cp x ~p in
Cp n TGL p; the Chow ring A *Cpx~pis Z[,, j]=(p,, pj).
Here are the facts about A *CpnTGLpthat we are going to need.
Proposition 6.4.
(a)The image of the restriction homomorphism
(A *TGLp)Cp -! A*~p= Z[j]=(pj)
is the subring generated by jp, which is the imagePof oep. The kernel is
the subgroup of (A *TGLp)Cp generated by the s2Cpsm, where m is a free
monomial, and by poep.
Cp
(b)The ring homomorphism A*CpnTGLp ! A*TGLp induced by the embed-
ding TGL p,! Cp n TGL p is surjective, and admits a canonical splitting
Cp
OE: A*TGLp ! A*CpnTGLp, which is a ring homomorphism.
(c)As an algebra over A *CpnTGLp, the ring A *CpnTGLpis generated by the
element ,, while the ideal of relations is generated by the following: p, =
0, and OE(u), = 0 for all u in the kernel of the ring homomorphism
A *TGLp! A*~pinduced by the embedding ~ p,! TGL p.
(d)The ring homomorphism A *CpnTGLp! A *TGLpx A*Cpx~pinduced by the
embeddings TGL p,! Cp n TGL pand Cp x ~p ,! Cp n TGL pis injective.
22 ANGELO VISTOLI
(e)The restriction homomorphism A*CpnTGLp! (A *TGLp)Cp sends the kernel
of A *CpnTGLp! A*Cpx~p bijectively onto the kernel of A *TGLp! A*~p.
Proof.Let us prove part (a). All the xi in A *TGLpmap to j in A *~p, so oep
P
maps to jp, and all the s2Cp sm map to pjdegm = 0.
Cp
Let us prove (b). First of all let us construct the splitting OE: A*TGLp !
Cp
A*CpnTGLp as a homomorphism of abelian groups. The group A *TGLp is
P
free over the powers of oep and the s2Cp sm.
The restriction of the canonical representation V of Cp n TGL p to the
maximal torus TGL psplits are a direct sum of 1-dimensional representations
with first Chern characters x1, . . . , xp; hence the ith Chern class ci(V ) 2
AiCpnTGLp restricts to oei2 (A iTGLp). We define the splitting by the rules
(a)OE(oerp) = cp(V )r 2 A*CpnTGLp for each r > 0, and
P TGLp *
(b)OE( s2Cp sm) = tsfCpnTGLpm 2 ACpnTGLp for each free monomial m.
Notice that the transfer in the second part of the definition only depends
on the orbit of m; hence OE is well defined.
We need to check that OE is a ring homomorphism, by taking two basis
element u and v and showing that OE(uv) equals OE(u)OE(v). This is clear when
both u and v are powers of oep.P P
Consider the product oerp s2Cpsm = s2Cps(oerpm); we have
i X j i X j
OE oerp sm= OE s(oerpm)
s2Cp s2Cp
TGLp r r
= tsfCpnTGLp(oepm) (because oepm is still a free monomial)
TGLp
= cp(V )rtsfCpnTGLp(m) (by the projection formula)
i X j
= OE(oerp)OE sm .
s2Cp
Now the hardest case. Notice that if m is any monomial, not necessarily
free, we have the equality
i X j T
OE sm = tsfGLpCpnTGLpm.
s2Cp
When m is free thisiholds by definition, whereas when m = oerpwe have
X j
OE oerp= pOE(oerp)
s2Cp = p cp(V )r
TGLp CpnTGLp r
= tsfCpnTGLpresTGLp cp(V )
TGLp r
= tsfCpnTGLpoep.
ON THE CLASSIFYING SPACE OF PGLp 23
Take two free monomials m and n. We have
i X X j i X j
OE sm . sn= OE sm . tn
s2Cp s2Cp s,t2Cp
i X j
= OE sm . stn
s,t2Cp
X i X j
= OE s(m . tn)
t2Cp s2Cp
X TGL
= tsfCppnTGLp(m . tn)
t2Cp
TGLp i X j
= tsfCpnTGLp m . tn
t2Cp
TGLp i CpnTGLp TGLp j
= tsfCpnTGLp m . resTGLp tsfCpnTGLpn
TGLp TGLp
= tsfCpnTGLp(m) tsfCpnTGLp(n)
i X j i X j
= OE sm OE sn
s2Cp s2Cp
as claimed. This ends the proof of part (b).
For parts (c) and (d), notice the following fact: since the restriction of ,
TGLp
to A*TGLpis 0, from the projection formula it follows that , tsfCpnTGLp(m) =
P
0 2 A *CpnTGLp for any m 2 A *TGLp; hence we get that OE( s2Cp m), =
0 2 A *CpnTGLp, as claimed. Thus, the relations of the statement of the
Proposition hold true.
Denote by A+TGLpthe ideal of A*TGLpgenerated by homogeneous elements
+ Cp
of positive degree. Then the image of ATGLp in A*CpnTGLpvia OE maps to
CpnTGLp * *
0 under the restriction homomorphism resCp : A CpnTGLp ! A Cp. In
TGLp
fact, the image of A+TGLpis generated by elements of the form tsfCpnTGLpm,
where m 2 A *TGLpis a monomial of positive degree, and by positive pow-
ers cp(V )r of the top Chern class of V . The fact that the restriction of
TGLp
tsfCpnTGLpm is 0 follows from Mackey's formula. On the other hand, the
restriction of V to Cp is a direct sum of 1-dimensional representations with
first Chern classes 0, ,, 2,, . . . , (p - 1),, so the restriction of V has tri*
*vial
top Chern class.
Lemma 6.5. The kernel of the restriction homomorphism
Cp nTGLp * *
resCp : ACpnTGLp - ! ACp
24 ANGELO VISTOLI
consists of the sum the image of (A +TGLp)Cp in A *CpnTGLpvia OE, and of the
ideal cp(V ) A*CpnTGLp.
Proof.Consider the hyperplane Hi in the canonical representation V = Ap
defined by the vanishing of the ith coordinate. Denote by H = [pi=1Hi V
the union. If V0 = V \ H we have an exact sequence
A *CpnTGLp(H) -! A*CpnTGLp(V ) -! A*CpnTGLp(V0) -! 0.
We identify A *CpnTGLp(V ) with A *CpnTGLpvia the pullback A *CpnTGLp!
A*CpnTGLp(V ), which is an isomorphism. The action of TGL p on V0 is free
and transitive, and the stabilizer of the point (1, 1, . .,.1) is Cp Cp n TGL*
* p.
Hence we have an isomorphism of A*CpnTGLp(V0) with A*Cp, and the pullback
A*CpnTGLp(V ) ! A *CpnTGLp(V0) is identified with the restriction homomor-
phism A *CpnTGLp! A *Cp. So the kernel of this restriction is the image of
A*CpnTGLp(H).
` p
Denote by eH the disjoint union i=1Hit {0} of the Hi with the origin
{0} V . I claim that the proper pushforward A*CpnTGLp(He) ! A*CpnTGLp(H)
is surjective. This follows from Proposition 4.5: we need to check that ev-
ery Cp n TGL p-invariant closed subvariety of H is the birational image of a
Cp n TGL p-invariant subvariety of eH. Denote by W a Cp n TGL p-invariant
closed subvariety of H. If W = {0} we are done. Otherwise it is easy to
see that W will be the union of p TGL p-invariant irreducible components
W1,`. . . , Wp, such that each Wi is contained in Hi. Then the disjoint union
p ` p e
i=1Wi i=1Hi H is Cp n TGL p-invariant and maps birationally onto
W . Hence we conclude that the kernel of the restriction homomorphism is
the sum of the images of the proper pushforwards
A *CpnTGLp({0}) -! A*CpnTGLp(V )
and
pa *
A*CpnTGLp Hi - ! ACpnTGLp (V ).
i=1
After identifying A *CpnTGLp(V ) with A *CpnTGLp, the first pushforward is
just multiplication by cp(V ), so its image is the ideal cp(V ) A*CpnTGLp.
