angewandte Mathematik (Walter de Gruyter
Berlin New York 2007
On the cohomology and the Chow ring
of the classifying space of PGL
p
By Angelo Vistoli at Pisa
Abstract. We investigate the integral cohomology ring and the Chow ring of the classifying space of the complex projective linear group PGLp, 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
2. Notations and conventions 3. The main results
4. Preliminaries on equivariant intersection theory 5. On Cp mp 6. On CpyTGLp 7. On CpyTPGLp 8. On SpyTPGLp 9. Some results on APGL p 10. Localization
11. The classes r and b 12. The splitting
13. The proofs of the main theorems 14. On the ringðATPGLpÞ
Sp
References
1. Introduction
Let G be a complex linear group. One of the main invariants associated with G is the integral cohomology ring HG of the classifying space BG. B. Totaro (see [19]) has also
in-Partially supported by the University of Bologna, funds for selected research topics, and by funds FIRB and PRIN from M.I.U.R.
troduced an algebraic version of the cohomology of the classifying space of an algebraic group G over a field k, the Chow ring AG of the classifying space of G. When k¼ C there is a cycle ring homomorphism AG ! HG. Chow rings are normally infinitely harder to study than cohomology; it is remarkable that, in contrast, AG seems to be better behaved, and easier to study, than HG. For example, when G is a finite abelian group, AG is the sym-metric algebra over Z of the dual group ^GG; while the cohomology ring contains this sym-metric algebra, but is much more complicated (for example, will contain elements of odd degree), unless G is cyclic.
This ring AG has also been computed for G¼ GLn;SLn;Spn by Totaro ([19]), for
G¼ On and G¼ SO2nþ1 by Totaro and R. Pandharipande ([19] and [16]), for G¼ SO2n
by R. Field ([4]), and for the semisimple simply connected group of type G2 by N. Yagita
([24]). Also, a lot of work has been done on the case of finite groups, by Totaro himself, by Guillot ([10] and [11]) and by Yagita ([23]). However, not much was known for the PGLn
series. Even the cohomology of B PGLnwas mysterious. Algebraic topologists tend to work
with cohomology with coe‰cients in a field, the case in which their extremely impressive toolkits work the best. When p does not divide n, the cohomology ring HðB PGLn;Z=pZÞ
is a well understood polynomial ring. Also, since PGL2¼ SO3, the ring HðB PGL2;Z=2ZÞ
is also well understood. To my knowledge, the other results on HðB PGLn;Z=pZÞ that
were known before this article was posted are the following.
(1) In [14], the authors compute HðB PGL3;Z=3ZÞ as a ring, by presentations and
relations.
(2) The ring HðB PGLn;Z=2ZÞ is known when n 1 2 ðmod 4Þ ([13] and [18]).
(3) In [20] the authors show that three conjectures on the mod p cohomology of classifying spaces of compact Lie groups, due to Adams, Kono-Yagita, and Dwyer-Miller-Wilkerson-Notbohm respectively, hold for B PGLpwhen p is a prime.
On the other hand, to my knowledge no one had studied the integral cohomology ring HPGL n.
In the algebraic case, the only known results about APGL n, apart from the case of PGL2¼ SO3, concern PGL3 and were proved by Vezzosi in [21]. Here he determines
al-most completely the structure of APGL 3 by generators and relations; the only ambiguity is about one of the generators, denoted by w and living in APGL6 3, about which he knows that it is 3-torsion, but is not able to determine whether it is 0. This w maps to 0 in the cohomo-logy ring HPGL 3; according to a conjecture of Totaro, the cycle map APGL 3 ! HPGL 3should be injective; so, if the conjecture is correct, w should be 0.
Despite this only partial success, the ideas in [21] 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 ([3]) have extended Totaro’s ideas to give a full-fledged equivariant in-tersection theory. Let V be a representation of a group G; then we have AG ¼ AGðV Þ. Sup-pose that we have a stratification V0; . . . ; Vt of V by locally closed invariant subvarieties,
such that each Vei ¼ def T
jei
Vj is open in V , each Vi is closed in Vei, and Vt¼ V nf0g. If we
can determine generators for AGðViÞ for each i, then we can use the localization sequence
AGðViÞ ! AGðVeiÞ ! AGðVei1Þ ! 0
and induction to get generators for AGðV nf0gÞ; and since AGðV nf0gÞ ¼ AG=crðV Þ
, where crðV Þ A AGr is the rth Chern class of V , we obtain that A
G is generated by lifts to
AG of the generators for AGðV nf0gÞ, plus crðV Þ.