` p
Notice that the disjoint union i=1Hi is canonically isomorphic, as a
Cp n TGL p-scheme, to (Cp n TGL p) xTGLp H1; hence there is a canonical
isomorphism `
ap '
A*CpnTGLp Hi ' A*TGLp(H1).
i=1
The pushforward A*TGLp(H1) ! A*CpnTGLp(V ) is the composite of the proper
pushforward A *TGLp(H1) ! A *TGLp(V ), followed by the transfer homomor-
phism A*TGLp(V ) ! A*CpnTGLp(V ). After identifying A*TGLp(H1) and A*TGLp(V )
ON THE CLASSIFYING SPACE OF PGLp 25
with A*TGLp, A*CpnTGLp(V ) with A*CpnTGLp, we see that this implies that the
` p
image of A *CpnTGLp i=1Hi in A *CpnTGLp(V ) = A *CpnTGLpis the image of
the ideal (x1) A *TGLpunder the transfer map A *TGLp! A *CpnTGLp. So
` p
each element of the image of A*CpnTGLp i=1Hi can be written as a linear
combination with integer coefficients of transfers of monomials of positive
degree: and this completes the proof of the Lemma.
Cp
Now we show that A*CpnTGLp is generated, as an algebra over A*TGLp ,
by the single element ,. Take an element ff of A *CpnTGLpof degree d, and
write its image in A *Cp= Z[,]=(p,) in the form m,d, where m is an integer.
Then ff - m,d 2 A*CpnTGLp maps to 0 in A *Cp, so according to Lemma 6.5 it
Cp d-p
is of the form fi + oepfl, where fi is in A*TGLp and fl 2 ACp . The proof is
concluded by induction on d.
Now we prove that the relations indicated generate the ideal of relations,
and, simultaneously, part (d).
Take an element ff 2 A dCpnTGLp; using the given relations, we can write
Cp
ff in the form ff0 + ff1, + ff2,2 + . .,.where ff0 2 AdTGLp , while for each
i > 0 the element ffiis of the form dioerp, where 0 di p-1, and rp = d-i,
when p divides d - i, and 0 otherwise.
Assume that the image of ff in A *TGLpx A*Cpx~pis 0. The image of ff in
A*TGLpis ff0, hence ff0 = 0.
Lemma 6.6. The restriction of OE(oep) = cp(V ) to A *Cpx~p= Z[,, j](p,, pj)
equals jp - j,p-1.
Proof.The restriction of V to Cp x ~p decomposes as a direct sum of 1-
dimensional representations with first Chern classes j, j - ,, j - 2,, . . . , *
*j -
(p - 1),, and
p p-1
j(j - ,)(j - 2,) . .j.- (p - 1), = j - j, .
Since , and jp - j,p-1 are algebraically independent in the polynomial
ring Fp[,, j], it follows that all the ffi are all 0. This finishes the proof o*
*f (c)
and (d).
Finally, let us prove part (e).
Injectivity follows immediately from part (d). To show that the restriction
homomorphism is surjective, it is sufficient to show that if u is in the kernel
of the homomorphism A *TGLpCp ! A *~p, then OE(u) is in the kernel of
Cp P
A*CpnTGLp ! A *Cpx~p. Each element of A *TGLp of the form s2Cp sm
goes to 0 in A *Cp, whilePoep goes to jp; hence u is a linear combination of
elements of the form s2Cp m and poerp. So OE(u) is a linear combination
26 ANGELO VISTOLI
TGLp
of element of A *CpnTGLpof the form p cp(V )r and tsfCpnTGLpm; from the
following Lemma we see that all these elements to A *Cpx~pis 0.
Lemma 6.7. If u is an element of positive degree in A *TGLp, the restriction
TGLp *
of tsfCpnTGLpu to A Cpx~p is 0.
Proof.The double coset space (Cp x ~p )\(Cp n TGL p)=TGL p consists of a
single point and (Cp x ~p ) \ TGL p= ~p, so we have
CpnTGLp TGLp ~p TGLp
resCpx~p tsfCpnTGLpu = tsfCpx~pres~p u.
However, I claim that the transfer homomorphism
~p * *
tsfCpx~p: A~ p-! ACpx~p
is 0 in positive degree. In fact, the restriction homomorphism
Cpx~p * *
res~p : ACpx~p - ! A~ p
is surjective, because the embedding ~p ,! Cp x ~p is split by the projection
Cp x ~p ! ~p. It follows immediately, again from Mackey's formula, that
~ p Cpx~p
the composition tsfCpx~pres~p is multiplication by p; and all classes in
A*Cpx~p in positive degree are p-torsion.
This concludes the proof of Proposition 6.4.
Remark 6.8. When k = C, Propositions 6.1 and 6.4 give a description of
the cohomology H *CpnTGLp. This can be proved directly, by studying the
Hochschild-Serre spectral sequence
j i+j
Eij2= Hi Cp, HTGLp =) HCpnTGLp .
7.On Cp n TPGL p
In this section we study the Chow ring of the classifying space of the group
Cp n TPGL p. Here is our main result. Consider the subgroup ~ p TPGL p
defined, as in the Introduction, by the formula i 7! [i, i2, . .,.ip-1, 1]. This
defines a homomorphism of rings A *TPGLp! A*~p.
Proposition 7.1.
(a)The image of the restriction homomorphism
(A *TPGLp)Cp -! A*~p= Z[j]=(pj)
is the subring generated by jp.
(b)The ring homomorphism A *CpnTPGLp! A*TPGLpCp induced by the em-
bedding TPGL p ,! Cp n TPGL p is surjective, and admits a canonical
Cp
splitting OE: A*TPGLp ! A*CpnTPGLp, which is a ring homomorphism.
ON THE CLASSIFYING SPACE OF PGLp 27
Cp
(c)As an algebra over A *TPGLp , the ring A *CpnTPGLpis generated by
the element ,, while the ideal of relations is generated by the following:
p, = 0, and OE(u), = 0 for all u in the kernel of the ring homomorphism
A *TPGLp! A*~pinduced by the embedding ~ p,! TPGL p.
(d)The ring homomorphisms
A *CpnTPGLp-! A*TPGLpx A*Cpx~p
and
H *CpnTPGLp-! H*TPGLpx H*Cpx~p
induced by the embeddings TPGL p,! Cp n TPGL pand Cp x ~p ,! Cp n TPGL p
is injective.
(e)The restriction homomorphism A*CpnTPGLp! (A *TPGLp)Cp sends the ker-
nel of A *CpnTPGLp! A *Cpx~pbijectively onto the kernel of A *TPGLp!
A *~p.
Proof.One of the main ideas in the paper is to exploit the fact, already used
in [15] and rediscovered in [14], that there is an isomorphism of tori
: TPGL p' TSLp
defined by
(t1, . .,.tp) = [t1=tp, t2=t1, t2=t2, . .,.tp-1=tp-2, tp=tp-1].
This isomorphism is not Sp-equivariant, but it is Cp -equivariant; therefore
it induces an isomorphism
: Cp n TPGL p' Cp n TSLp.
The composite of the embedding ~ p,! TPGL p with the isomorphism
is the embedding ~p ,! TSLp defined by i 7! [i, i, . .,.i].
Now, take an open subset U of a representation of Cp n TGL p on which
Cp n TGL p acts freely. The projection U=Cp n TSLp ! U=Cp n TGL p is
a Gm -torsor, coming from the determinant det: Cp n TGL p! Gm of the
canonical representation V of TGL p. Lemma 4.1 implies that there is an
exact sequence
c1(V ) * *
A*CpnTGLp- --! A CpnTGLp -! ACpnTSLp -! 0
and a ring isomorphism A *CpnTSLp' A*CpnTGLp= c1(V ) .