The stratification method gives a unified approach for all the known calculations of AG for classical groups (see [15]).
Vezzosi applies the method to the adjoint representation space V ¼ sl3consisting of
matrices with trace 0. The open subscheme V0 is the subscheme of matrices with distinct
eigenvalues; its Chow ring is related with the Chow ring of the normalizer N3of a maximal
torus TPGLp in PGL3. In order to get relations, Vezzosi uses an unpublished result of
To-taro, implying that the restriction homomorphism APGL 3 ! AN3 is injective. The reason why he is not able to determine whether w is 0 or not is that he does not have a good de-scription of the 3-torsion in AN3.
In this paper we refine Vezzosi’s approach, and extend it to the case of PGLp, where p
is an odd prime.
Let TPGLp be the standard maximal torus in PGLp, consisting of classes of diagonal
matrices, Npits normalizer, Sp¼ Np=TPGLp its Weyl group. Here are our main results (see
Section 3 for details):
(1) The natural homomorphism AT
PGLp ! ðA
TPGLpÞ
Sp is surjective, and has a natural
splittingðAT
PGLpÞ
Sp ! A
TPGLp, which is a ring homomorphism.
(2) The ring APGL p is generated as an algebra over ðATPGLpÞ
Sp by a single p-torsion
element r A APGLpþ1p; we also describe the relations.
(3) The ring HPGL p is generated as an algebra over ðATPGLpÞ
Sp by two elements: the
image r A HPGL2pþ2p of the class above and the Brauer class b A HPGL3 p; we also describe the relations.
(4) Using (2) and (3) above, we describe completely the additive structures of APGL p
and HPGL p.
(5) For p¼ 3 we give a presentation of ðATPGLpÞ
S3 by generators and relations (this is
already in [21]); and this, together with (2) and (3) above, gives presentations of APGL 3 and HPGL 3, completing the work of [21].
(6) The cycle homomorphism APGL p! Heven
PGLp into the even-dimensional
cohomo-logy is an isomorphism. The ringðAT
PGLpÞ
Sp is complicated when p > 3; see the discussion in Section 14.
The class r in (2) seems interesting, and gives a new invariant for sheaves of Azumaya algebras of prime rank (Remark 11.3). In [17], Elisa Targa shows that r is not a polynomial in Chern classes of representations of PGLp.
Many of the ideas in this paper come from [21]. The main new contributions here are the contents of Sections 6 and 7 (the heart of these results are Proposition 6.1, and the proof of Lemma 6.6), which substantially improve our understanding of the cohomology and Chow ring of the classifying space of Np, and Proposition 10.1, which gives a way of
show-ing that in the stratification method no new generators come from the strata correspond-ing to non-zero matrices with multiple eigenvalues, thus avoidcorrespond-ing the painful case-by-case analysis that was necessary in [21].
Recently I received a preprint of M. Komeko and N. Yagita ([12]) who also calculate the additive structure of APGL p and HPGL p, with completely di¤erent methods.
Acknowledgments. I would like to thank Nitin Nitsure and Alejandro Adem for pointing out references [13], [18] and [20] 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.
Finally, I would like to acknowledge the very interesting discussions I have had with Nobuaki Yagita and Andrzej Weber on the subject of this paper.
2. Notations and conventions
All algebraic groups and schemes will be of finite type over a fixed field k of charac-teristic 0. Furthermore, we will fix an odd prime p, and assume that k contains a fixed pth
root of 1, denoted by o. When k¼ C, we take o ¼ e2pi=p.
The hypothesis that the characteristic be 0 is only used in the proof of 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 di¤erent from p.