Cp
Consider the splitting OE: A*TGLp ! A*CpnTGLp constructed in the pre-
TGLp
vious section. I claim that c1(V ) coincides with OE(oe1) = tsfCpnTGLp x1.
To prove this it is enough, according to Proposition 6.4 (d), to show that
these two classes coincide after restriction to A *TGLpand to A *Cpx~p. The
restrictions of both classes to A *TGLpcoincide with x1 + . .+.xp.
28 ANGELO VISTOLI
The action of Cp x ~p on V splits as a direct sum of 1-dimensional repre-
sentations with characters j +,, j +2,, . . . , j +(p-1),, j, so the restriction
of c1(V ) to A *Cpx~pis
p(p - 1)
j + , + j + 2, + . .+.j + (p - 1), + j = pj + ________, = 0.
2
TGLp *
So we need to show that the restriction of tsfCpnTGLpx1 to A Cpx~p is also
0. This is a particular case of Lemma 6.7.
There is also an exact sequence
0 -! A*TGLp-oe1!A*TGLp-! A*TSLp-! 0,
so A *TSLpis the quotient A *TGLp=(oe1).
G G
Lemma 7.2. If G is a subgroup of Sp, the projection A*TGLp ! A*TSLp
induces an isomorphism
* G * G
A TSLp =(oe1) ' ATGLp .
Proof.This is equivalent to saying that the exact sequence above stays exact
after taking G-invariants; but we have that H1 G, A*TGLp = 0, because
A*TGLpis a torsion-free permutation module under G.
Part (a) comes from the surjectivity of the restriction homomorphism
A*TGLp Cp! A*TSLpCp and Proposition 6.4 (a).
Cp
We construct the splitting A*TSLp ! A*CpnTSLpby taking the splitting
Cp
A*TGLp ! A *CpnTGLpconstructed in the previous section, tensoring it
Cp Cp
with A *TGLp =(oe1) over A *TGLp , to get a ring homomorphism
* Cp *
ATGLp =(oe1) -! ACpnTGLp =(oe1)
and using the isomorphisms
* Cp * Cp
A TSLp ' ATGLp =(oe1)
and
A*CpnTGLp=(oe1) ' A*CpnTSLp
constructed above. This proves part (b). Part (c) follows from Proposi-
tion 6.4 (c).
To prove part (e) consider the diagram of restriction homomorphisms
A*CpnTGLp _____//_A*CpnTSLp___//_A*Cpx~p
| | |
| | |
fflffl| fflffl| fflffl|
A *TGLpCp _____//A*TSLpCp______//_A*~p.
ON THE CLASSIFYING SPACE OF PGLp 29
The surjectivity of the map in the statement follows from Proposition 6.4 (e)
and from the fact that the first arrow in the bottom row is surjective.
To prove injectivity take an element u of A *CpnTSLpthat maps to 0 in
Cp
A*Cpx~pand in A*TSLp . Let v be an element of A *CpnTGLpmapping to u.
Cp Cp
Since the kernel of the homomorphism A*TGLp ! A*TSLp is generated
Cp
by oe1, we can write the image of v in A *TGLp as oe1w for some w 2
Cp Cp
A*TGLp . Then the element v-OE(oe1w) maps to 0 in A*~pand in A*TGLp ;
hence, by Proposition 6.4 (d), it is 0. So v = OE(oe1)OE(w) maps to 0 in
A*TSLpCp, as claimed.
Let us prove part (d). The statement on Chow rings is an immediate
consequence of part (e).
For the cohomology, we will argue as follows. We have a long exact
sequence
. . .__________//_Hi-1CpnTSLpg
ggggg
g@gggggggggg
gggggg
i-2 ssggggggg i i
H CpnTGLp _________//HCnT ________//_HCnT
c1(V ) p GLp ggggggg p SLp
g@ggggggggg
gggggg
i-1 ssggggggg
H CpnTGLp ____________//. . .
By Proposition 6.1, the cycle homomorphism A *CpnTGLp! H *CpnTGLpis
an isomorphism. Hence, for each i we have a commutative diagram with
exact rows
Ai-1CpnTGLp___//_AiCpnTGLp___//AiCpnTSLp__________//0
| | |
| | |
fflffl| fflffl| fflffl|
H2i-2CpnTGLp__//_H2iCpnTGLp__//H2iCpnTSLp___//H2i-1CpnTGLp= 0
in which the first two columns are isomorphisms. This implies that the third
column is also an isomorphism: so the cycle homomorphism A *CpnTSLp!
HevenCpnTSLpis an isomorphism. Therefore the homomorphism H evenCpnTSLp!
HevenTSLpx HevenCpx~pis injective.
When i is odd, we have an exact sequence
c1(V ) i+2
0 = HiCpnTGLp- ! HiCpnTSLp-@! H iCpnTGLp---! HCpnTGLp ;
hence the boundary homomorphism @ :H oddCpnTSLpHevenCpnTGLpyields an iso-
morphism of HoddCpnTSLpwith the annihilator of the element c1(V ) of HevenCpnTG*
*Lp=
30 ANGELO VISTOLI
A*CpnTGLp. From the description of the ring A *CpnTSLpin (c), it is easy to
conclude that this annihilator is the ideal generated by ,.
Consider a free action of Cp n TGL pon an open subscheme U of a repre-
sentation. The diagram of embeddings
Cp x ~p _______//Cpx Gm
| |
| |
fflffl| fflffl|
Cp n TSLp _____//Cpn TGL p
induces a cartesian diagram
U=Cp x ~p _______//U=Cp n Gm
| |
| |
fflffl| fflffl|
U=Cp n TSLp _____//U=Cp n TGL p
in which the rows are principal Gm -bundles, and the columns are Gm -
equivariants. This in turn induces a commutative diagram
HoddCpnTSLp@__//HevenCpnTGLp
| |
| |
fflffl| fflffl|
HoddCpx~p_____//HevenCpxGm
in which the top row is injective, and has as its image the ideal (,)
HevenCpnTGLpas we have just seen. Furthermore, every element of (,)
HevenCpnTGLpmaps to 0 in H *TGLp, because it is torsion: hence (,) injects
into H evenCpx~p, by Proposition 6.4 (d). Since Cp x ~p is contained into Cp x
Gm , it follows that (,) also injects into H evenCpxGm. So the composite arrow
HoddCpnTSLp! H evenCpxGmin the commutative diagram above is injective. It
follows that the left hand column is injective.
This ends the proof of Proposition 7.1.
Proof of Proposition 3.1.We need to analyze the action of the normalizer
Np of Cp in Sp on the Chow ring A *CpnTPGLp. If we identify {1, . .,.p} with
the field Fp with p elements, by sending each i into its class modulo p, then
Cp can be identified with the additive group Fp itself, acting by translations.
There is also the multiplicative subgroup F*pof Sp, acting via multiplication.
This is contained in the normalizer of Cp = Fp, and, since p is a prime, it
is easy to show that the normalizer of Cp inside Sp is in fact the subgroup
generated by Fp and F*p, which is the semi-direct product F*pn Fp.
The subgroup Cp = Fp acts trivially, so in fact the action is through F*p.
The action of F*pleaves ~p invariant, and the result of the action of a 2 F*p
on i 2 ~ p is ia: hence a acts on A *~p= Z[j]=(pj) by sending j to aj,
ON THE CLASSIFYING SPACE OF PGLp 31
and the ring of invariants is the subring generated by jp-1. The image of
A*TPGLpinto A *~pis the subring generated by jp, by Proposition 7.1, and its
intersection with the ring of invariants in A *~pis the subring generated by
Sp
jp(p-1). This shows that the image of A *TPGLp into A *~pis contained in
the subring generated by jp(p-1). The opposite inclusion is ensured by the
fact that the discriminant ffi 2 A*TPGLpSp maps to -jp(p-1).
8. On Sp n TPGL p
The group Sp does not act on Cp n TPGL p, only the normalizer F*pn Fp of
Sp
Cp does. Nevertheless, we define the subring A *CpnTPGLp of A *CpnTPGLp
consisting of all the elements that are invariant under F*pnFp, and whose im-
ages in A*TPGLpare Sp-invariant. The restriction homomorphism A*SpnTPGLp!
Sp
A*CpnTPGLp has its image in A *CpnTPGLp .
The result we need about Sp n TPGL pis the following.
Proposition 8.1. The localized restriction homomorphism
* Sp
A*SpnTPGLp Z[1=(p - 1)!] -! ACpnTPGLp Z[1=(p - 1)!]
is an isomorphism.
Of course the statement can not be correct without inverting (p - 1)!,
because the torsion part of A *CpnTPGLpis all p-torsion, while A *SpnTPGLp
contains a lot of (p - 1)!-torsion coming from A*Sp. This is complicated, but
fortunately we do not need to worry about it.
Proof.Injectivity is clear: because of the projection formula, the composite
SpnTPGLp CpnTPGLp * *
tsfCpnTPGLp resSpnTPGLp:A SpnTPGLp -! ACpnTPGLp
SpnTPGLp
is multiplication by tsfCpnTPGLp= (p - 1)!.
Sp
To show surjectivity, take a class u 2 A*CpnTPGLp , and set
CpnTPGLp *
v def=tsfSpnTPGLpu 2 ASpnTPGLp .
We apply Mackey's formula (Proposition 4.4). The double quotient
Cp n TPGL p\Sp n TPGL p=Cp n TPGL p = Cp\Sp=Cp
consists of p - 1 elements coming from the normalizer F*pn Fp, and (p -
1)(p-2)!-1_pelements with the property that, if we call s a representative in
Spn TPGL p, we have
s(Cp n TPGL p)s-1 \ Cp n TPGL p= TPGL p.
32 ANGELO VISTOLI
Therefore
SpnTPGLp (p - 2)! - 1 TPGLp CpnTPGLp
resCpnTPGLpv = (p - 1)u + (p - 1)___________ tsfCnT resT u;
p p PGLp PGLp
hence it is enough to show that an element in the image of the trasfer map
TPGLp * Sp * Sp
tsfCpnTPGLp: A TPGLp - ! ACpnTPGLp
is in the image of A *TPGLp, up to a multiple of (p - 1)!. But again an easy
application of Mackey's formula reveals that
CpnTPGLp TPGLp
resTPGLp tsfSpnTPGLpw = (p - 1)!w
Sp
for all w 2 A*TPGLp , and this finishes the proof.
9. Some results on A *PGLp
In this section we prove some auxilliary results, which play an important
role in the proof of the main theorems.
The following observation is in [15, Corollary 2.4].
Proposition 9.1. If , is a torsion element of A *PGLp, or H *PGLp, then p, =
0.
Proof.Suppose that , 2 AmPGLp. Take a representation V of PGL p with an
open subset U on which PGL p acts freely, such that the codimension of V \U
has codimension larger than m, so that AmPGLp= Am (B), where we have set
B def=U=PGL p. Let ss :E ! B be the Brauer-Severi scheme associated with
the PGL p-torsor U ! B: this is the projection U=H ! U=PGL p, where
H is the parabolic subgroup of PGL p consisting of classes of matrices (aij)
with ai1= 0 when i > 1. The embedding H ,! PGL p lift to an embedding
H ,! GL p, as the subgroup of matrices (aij) with ai1= 0 when i > 1, and
a11 = 1; hence the pullback A m(B) ! Am (E) factors through A mGLp, which
is torsion-free. It follows that , maps to 0 in A m(E).
Now consider the Chern class cp-1(TE=B ) 2 Ap-1(E) of the relative tan-
gent bundle. This has the property that ss* cp-1(TE=B ) = p[B] 2 A 0(B);
hence, by the projection formula we have
p, = , . ss* cp-1(TE=B )
*
= ss* ss , . cp-1(TE=B )
= 0.
The proof for cohomology is identical, except for notation.
Proposition 9.2. The restriction homomorphisms A *PGLp! A *CpnTPGLp
and H *PGLp! H*CpnTPGLp are injective.
ON THE CLASSIFYING SPACE OF PGLp 33
Proof.By a classical result of Gottlieb ([5]) the homomorphism H *PGLp!
H*SpnTPGLp is injective; while the injectivity of A *PGLp! A *SpnTPGLpis a
recent result of Totaro. This is unpublished: a sketch of proof is presented
in [15].
Theorem 9.3 (Totaro). If G is a linear algebraic group over a field k of
characteristic 0 acting on a scheme X of finite type over k, and N is the
normalizer of maximal torus, then the restriction homomorphism A*G(X) !
A*N(X) is injective.
Now, the kernels of the homomorphisms in the statement are p-torsion, by
Proposition 9.1, while the kernels of A*SpnTPGLp! A*CpnTPGLpand H*SpnTPGLp!
H*CpnTPGLp are (p - 1)!-torsion, by the projection formula, so the statement
follows.
Here is the basic result that we are going to use in order to verify that a
given relation holds in A *PGLpand H *PGLp.
Proposition 9.4. The homomorphisms
A *PGLp-! A*TPGLpx A*Cpx~p
and
H *PGLp-! H*TPGLpx H*Cpx~p
obtained from the embeddings TPGL p ,! PGL pand Cp x ~p ,! PGL pare
injective.
Proof.This follows from Propositions 9.4 and 7.1 (d).
Here is another fundamental fact, which is one of the cornerstones of
the treatment of PGL 3 in [15]. In the Lie algebra slpof matrices of trace
0 consider the Zariski open subset sl0pconsisting of matrices with distinct
eigenvalues; this is invariant by the action of PGL p. Furthermore, we will
consider the subspace Dp slp of diagonal matrices with trace equal to
zero, and D0p= Dp \ sl0p. The subspaces Dp and D0pare invariant under the
action of Sp n TPGL p PGL p.
Proposition 9.5 (see [15], Proposition 3.1). The composites of restriction
homomorphisms
A *PGLp(sl0p) -! A*SpnTPGLp(sl0p) -! A*SpnTPGLp(D0p)
and
H *PGLp(sl0p) -! H*SpnTPGLp(sl0p) -! H*SpnTPGLp(D0p)
are isomorphisms.
Proof.The Spn TPGL p-equivariant embedding D0p sl0pinduces a PGL p-equi-
variant morphism PGL pxSpnTPGLp D0p! sl0p, which sends the class of a pair
(A, X) into AXA-1. This morphism is easily seen to be an isomorphism,
and the proof follows.
34 ANGELO VISTOLI
Corollary 9.6. The restriction homomorphisms
A*PGLp! A*TPGLp and A*SpnTPGLp! A*TPGLp
have the same image.
Proof.In the commutative diagram of restriction homomorphisms
A *PGLp_____//_A*PGLp(sl0p)
| |
| |
fflffl| fflffl|
A*TPGLp_____//A*TPGLp(sl0p)
the top row is surjective. On the other hand, the action on TPGL p on sl0pis
trivial and sl0pis an open subscheme of an affine space, so the bottom row is
an isomorphism. It follows that the image of A*PGLpin A*TPGLpmaps isomor-
phically onto the image of A *PGLp(sl0p) in A *TPGLp(sl0p). A similar argument
shows that the image of A*SpnTPGLpin A*TPGLpmaps isomorphically onto the
image of A *SpnTPGLp(D0p) in A *TPGLp(D0p). By we also have a commutative
diagram
A*PGLp(sl0p)___//_A*SpnTPGLp(D0p)
| |
| |
fflffl| fflffl|
A *TPGLp(sl0p)____//_A*TPGLp(D0p)
where the top row is an isomorphism, by Proposition 9.5, and this concludes
the proof.
10. Localization
Consider the top Chern classes
2-1 p-1
cp2-1(slp) 2 ApPGLp and cp-1(Dp) 2 ASpnTPGLp .
We have the following fact.