Our main tool is Edidin and Graham’s equivariant intersection theory (see [3]), which works over an arbitrary field; when we discuss cohomology, instead, we will always assume that k¼ C. All finite groups will be considered as algebraic groups over k, in the usual fashion. We denote by Gm the multiplicative group of non-zero scalars over k, by mn the
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 V4
. 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 TGLp, TSLp and TPGLp 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 TPGLp in PGLpby
SpyTPGLp.
If a1; . . . ; ap are elements of k, we will denote by½a1; . . . ; ap the diagonal matrix in
GLp with entries a1; . . . ; ap, and also its class in PGLp. In general, we will often use the
same symbol for a matrix in GLp and its class in PGLp; this should not give rise to
confusion.
It is well known that the arrows AGL p ! ðA TGLpÞ Sp ; HGL p ! ðA TGLpÞ Sp and ASL p ! ðATSLpÞ Sp ; HGL p ! ðATSLpÞ Sp
induced by the embeddings TGLp ,! GLpand TSLp,! SLpare isomorphisms. If we denote
by xiAATGLp ¼ H
TGLp the first Chern class of the ith projection TGLp ! Gm, or its
restric-tion to TSLp, then A
TGLp ¼ HTGLp is the polynomial ring Z½x1; . . . ; xp, while ATSLp ¼ HTSLp
equals Z½x1; . . . ; xp=ðx1þ þ xpÞ. If we denote by s1; . . . ;sp the elementary symmetric
functions in the xi, then we conclude that
AGL p ¼ HGL p ¼ Z½s1; . . . ;sp while ASL p ¼ H SLp ¼ Z½s1; . . . ;sp=ðs1Þ ¼ Z½s2; . . . ;sp: The ring AT PGLp ¼ H TPGLp is the subring of A
TGLp generated by the di¤erences xi xj.
In particular it contains the element d¼ Q
i3j
ðxi xjÞ, 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 elements of R,
f1; . . . ; fr are polynomials in Z½x1; . . . ; xn, we write
R¼ Z½t1; . . . ; tn=
f1ðt1; . . . ; tnÞ; . . . ; frðt1; . . . ; tnÞ
to indicate that the ring R is generated by t1; . . . ; tn, and the kernel of the evaluation map
Z½x1; . . . ; xn ! R sending xi to ti is 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 mp,! TPGLp defined by z7! ½z; z
2
; . . . ;zp1;1. This induces a restriction homomorphism
ATPGLp ! Amp ¼ Z½h=ðphÞ;
where h is the first Chern class of the embedding mpL Gm.
The restriction of the discriminant d AðATp2p
PGLpÞ
Sp to m
Q i3j ðih jhÞ ¼ Q i3j ði jÞ hp2p
of Z½h=ðphÞ; this is non-zero multiple of hp2p
(in fact, it is easy to check that it equals hp2p
).
Proposition 3.1. The image of the restriction homomorphism ðAT
PGLpÞ
Sp ! Z½h=ðphÞ
is the subring generated by hp2p
.
This is proved at the end of Section 7.
Theorem 3.2. There exists a canonical ring homomorphism ðAT
PGLpÞ
Sp ! A
PGLp
whose composite with the restriction homomorphism APGL p ! ðATPGLpÞ
Sp is the identity.
This is proved in Section 12.
As a consequence, APGL p and HPGL p can be regarded asðATPGLpÞ
Sp-algebras.
Theorem 3.3. The ðATPGLpÞ Sp
-algebra APGL p is generated by an element r A A
pþ1 PGLp,
and the ideal of relations is generated by the following: (a) pr¼ 0, and
(b) ru¼ 0 for all u in the kernel of the homomorphism ðATPGLpÞ Sp
! Amp.
There is a similar description for the cohomology: besides the element r, now con-sidered as living in HPGL2pþ2p, we need a single class b in degree 3. This class is essentially the tautological Brauer class. That is, if we call C the sheaf of complex valued continuous func-tions and C the sheaf of complex valued nowhere vanishing continuous functions on the classifying space B PGLp, the tautological PGLpprincipal bundle on B PGLphas a class in
the topological Brauer group H2ðB PGLp; CÞtors(see [8]). On the other hand, the
exponen-tial sequence
0! Z !2piC ! C! 1 induces a boundary homomorphism
H2ðB PGLp; CÞ ! H3ðB PGLp;ZÞ ¼ HPGL3 p;
which is an isomorphism, since B PGLpis paracompact, hence
HiðB PGLp; CÞ ¼ 0
for all i > 0. Our class b is, up to sign, the image under this boundary homomorphism of the Brauer class of the tautological bundle.