Proposition 10.1. The restriction homomorphism A *PGLp! A *SpnTPGLp
carries cp2-1(slp) into the ideal cp-1(Dp) A *SpnTPGLp. The induced ho-
momorphism
ffi *
A *PGLp cp2-1(slp) -! ASpnTPGLp = cp-1(Dp)
becomes an isomorphism when tensored with Z[1=(p - 1)!].
Proof.The representation Dp of Sp n TPGL pis naturally embedded in slp,
so we have that
2-1
cp2-1(slp) = cp-1(Dp) cp+1(slp=Dp) 2 ApSpnTPGLp,
and this proves the first statement.
ON THE CLASSIFYING SPACE OF PGLp 35
The pullbacks
A *PGLp-! A*PGLp(slp\ {0}) and A *SpnTPGLp-! A*SpnTPGLp(Dp \ {0})
are surjective, and their kernels are the ideals generated by cp2-1(slp) and
cp-1(Dp) respectively: so it enough to show that the homomorphism
A *PGLp(slp\ {0}) -! A*SpnTPGLp(Dp \ {0})
obtained by restricting the groups, and then pulling back along the embed-
ding Dp \ {0} ,! slp\ {0} becomes an isomorphism after inverting (p - 1)!.
Now, consider the diagram
A *PGLp(slp\ {0})_______//_A*PGLp(sl*p)
| |
| |
fflffl| fflffl|
A *SpnTPGLp(Dp \ {0})___//_A*SpnTPGLp(D*p)
where all the arrows are the obvious ones. The rows are surjective, while
the right hand column is an isomorphism, by Proposition 9.5: hence it is
enough to show that the rows are injective, after inverting (p - 1)!.
The first step is to observe that the restriction homomorphism A*PGLp(slp\
{0}) ! A*SpnTPGLp(slp\{0}) is injective, by Totaro's Theorem 9.3. Next, the
restriction homomorphisms A*SpnTPGLp(slp\{0}) ! A*CpnTPGLp(slp\{0}) and
A*SpnTPGLp(D*p) ! A*CpnTPGLp(D*p) become injective after inverting (p - 1)!.
So it is enough to show that the restriction homomorphisms A*CpnTPGLp(slp\
{0}) ! A *CpnTPGLp(sl*p) and A *CpnTPGLp(Dp \ {0}) ! A *CpnTPGLp(D*p) are
injective.
Lemma 10.2. Suppose that W is a representation of Cp n TPGL p, and U
an open subset of W \ {0}. Assume that
(a)the restriction of W to Cp x ~p splits as a direct sum of 1-dimensional
representations W = L1 . . .Lr, in such a way that the characters
Cp x ~p ! Gm describing the action of Cp x ~p on the Li are all dis-
tinct, and each Li\ {0} is contained in U, and
(b)U contains a point that is fixed under TPGL p.
Then the restriction homomorphism A*CpnTPGLp(W \{0}) ! A*CpnTPGLp(U)
is an isomorphism.
Proof.First of all, let us show that A *Cpx~p(W \ {0}) ! A *Cpx~p(U) is an
isomorphism. Denote by D the complement of U in W \{0}, with its_reduced__
scheme structure. Let P be the projectivization of W , and call U and D the
(respectively open and closed) subschemes of P corresponding to U and D.
36 ANGELO VISTOLI
We have a commutative diagram
__ * __
A*Cpx~p(D )_______//_ACpx~p(P )_______//A*Cpx~p(U )___//_0
| | |
| | |
fflffl| fflffl| fflffl|
A*Cpx~p(D) ____//_A*Cpx~p(W \ {0})____//A*Cpx~p(U)____//_0
where the columns are surjective pullbacks, and the rows are exact. It follows
that is enough to show that the composite
__ * *
A*Cpx~p(D ) -! ACpx~p (P ) -! ACpx~p (W \ {0})
is 0, or, equivalently, that any element of the kernel of A *Cpx~p(P ) !
__
A*Cpx~p(U ) maps to 0 in A *Cpx~p(W \ {0}). Denote by qi 2 P the ratio-
nal point corresponding to Li.
Denote by `i 2 A1Cpx~p the first Chern class of the character Cp x ~p !
Gm describing the action of Cp x ~p on Li, and h 2 A1Cpx~pthe first Chern
class of the sheaf O(1) on P . We have presentations
A*Cpx~p(P ) = Z[,, j, h]= p,, pj, (h - `1) . .(.h - `r)
and
A *Cpx~p(W \ {0}) = Z[,, j]=(p,, pj, `1 . .`.r),
and a commutative diagram
Z[,, j, h]= p,, pj, (h - `1) . .(.h_-_`r)//_Z[,, j]=(p,, pj, `1 . .`.r)
| |
| |
fflffl| fflffl|
Fp[,, j, h]= (h - `1) . .(.h -_`r)______//Fp[,, j]=(`1 . .`.r)
in which the first row is the map that sends h to 0, and corresponds to the
pullback.
The restriction homomorphism A *Cpx~p(P ) ! A*Cpx~p(qi) = A*Cpx~p
__
sends h into `i. But U contains all_the qi, so the kernel K of the re-
striction A *Cpx~p(P ) ! A *Cpx~p(U ) is contained in the intersection of the
ideals (h - `i). In the polynomial ring Fp[,, j, h], however, the intersection
of the ideals (h - `i) is the ideal generated by the product of the h - `i,
because Fp[,, j, h] is a unique factorization domain, and the h - `i are pair-
wise non-associated primes. Hence the image of an element of K is 0 in
Fp[,, j]=(`1 . .`.r); but the homomorphism
A *Cpx~p(W \ {0}) -! Fp[,, j]=(`1 . .`.r)
is an isomorphism in positive degree, and from this the statement follows.
Now consider the restriction homomorphism
A*CpnTPGLp(W \ {0}) -! A*CpnTPGLp(U).
ON THE CLASSIFYING SPACE OF PGLp 37
Denote by fl the top Chern class of W in A *CpnTPGLp; the kernel of the
surjective pullback A*CpnTPGLp! A*CpnTPGLp(W \{0}) is the ideal generated
by fl, and we need to show that the kernel of the pullback A *CpnTPGLp!
A*CpnTPGLp(U) is also the ideal generated by fl.
Denote by R the image of A *CpnTPGLpin A *Cpx~p= Z[,, j]=(p,, pj); this
is the subring generated by , and the image of oep, that is jp - ,p-1j, by
Lemma 6.6.
Take some u in the kernel of A *CpnTPGLp! A*CpnTPGLp(U). Since TPGL p
has a fixed point in U, the pullback A*TPGLp! A*TPGLp(U) is an isomorphism;
hence u is contained in the kernel of the restriction A *CpnTPGLp! A *TPGLp.
This kernel is the ideal , A*CpnTPGLp, which is a vector space over the field
Fp, with a basis consisting of the elements ,ioejp, with i > 0 and j 0. The
homomorphism A *CpnTPGLp! A*Cpx~p sends ,ioejpinto ,i(jp - ,p-1j)j. The
two elements , and jp - ,p-1j are algebraically independent in A *Cpx~p, so
the ideal , A*CpnTPGLp map isomorphically onto the ideal ,R. Hence it is
enough to show that u maps into the ideal flR. But u maps into the ideal
fl A*Cpx~p, because by hypothesis maps into 0 in A *Cpx~p(W \ {0}), so we
will be done once we have shown that flR = R \ fl A*Cpx~p.
For this purpose, consider the diagram
*
R //_________//ACpx~p
| |
| |
fflffl| fflffl|
Fp[,, jp - ,p-1j]//__//_Fp[,, j]
where the horizontal arrows are inclusions and the vertical arrows are iso-
morphisms in positive degree. It suffices to prove that
flFp[,, jp - ,p-1j] = Fp[,, jp - ,p-1j] \ flFp[,, j];
but this follows from the fact that the extension Fp[,, jp - ,p-1j] Fp[,, j]
is faithfully flat, since it is a finite extension of regular rings.
This concludes the proof of the Lemma.
The Lemma applies to the case W = Dp and W = slp. In the first case
this is straightforward; in the second one it follows from Lemma 5.1.
11. The classes ae and fi
In this section we construct the classes ae 2 Ap+1PGLpand fi 2 H3PGLp.
Proposition 11.1. There exists a unique torsion class ae 2 A p+1PGLp, whose
image in A p+1Cpx~pequals r = ,j(,p-1 - jp-1).
Furthermore we have aep-1 = cp2-1(slp) 2 A*PGLp.
38 ANGELO VISTOLI
Proof.Uniqueness is obvious from Proposition 9.4.
Let us construct a p-torsion element __ae2 A p+1SpnTPGLpthat maps to r in
A*Cpx~p.
Consider the element -,OE(oep) = , cp(V ) 2 A p+1CpnTPGLp; by Lemma 6.6,
its restriction to A*Cpx~p is r. It is p-torsion, because , is p-torsion; hence*
* it
maps to 0 in A*TPGLp. Since the torsion part of A*Cpx~p injects into A*Cpx~p,
and the image of , cp(V ) in A *Cpx~pis invariant under F*pn Fp, it follows
that , cp(V ) is also invariant under F*pn Fp.
By Proposition 8.1, there exists a p-torsion class __ae2 A p+1SpnTPGLpwhose
image in A *Cpx~pis r. By Proposition 10.1, there exists a p-torsion element
ae 2 A p+1PGLpwhose image in A p+1SpnTPGLphas the form __ae+ cp-1(Dp)oe for a
certain class oe 2 A2SpnTPGLp.
SL2(Fp)
The image of ae in A *Cpx~p = Z[q, r] must be an integer multiple
ar of r, for reasons of degree. The image of cp-1(Dp) is -,p-1; hence by
mapping into A *Cpx~pwe get an equality
ar = r - ,p-1h 2 A*Cpx~p,
where h 2 A 2Cpx~pis the image of oe. From this equality it follows easily
that a is 1 and h is 0, and therefore ae maps to r.
To check that aep-1 = cp2-1(slp), observe that both members of the equal-
ity are 0 when restricted to TPGL p; hence, by Proposition 9.4, it is enough to
2-1
show that the restriction of cp2-1(slp) = cp2-1(glp) to A pCpx~pequals rp-1;
and this follows from Lemma 5.6.
SL2(Fp)
Corollary 11.2. The restriction homomorphism A*PGLp! A*Cpx~p
is surjective.
SL2(Fp)
Proof.The ring A*Cpx~p is generated by q and r. The class - cp2-p(slp)
restricts to q, by Lemma 5.6, while ae restricts to r.
Remark 11.3. The class ae gives a new invariant for sheaves of Azumaya
algebras of prime degree. Let X be a scheme of finite type over k, and
let A be a sheaf of Azumaya algebras of degree p. This corresponds to a
PGL p-torsor E ! X; and according to a result of Totaro (see [13] and [1]),
we can associate to the class ae 2 A p+1PGLpand the PGL p-torsor E a class
OE(A) 2 A p+1(X) (where by A *(X) we mean the bivariant ring of X, see
[3]). Since by definition A is the vector bundle associated with E and the
representation glpof PGL p, we have the relation
2-1
ae(A)p-1 = cp2-1(A) 2 Ap (X).
ON THE CLASSIFYING SPACE OF PGLp 39
Remark 11.4. The class ae depends on the choice of the primitive pth root
of 1 that we have denoted by !. If we substitute !i for !, then the new
class ae is iae.
For the class fi, one possibility it to obtain it as the Brauer class of the
canonical PGL p-principal bundle, as explained in the Introduction. Another
possibility is to define it via a transgression homomorphism, as follows.
There is a well known Hochshild-Serre spectral sequence
Eij2= HiPGLp HjGm=) Hi+jGLp
from which we get an exact sequence
H2GLp-! H2Gm -! H3PGLp- ! H3TPGLp= 0;
and H 2Gmis the infinite cyclic group generated by the first Chern class t of
the identity character Gm = Gm , while H 2GLpis the cyclic group generated
by the first Chern class of the determinant GL p= Gm , whose image in H2Gm
is pt. Hence H 3PGLpis the cyclic group of order p generated by the image of
t. We define fi 2 H3PGLpto be this image.
The odd dimensional cohomology H oddPGLpmaps to 0 in H *TPGLp; hence,
according to Proposition 9.4, maps injectively into H*Cpx~p. By the results of
Section 5, we have that H3Cpx~pis isomorphic to Z=pZ, hence the restriction
homomorphism H 3PGLp! H 3Cpx~pis an isomorphism; and the image of fi
generated H 3Cpx~p. From Proposition 5.9 we obtain the following.
SL2(Fp)
Corollary 11.5. The restriction homomorphism H*PGLp! H*Cpx~p
is surjective.
12. The splitting
In this section we prove Theorem 3.2.
Consider the embeddings
~ pO_"_______//"TPGL
`" `p
| |
| |
fflffl|" fflffl|
Cp x ~pO____//_Spn TPGL p
which induce a diagram of restriction homomorphisms
S
A *SpnTPGLp____//_A*TPGL p
p
| |
| |
fflffl| fflffl|
A*Cpx~p__________//A*~p.
40 ANGELO VISTOLI
Lemma 12.1. The induced homomorphism
* * * Sp *
ker ASpnTPGLp ! ACpx~p - ! ker ATPGLp ! A~ p
is surjective.
Proof.We will prove surjectivity in two steps; first we will show that the
map is surjective when tensored with Z[1=p], then that is surjective when
tensored with Z[1=(p - 1)!].
For the first case, notice that A*Cpx~p Z[1=p] is 0 in positive degree, while
in degree 0 there is nothing to prove; so what we are really trying to show
is that A *SpnTPGLp Z[1=p] ! A*TPGLpSp Z[1=p] is surjective.
Consider the subgroup Sp-1 Sp of the Weyl group of PGL p, consisting
of permutations of {1, . .,.p} leaving p fixed.
Lemma 12.2. The restriction homomorphism A*Sp-1nTPGLp! (A *TPGLp)Sp-1
is surjective.
Proof.There is an isomorphism TGL p-1' TPGL p, defined by
(t1, . .,.tp-1) 7! (t1, . .,.tp-1, 1)
that is Sp-1-equivariant, and therefore induces an isomorphism of the semi-
direct product Sp-1nTPGL p with the normalizer Sp-1n TGL p-1 of the max-
imal torus in GL p-1. Hence it is enough to show that A *Sp-1nTGLp-1!
(A *TGLp-1)Sp-1 is surjective; but the composite
A *GLp-1-! A*Sp-1nTGLp-1-! (A *TGLp-1)Sp-1
is an isomorphism, and this proves what we want.
Sp
Take an element u 2 A*TPGLp ; according to the Lemma above, there is
Sp-1nTPGLp
some v 2 A*Sp-1nTPGLpsuch that resTPGLp v = u. Consider the element
Sp-1nTPGLp *
w def=tsfSpnTPGLpv 2 ASpnTPGLp ;
to compute its restriction to A *TPGLpwe use Mackey's formula (Proposi-
tion 4.4). The double quotient TPGL p\Sp n TPGL p=Sp-1 n TPGL phas p el-
ement, and we may take Cp as a set of representatives. Then the formula
gives us that the restriction of w to A *TPGLpis
X Sp-1nTPGL
s resTPGLp p v = pu.
s2Cp
If we invert p, this shows that u is in the image of A*SpnTPGLp, and completes
the proof of the first step.
ON THE CLASSIFYING SPACE OF PGLp 41
For the second step, take some u 2 L. According to Proposition 7.1 (e)
there exists v in the kernel of the restriction homomorphism A *CpnTPGLp!
A*Cpx~p whose restriction to A *TPGLpis u. Consider the element
CpnTPGLp
w def=tsfSpnTPGLpv.