Theorem 3.4. The ring HPGL p is the commutativeðATPGLpÞ
Sp-algebra generated by an
element b of degree 3 and the element r of degree 2pþ 2. The ideal of relations is generated by the following:
(a) b2¼ 0,
(b) pr¼ pb ¼ 0, and
(c) ru¼ bu ¼ 0 for all u in the kernel of the homomorphism ðATPGLpÞ Sp
! Amp.
Corollary 3.5. The cycle homomorphism induces an isomorphism of APGL p with HPGLevenp.
From here it is not hard to get the additive structure of APGL p and HPGL p. For each integer m, denote by rðm; pÞ the number of partitions of m into numbers between 2 and p. If we denote by pðm; pÞ the number of partitions of m with numbers at most equal to p (a more usual notation for this is pðm; pÞ, which does not look very good), then rðm; pÞ ¼ pðm; pÞ pðm 1; pÞ.
We will also denote by sðm; pÞ the number of ways of writing m as a linear com-bination ðp2 pÞi þ ðp þ 1Þ j, with i f 0 and j > 0; and by s0ðm; pÞ the number of ways
of writing m as the same linear combination, with i f 0 and j f 0. Obviously we have s0ðm; pÞ ¼ sðm; pÞ, unless m is divisible by p2 p, in which case s0ðm; pÞ ¼ sðm; pÞ þ 1.
Theorem 3.6. (a) The group APGLm p is isomorphic to Zrðm; pÞlðZ=pZÞsðm; pÞ:
(b) The group HPGLm p is isomorphic to APGLm=2p when m is even, and is isomorphic to ðZ=pZÞs0 m3ð 2 ; pÞ
when m is odd.
When p¼ 3 we are able to get a description of APGL 3 and HPGL 3 by generator and relations, completing the work of [21].
Theorem 3.7. (a) APGL 3 is the commutative Z-algebra generated by elements g2, g3, d, r, of degrees 2, 3, 6 and 4 respectively, with relations
27d ð4g3 2þ g
2
3Þ; 3r; g2r; g3r:
(b) HPGL 3 is the commutative Z-algebra generated by elements g2, g3, d, r and b of degrees 4, 6, 12, 8 and 3 respectively, with relations
27d ð4g3 2þ g 2 3Þ; 3r; 3b; b 2 ; g2r; g3r; g2b; g3b:
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 [19], [3] and [21] for the definitions and the basic properties of the Chow ring AG of the classifying space of an algebraic group G over a field k, and of the Chow group AGðX Þ 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 AGðX Þ is a commutative ring; the single exception will be in the proof of Lemma 6.6.
The connection between these two notions is that AG ¼ AGðSpec kÞ.
Recall that AGðX Þ is contravariant for equivariant morphisms of smooth varieties; that is, if f : X ! Y is a G-equivariant morphism of smooth G-schemes, there is an in-duced ring homomorphism f:AGðX Þ ! AGðY Þ.
If k¼ C, and X is a smooth algebraic variety on which G acts, there is a cycle ring homomorphism AGðX Þ ! HGðX Þ from the equivariant Chow ring to the equivariant
co-homology ring; this is compatible with pullbacks.
Furthermore, if f is proper there is a pushforward f:AGðY Þ ! A
GðX Þ; this is not a
ring homomorphism, but it satisfies the projection formula fðx fhÞ ¼ fx h
for any x A AGðX Þ and h A AGðY Þ.
When Y is a closed G-invariant subscheme of X and we denote by i : Y ,! X the embedding, then we have a localization sequence
AGðY Þ !i AGðX Þ ! AGðX nY Þ ! 0:
The analogous statement for cohomology is di¤erent: here the restriction homomorphism HGðX Þ ! HGðX nY Þ is not necessarily surjective. However, when X and Y are smooth we have a long exact sequence
! HGi1ðX nY Þ HGi2rðY Þ ! i HGiðX Þ ! H i GðX nY Þ HGi2rþ1ðY Þ ! q q
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 composing with the given homo-morphism H ! G. Then we have a restriction homomorphism
resHG :AGðX Þ ! AHðX Þ:
Here is another property that will be used often. Suppose that H is an algebraic sub-group of G. We can define a ring homomorphism AGðG=HÞ ! AH by composing the
re-striction homomorphism AGðG=HÞ ! AHðG=HÞ with the pullback
AHðG=HÞ ! AHðSpec kÞ ¼ AH
obtained by the homomorphism Spec k! G=H whose image is the image of the identity in GðkÞ. Then this ring homomorphism is an isomorphism.
More generally, suppose that H acts on a scheme X . We define the induced space GHX as usual, as the quotient ðG X Þ=H by the free right action given by the
for-mula ðg; xÞh ¼ ðgh; h1xÞ. This carries a natural left action of G defined by the formula
g0ðg; xÞ ¼ ðg0g; xÞ. There is also an embedding X F H H X ,! G HX that is
H-equivariant: and the composite of the restriction homomorphism AGðG HXÞ ! A
HðG HXÞ
with the pullback AHðG HXÞ ! A
HðX Þ is an isomorphism.
Furthermore, if V is a representation of G, then there are Chern classes ciðV Þ A AGi,
satisfying the usual properties. More generally, if X is a smooth scheme over k with an action of G, every G-equivariant vector bundle E ! X has Chern classes ciðEÞ A AGiðX Þ.
The following fact will often be used.
Lemma 4.1. Let E ! X be an equivariant vector bundle of constant rank r; s : X ! E the 0-section, E0L E the complement of the 0-section. Then the sequence
AGðX Þ !crðEÞ AGðX Þ ! AGðE0Þ ! 0;
where the second arrow is the pullback along E0! Spec k, is exact.
Furthermore, when k¼ C we also have a long exact sequence ! HGi1ðE0Þ HGi2rðX Þ ! crðEÞ HGiðX Þ ! HGiðE0Þ HGi2rþ1ðX Þ ! : q q
Proof. This follows from the following facts:
(1) the pullbacks AGðX Þ ! AGðEÞ and HGðX Þ ! HGðEÞ are isomorphisms, (2) the self-intersection formula, that says that the homomorphisms
ss:AGðX Þ ! A GðX Þ and s s :HGðX Þ ! H GðX Þ
are multiplication by crðEÞ, and
(3) the localization sequences for Chow rings and cohomology. r Let us recall the following results from [19].
(1) If T ¼ Gmn is a torus, and we denote by xi AA1T the first Chern class of the ith
projection T ! Gm, considered as a representation, then
AT ¼ Z½x1; . . . ; xn:
(2) If TGLn is the standard maximal torus in GLn consisting of diagonal matrices,
then the restriction homomorphism AGL n ! ATGLn induces an isomorphism
AGL nF Z½x1; . . . ; xnSn
¼ Z½s1; . . . ;sn
where the si are the elementary symmetric functions of the xi.
(3) If TSLn is the standard maximal torus in SLnconsisting of diagonal matrices, and
we denote by xi the restriction to ASL n of xi AA
GLn, then we have
AT
SLn ¼ Z½x1; . . . ; xn=ðs1Þ;
furthermore the restriction homomorphism ASL n ! A
TSLn induces an isomorphism ASL nFZ½x1; . . . ; xn=ðs1Þ Sn ¼ Z½s1;s2; . . . ;sn=ðs1Þ: (4) If t A Am
n is the first Chern class of the embedding mn,! Gm, considered as a
1-dimensional representation, then we have
Amn ¼ Z½tðntÞ:
Furthemore, if G is any of the groups above and k¼ C, then the cycle homomor-phism AG ! HG is an isomorphism.
The following result is implicit in [19]. Let G be a finite algebraic group that is a product of copies of mn, for various n. This is equivalent to saying that G is a finite
nalizable group scheme, or that G is the Cartier dual of a finite abelian group G, considered as a group scheme over k. By Cartier duality, we have that G is the character group
^ G
Gdef¼ HomðG; GmÞ.