I claim that w is in K. In fact the restriction of w to A*TPGLpis (p-1)!v = -v,
and thefore further restricting it to Cp n TPGL p sends it to 0.
The double quotient TPGL p\Sp n TPGL p=Sp-1 n TPGL p has (p - 1)! ele-
ments, and a set of representatives is given by Sp-1. Hence according to
Mackey's formula we have that the restriction of w to A *TPGLpis
X CpnTPGL
s resTPGLp p v = (p - 1)!u
s2Sp-1
and this completes the second step in the proof of Lemma 12.1.
Similarly, there is a diagram of restriction homomorphisms
Sp
A *PGLp_____//_A*TPGLp
| |
| |
fflffl| fflffl|
A*Cpx~p________//A*~p.
Lemma 12.3. The homomorphism
* * * Sp *
kerA PGLp ! ACpx~p - ! ker ATPGLp ! A~ p
induced by restriction is an isomorphism.
Proof.Injectivity follows from Proposition 9.4.
As in the previous case, we show surjectivity first after inverting p, and
then after inverting (p - 1)!.
As before, we have A *Cpx~p Z[1=p] = Z[1=p], so we only need to check
Sp
that A *PGLp Z[1=p] ! A*TPGLp Z[1=p] is surjective. This follows from
Lemma 12.1 and from Corollary 9.6.
Now we invert (p - 1)!. Choose an element
i S j
u 2 ker A*TPGLp p! A*~p Z[1=(p - 1)!];
by Lemma 12.1, we can choose
* *
u02 ker ASpnTPGLp ! ACpx~p Z[1=(p - 1)!]
mapping to u in A *TPGLp. By Proposition 10.1, we can write
u0= v + cp-1(Dp)w,
where v is in A *PGLp Z[1=(p - 1)!] and w is in A *SpnTPGLp Z[1=(p - 1)!].
The image of cp(Dp) in A *TPGLpis 0, because TPGL p acts trivially on Dp; so
42 ANGELO VISTOLI
the image of v in A*TPGLpequals u. But there is no reason why v should map
to 0 in A *Cpx~p.
Let us denote by __vand __wthe images of v and w respectively in
A*Cpx~p Z[1=(p - 1)!] = Z[1=(p - 1)!][,, j]=(p,, pj);
the restriction of cp-1(Dp) equals -,p-1, so we have __v- ,p-1__w= 0. On the
other hand __vis contained in
* SL2(Fp)
ACpx~p Z[1=(p - 1)!] = Z[1=(p - 1)!][q, r]=(pq, pr);
since __vis contained in the ideal of Z[1=(p - 1)!][,, j]=(p,, pj) generated by
,, and the images of q and r in
Z[1=(p - 1)!][,, j]=(,, pj) = Z[1=(p - 1)!][j]=(pj)
are jp2-p and 0, we see that __vis a multiple of r; hence we can write __v=
rOE(q, r), where OE is a polynomial with coefficients in Z[1=(p - 1)!]. Set
v0= v - aeOE(- cp2-p, ae);
then v0 restricts to 0 in A *Cpx~p, and its image in A *TPGLpequals the image
of v, which is u, because ae maps to 0.
This concludes the proof of Lemma 12.3.
Sp
Set K = ker A*PGLp! A *Cpx~p and L = ker A*TPGLp ! A *~p. The
induced homomorphism K ! L is an isomorphism, according to Lemma 12.3.
Consider the subring Z L A*TPGLpSp; Proposition 7.1 (e) gives us a
copy Z K of it inside A *PGLp. To finish the proof of Theorem 3.2 we need
Sp
to extend this splitting to all of A *TPGLp . According to Proposition 3.1,
Sp
we have that A*TPGLp is generated as an algebra over Z L by the single
element ffi. We need to find a lifting for ffi; this is provided by the followi*
*ng
lemma.
2-p
Lemma 12.4. The restriction of cp2-p(slp) 2 A*PGLp to A pTPGLpequals ffi.
Proof.We use the notation in the beginning of Section 3. The representa-
tions slpand glp= slp k of PGL p have the same Chern classes. If V = kn
is the standard representation of GL p, then glp= V V _ has total Chern
class Y Y
ct(glp) = 1 + t(xi- xj) = 1 + t(xi- xj)
i,j i6=j
in A *TGLp; but A *TPGLp A*TGLp, so the thesis follows.
Set ffi1 = cp2-p(slp) 2 A *PGLp. We consider the subring (Z K)[ffi1] of
A*PGLp; to finish the proof of the theorem we have left to show that it maps
Sp
injectively into (Z L)[ffi] = A*TPGLp .
ON THE CLASSIFYING SPACE OF PGLp 43
Let us take a homogeneous element x 2 (Z K)[ffi1] that maps to 0 in
A*TPGLpSp; according to Proposition 9.4, to check that it is 0 it is enough
to prove that it restricts to 0 into A *Cpx~p. Write
x = a0 + a1ffi1 + a2ffi21+ a3ffi31+ . ...
The ai of positive degree are in K, and therefore map to 0 in A *Cpx~pby
definition; so there can be at most one term that does not map to zero, and
that has to be of the form hffid1, where h is an integer. However, the restrict*
*ion
of x to A *~p= Z[j]=(pj) is zero, and since ffi1 restricts to a nonzero multiple
of jp2-p we see that h must be divisible by p. This proves that hffid1also
restricts to 0 in A *Cpx~p, and completes the proof of the theorem.
Sp
Remark 12.5. The splitting A *TPGLp ! A*PGLp that we have con-
Cp
structed is not compatible with the splitting A *TPGLp ! A *CpnTPGLp
constructed in Section 7, in the sense that the diagram
Sp *
A *TPGLp _______//APGLp
| ||
| |
fflffl| fflffl|
A *TPGLpCp ____//_A*CpnTPGLp,
where the rows are the splittings and the columns are restrictions, does not
commute.
13. The proofs of the main Theorems
Let us prove Theorem 3.3.
First of all, let us check that ae generates A *PGLpas an algebra over
Sp
A*TPGLp . Take a homogeneous element ff 2 A*PGLp. The image of
Sp
ffi 2 A *TPGLp in A *PGLpis cp2-p(slp), by construction; and this maps to
-q in A *Cpx~p, by Lemma 5.6. So there is a polynomial OE(x, y) with integer
coefficients such that ff - OE(ffi, ae) is in the kernel of the restriction hom*
*o-
morphism A *PGLp! A*Cpx~p; but this kernel is in the image of A *TPGLpSp,
again by construction; and this shows that A *PGLpis generated by ae as an
Sp
algebra over A *TPGLp .
The relations given in the statement are satisfied. We have pae = 0 by
construction. Furthermore, by construction the splitting A *TPGLpSp !
Sp
A*PGLp sends the kernel of the homomorphism A *TPGLp ! A *~pinto the
Sp
kernel of A *PGLp! A *Cpx~p; hence, if u 2 ker( A *TPGLp ! A *PGLp) we
44 ANGELO VISTOLI
have that uae 2 A *Cpx~pgoes to 0 in A *TPGLp, because ae is torsion, and to
A*Cpx~p. Hence uae = 0, because of Proposition 9.4.
Let x be an indeterminate, I the ideal in the polynomial algebra
* Sp
A TPGLp [x]
generated by px and by the polynomials ux, where u is in the kernel of the
restriction homomorphism A *TPGLpSp ! A *~p; we need to show that the
Sp
homomorphism A *TPGLp [x]=I ! A *PGLpthat sends x to ae is an isomor-
Sp
phism. Pick a polynomial OE 2 A*TPGLp [x] such that OE(ffi) = 0 in A *PGLp.