Proposition 4.2. Consider the group homomorphism ^GG! AG1 that sends each
char-acter w : G! Gm into c1ðwÞ. The induced ring homomorphism SymZGG^! AG is an
isomor-phism.
A more concrete way of stating this is the following. Set G¼ mn1 mnr:
For each i¼ 1; . . . ; n call withe character obtained by composing the ithprojection G! mni
with the embedding mni ,! Gm, and set xi ¼ c1ðwiÞ A A
1 G. Then
AG ¼ Z½x1; . . . ;xr=ðn1x1; . . . ; nrxrÞ:
Proof. When G ¼ mn, this follows from Totaro’s calculation cited above. The
gen-eral case follows by induction on r from the following lemma.
Lemma 4.3. If H is a linear algebraic group over k, the ring homomorphism AH n
ZAmn ! A
Hmn
induced by the pullbacks AH ! AHm n and Am
n ! A
Hmn along the two projections
H mn! H and H mn! mnis an isomorphism.
Proof. This follows easily, for example, from the Chow-Ku¨nneth formula in [19], Section 6, because mnhas a representation V ¼ kr on which it acts by multiplication, with
an open subscheme U def¼ Vnf0g on which it acts trivally; and the quotient U=mnis the total
space of a Gm-torsor on Pr1, and, as such, it is a union of open subschemes of a‰ne
spaces.
It is also not hard to prove directly, as in [15]. r
There is also a very important transfer operation (sometimes called induction). Sup-pose that H is an algebraic subgroup of G of finite index. The transfer homomorphism
trGH :AH ! AG
(see [21]) is the proper pushforward from AH F AGðG=HÞ to AGðSpec kÞ ¼ A G.
This is not a ring homomorphism; however, the projection formula holds, that is, if x A AGðX Þ and h A AHðX Þ, we have
trGHðx resG
HhÞ ¼ x tr H G h
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 automat-ically verified when k is algebraautomat-ically closed of characteristic 0). Then it is easy to see that the double quotient KnG=H is also the disjoint union of copies of Spec k. Furthermore, we assume that every element of ðKnG=HÞðkÞ is in the image of some element of GðkÞ. 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 GðkÞ for classes in ðKnG=HÞðkÞ. For each s A C, set
Ks ¼ def
K X sHs1L G:
Obviously Ksis a subgroup of finite index of K; there is also an embedding Ks,! H defined
by k7! s1ks.
Proposition 4.4 (Mackey’s formula). resKGtrGH ¼ P s A C trKs K res H Ks :A H ! AK:
Proof. This is standard. We have that the equivariant cohomology rings AGðG=HÞ and AGðG=KÞ are canonically isomorphic to AH and AK, respectively. The restriction ho-momorphism AG ! AK corresponds to the pullback A
GðSpec kÞ ! A
GðG=KÞ, and the
transfer homomorphism corresponds to the proper pushforward AGðG=HÞ ! AGðSpec kÞ.
Since proper pushforwards and flat pullbacks commute, from the cartesian diagram G=K G=H !pr2 G=H ? ? ? ypr1 ? ? ? yp G=K !r Spec k
we get the equality
resKGtrGH ¼ rp¼ pr1pr2:A H ! A
K:
Now we need to express G=K G=H as a disjoint union of orbits by the diag-onal action of G. There is a G-invariant morphism G G ! G, defined by the rule ða; bÞ 7! a1b, that induces a morphism G=K G=H ! KnG=H. For each s A C, call W
s
the inverse image of s AðKnG=HÞðkÞ, so that G=K G=H is a disjoint union ‘
s A C
Ws. It
is easy to verify that Ws is the orbit of the class ½1; s A ðG=K G=HÞðkÞ of the element
ð1; sÞ A ðG GÞðkÞ, and that the stabilizer of ½1; s is precisely Ks. From this we get an
iso-morphism
G=K G=H F ‘
s A C
G=Ks
from which the statement follows easily. r
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 L Y there exists a G-invariant closed subvariety of X mapping birationally onto W . Then the pushfor-ward f:AGX ! 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 [5], 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 Y0! Y is a G-equivariant morphism
and X0def¼ Y0Y X , the projection X0! Y0 is also an equivariant 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 idU : X U ! Y U is an equivariant Chow
envelope. But since G is smooth, it is easily seen that pullback fromðX UÞ=G to X U defines a bijective correspondence between closed subvarieties ofðX UÞ=G and closed in-variant subvarieties of X U; hence the ðX UÞ=G is a Chow envelope of ðY UÞ=G. So the proper pushforward AðX UÞ=G! AðY UÞ=G is surjective, and this com-pletes the proof. r
5. On CpD mp
A key role in our proof is played by a finite subgroup Cp mpL PGLp.