After modifying it by an element of I, we may assume that is of the form
ff + _(ffi, ae), where ff is in the the kernel of A *TPGLpSp! A *~p, while _ is
a polynomial in two variables with coefficients in Fp. Since the images of ffi
and ae in A *Cpx~p, that are q and r, are linearly independent in FP [,, j], we
Sp
see that _ must be 0. Hence ff = 0 in A *PGLp; but since A *TPGLp injects
inside A *PGLp, we have that OE(x) = 0, as we want.
Next we prove Theorem 3.4. We start by proving Corollary 3.5, that says
that the cycle homomorphism A *PGLp! HevenPGLpis an isomorphism.
Call K and L, respectively, the kernels of the restriction homomorphisms
A*PGLp! A *Cpx~pSL2(Fp)and H evenPGLp! H evenCpx~pSL2(Fp); we have a com-
mutative diagram
* SL2(Fp)
0 _____//K_____//A*PGLp___//_ACpx~ _____//0
| p
| | |
| | |
| | fflffl|
fflffl| efflffl|ven SL2(Fp)
0 _____//L_____//HPGLp____//_HevenCpx~p _____//0
with exact rows. The right hand column is an isomorphism, because of
Propositions 5.4 and 5.9. The group L injects into
i S j i S j
ker H *TPGLp p! H*~p = ker A*TPGLp p! A*~p ,
because of Proposition 9.4; on the other hand the restriction homomorphism
* Sp *
K -! ker ATPGLp ! A~ p
is an isomorphism, because of Lemma 12.3. This proves that K ! L is an
isomorphism, and this proves Corollary 3.5.
To show that ae and fi generate H *PGLpas an algebra over A *TPGLpSp,
take a homogeneous element ff 2 A *PGLp. The element ae generates H evenPGLp,
because of Theorem 3.3 and the fact above.
If ff is a homogeneous element of odd degree in H*PGLp, its image in HoddCpx~p
can be written in the form OE(q, r)s, where OE is an integral polynomial, by
Proposition 5.9. Then ff - OE(-ffi, ae)fi maps to 0 in H oddCpx~p. On the oth*
*er
ON THE CLASSIFYING SPACE OF PGLp 45
hand HoddPGLpinjects into HoddCpx~p, by Proposition 9.4, and this completes the
proof that ae and fi generated.
To prove that the given relations generated the ideal of relations is straigh*
*t-
forward, and left to the reader.
Finally, let us prove Theorem 3.6.
Since the homomorphisms A *PGLp Q ! A*SLp Q and A *PGLp Q !
A*SLp Q are isomorphisms, the ranks of A iPGLpand H iPGLpequal the ranks
of A iSLpand H iSLp. The ranks of the H iPGLpare 0 when i is odd; while for
any m 0 the rank of A mPGLp' H 2nPGLpequal the number of monomials of
degree m in oe2, . . . , oep. Such a monomial oed22. .p.dpcan be identified with
a partition <2d2. .p.dp> of m, so this rank is the numbero of partitions of m
with numbers between 2 and p.
On the other hand it follows from Theorem 3.3 that the torsion part of
A*PGLpis a vector space over the field Fp, with a basis given by the elements
ffiiaej, where i 0 and j > 0. Similarly, from Corollary 3.5 we see that the
same elements form a basis for H evenPGLp, while H oddPGLpis an Fp-vector space
with a basis formed by the elements ffiiaejfi, where i 0 and j 0.
The theorem follows easily from these facts.
Sp
14.On the ring A *TPGLp
If T is a torus, we denote by bTthe group of characters T ! Gm . We have
a homomorphism of bTinto the additive group A*Tthat sends each character
into its first Chern class: and this induced an isomorphism of the symmetric
algebra Sym ZTb with A *T.
In this section we study the ring of invariants A*TPGLpSp. It is convenient
Sp Sp
to view A*TPGLp as a subring of A*TGLp ; this last ring is generated by
the symmetric functions oe1, . . . , oep of the first Chern characters x1, . . *
*. , xp
of the projections TGL p! Gm .
Sp
If we tensor A *TPGLp with Z[1=p], then we get a polynomial ring; and
it is easy to exhibit generators. The homomorphism of groups of characters
bTPGL p-! bTSLp
induced by the projection TSLp ! TPGL p is injective, with cokernel Z=pZ;
hence it becomes an isomorphism when tensored with Z[1=p]. Hence
* Sp * Sp
A TPGLp Z[1=p] -! ATSLp Z[1=p]
is an isomorphism.
According to Lemma 7.2, the ring A *TSLpSpis a quotient
* Sp
ATGLp =(oe1) = Z[oe1, . .,.oep]=(oe1) = Z[o2, . .,.op],
46 ANGELO VISTOLI
where we have denoted by oi the image of oei in A*TSLp. One way to produce
Sp
elements of A*TPGLp is to write down explicitly the elements correspond-
ing to the oei in the isomorphism
* Sp
ATPGLp Z[1=p] ' Z[1=p][oe2, . .,.oep]
and then clear the denominators.
The composite
A *TPGLp Z[1=p]O_"___//A*TGLp Z[1=p]_________//_A*TSLp Z[1=p]
|| ||
Z[1=p][x1, . .,.xp]___//Z[1=p][x1, . .,.xp]=(oe1)
is an isomorphism, and the inverse Z[1=p][x1, . .,.xp]=(oe1) ! A*TPGLp Z[1=p]
is obtained by sending xi to xi- 1_poe1. We need to compute the image of the
oek in A *PGLp Z[1=p] Z[1=p][oe1, . .,.oep], and this is given by the followi*
*ng
formula (the one giving the Chern classes of the tensor product of a vector
bundle and a line bundle).
Lemma 14.1. If t is an indeterminate, we have
Xk ` p - k + i'
oek(x1 + t, . .,.xp +=t) tioek-i
i=0 i ` '
p - k + 2 2
= oek + (p - k + 1)toek-1 + t oek-2
2
` ' ` '
p - 1 k-1 p k
+ . .+. t oe1 + t .
k - 1 k
in Z[x1, . .,.xp, t], for k = 0, . . . , p.
Proof.This follows by comparing terms of degree k in the equality
Xp Yp
(1 + t)ioep-i= (1 + t + xi)
i=0 i=1
Xp
= oei(x1 + t, . .,.xp + t).
i=0
Sp
If we subtitute -1_poe1 for t we obtain the images of the ok in A*TPGLp
Sp
Z[1=p]; we denote them by fl0k. In order to get elements of A *TPGLp , we
can clear the denominators in the fl0i; by a straightforward calculation we
can check that
flk= pk-1fl0k
k-2X ` p - k + i' k - 1`p - 1'
= (-1)ipk-i-1 oek-ioei1+ (-1)k-1 _____ oek1
i=0 i k k - 1
ON THE CLASSIFYING SPACE OF PGLp 47
for k = 2, . . . , p - 1, while
flp= ppfl0p
p-2X
= (-1)ipp-ioep-ioei1+ (p - 1)oep1.
i=1
Sp
From the discussion above we get that A *TPGLp Z[1=p] is a poly-
nomial ring over Z[1=p] over fl2, . . . , flp. However, the fli cannot generate
Sp
A*TPGLp integrally, because all of them are in the kernel of the homomor-
Sp p2-p Sp
phism A *TPGLp ! A*~p, while ffi 2 ATPGLp is not.
When p = 3 the situation is simple. The following result was proved by
Vezzosi.
Theorem 14.2 ([15, Lemma 3.2]).
* S3 3 2
ATPGL3 = Z[fl2, fl3, ffi]= 27ffi - 4fl2 - fl3 .
Proof.What follows is essentially the argument given in the proof of Lemma 3.2
in [15]. We have
fl2= 3oe2 - oe21
and
fl3= 27oe3 - 9oe1oe2 + 2oe31.
Let us express ffi as a rational polynomial in fl2 and fl3. This is most easily
done after projecting into
A*TSLp= Z[x1, x2, x3]=(oe1)
since we know that A*TPGLpinjects inside A*TSLp. Since ffi is the opposite of t*
*he
Q 3
classical discriminant 1 i