We denote by CpL Spthe subgroup generated by the cycle s ¼ def
ð1 2 . . . pÞ. We embed Spinto PGLp as usual by identifying a permutation a A Spwith the corresponding
permu-tation matrix, obtained by applying a to the indices of the canonical basis e1; . . . ;ep of V
(so that sei¼ eiþ1, where addition is modulo p).
If we denote by t the generator
½o; . . . ; op1;1 of mpL PGLp, we have that
ts¼ ost in GLp;
so s and t commute in PGLp, and they generate a subgroup
Cp mpL PGLp:
We denote by a and b the characters Cp mp! Gm defined as
and
bðsÞ ¼ 1 and bðtÞ ¼ o: The following fact will be useful later.
Lemma 5.1. If i and j are integers between 1 and p, consider the matrix sitj in the
algebra glpof p p matrices. Then if ði; jÞ 3 ðp; pÞ, the matrix sitj has trace 0, and its
ei-genvalues are precisely the p-roots of 1.
Each sitj is a semi-invariant for the action of C
p mp, with character ajb
i.
Further-more the sitj form a basis of gl
p, and those withði; jÞ 3 ðp; pÞ form a basis of slp.
Proof. The fact that Cp mp acts on sitj via the character ajbi is an elementary
calculation, using the relation ts¼ ost. From this it follows that the sitj are linearly
in-dependent, and therefore form a basis of glp. The statement about the trace is also easy. Let us check that the sitj withði; jÞ 3 ðp; pÞ have the elements of m
pas eigenvalues.
When i¼ p we get a diagonal matrix with eigenvalues oj; . . . ;opj, which are all the
elements of mp, because p is a prime and j is not divisible by p. Assume that i 3 p. The numbers i; 2i; . . . ; pi, reduced modulo p, coincide with 1; . . . ; p. If l is a pthroot of 1, and e1; . . . ;epis the canonical basis of kn, then the vector
Pp t¼1
ltoijð Þet2 ti
is easily seen to be an eigenvector of sitj with eigenvalue l (using the fact that
t1 2 1 t2 2 ðmod pÞ
when t11t2ðmod pÞ, which holds because p is odd, and the relations sei ¼ eiþ1 and
tei¼ oiei). This concludes the proof of the lemma. r
Corollary 5.2. Any two elements in Cp mp di¤erent from the identity are conjugate
in PGLp.
Remark 5.3. It is interesting to observe that the proposition, and its corollary, are false for p¼ 2; then the matrix st has eigenvalues Gpffiffiffiffiffiffiffi1, which are not square roots of 1.
We will denote by x and h the first Chern classes in AC1pm
p of the characters a and b.
Then we have
ACpm
p ¼ Z½x; h=ðpx; phÞ:
We will identify Cp mpwith Fp Fp, by sending s toð1; 0Þ and t to ð0; 1Þ; this
iden-tifies the automorphism group of Cp mpwith GL2ðFpÞ.
We are interested in the action of the normalizer NCpmpPGLpof Cp mpin PGLpon
Cp mpand on the Chow ring ACpmp.
Proposition 5.4. Consider the homomorphism NCpmpPGLp! GL2ðFpÞ
defined by the action of NCpmpPGLp on Cp mp. Its kernel is Cp mp, while its image is
SL2ðFpÞ.
Furthermore, the ring of invariants ðACpmpÞ
SL2ðFpÞ
is the subring of ACpm
p generated by the two homogeneous polynomials
qdef¼ hp2pþ xp1ðxp1 hp1Þp1 ¼ xp2pþ hp1ðxp1 hp1Þp1 and
rdef¼ xhðxp1 hp1Þ:
The equality of the two polynomials that appear in the definition of q is not immedi-ately obvious, but is easy to prove, by subtracting them and using the identity
ðxp1 hp1Þp¼ xp2p hp2p
:
Much more is known: as I learnt from N. Yagita, the rings of invariants Fp½x1; . . . ; xnGLnðFpÞ and Fp½x1; . . . ; xnSLnðFpÞ were computed by L. E. Dickson, in [1]; the
first is known as the Dickson algebra.
Proof. First of all, let us show that the image of the homomorphism above is con-tained in SL2ðFpÞ. There is a canonical symplectic form
V2
ðCp mpÞ ! mp
defined as follows: if a and b are in Cp mpL PGLp, lift them to matrices a and b in GLp.
Then the commutator aba1b1 is a scalar multiple of the identity matrix I
p; it is easy to see
that the scalar factor, which we denote by ha; bi, is in mp, and that it only depends on a and
b, that is, it is independent of the liftings. The resulting function h; i : ðCp m
pÞ ðCp mpÞ ! mp
is the desired symplectic form.
Now, SL2ðFpÞ has pðp2 1Þ elements. According to Corollary 5.2, the action of
NCpmpPGLp is transitive on the non-zero vectors in F
2
p; so the order of the image of
NCpmpPGLpin SL2ðFpÞ has order divisible by p
2 1. It is easy to check that the diagonal
½1; o; o3; . . . ; oð Þ2i
|{z}
ithplace
; . . . ;o; 1
is also in NCpmpPGLp, acts non-trivially on Cp mp, and has order p. So the order of the
image of NCpmpPGLpis divisible by p; it follows that it is equal to all of SL2ðFpÞ.
It is not hard to check that the centralizer of Cp mp equals Cp mp; and this
com-pletes the proof of the first part of the statement. For the second part, see [1] or [22]. r
For later use, let us record the following fact. The image of the restriction homomor-phism APGL p ! ACpmp is contained inðA
CpmpÞ
SL2ðFpÞ. We are going to need formulas for
the restrictions of the Chern classes ciðslpÞ to ACpmp.
Lemma 5.5. Let i be a positive integer. Then the restriction of ciðslpÞ to ACpmp isq
if i¼ p2 p, is rp1 if i¼ p2 1, and is 0 in all other cases.
Proof. The total Chern class of glp coincides with the total Chern class of slp,
be-cause glp is the direct sum of slp and a trivial representation. From Lemma 5.1 we see
that this total Chern class, when restricted to ACpm
p, equals
Pp i; j¼1
ð1 þ ix þ jhÞ;
and then the result follows from Lemma 5.6 below. r Lemma 5.6.
Q
0ei; jep1
ð1 þ ix þ jhÞ ¼ 1 q þ rp1: Proof. Using the formula
Q
i A Fp
ða þ ibÞ ¼ ap abp1;
which holds for any two elements a and b of a commmutative Fp-algebra, we obtain
Q i; j A Fp ð1 þ ix þ jhÞ ¼ Q i A Fp ð1 þ ixÞp ð1 þ ixÞhp1 ¼ Q i A Fp ð1 hp1Þ þ iðxp xhp1Þ ¼ ð1 hp1Þp ð1 hp1Þðxp xhp1Þp1 ¼ 1 hp2pþ ðxp xhp1Þp1þ xp1hp1ðxp1 hp1Þp1 ¼ 1 q þ rp1: r
We will also need to know about the cohomology ring HCpmp. For any cyclic group
CnF mn, the homomorphism A Cn ! H
Cn is an isomorphism. This does not extend to
Cp mp; however, from the universal coe‰cients theorem for cohomology, for each index
k we have a split exact sequence
0! L iþj¼k HCipnHj mp ! H k Cpmp! L iþj¼kþ1 TorZ1ðHi Cp;H j mpÞ ! 0;
furthermore, since the exterior product homomorphism ACpnA
mp! A
Cpmp is an
isomor-phism, the image of the term L
iþj¼k
HCipnHj
mp into H
Cpmp is the image of the cycle
homo-morphism ACpm
p! H
Cpmp. From this it is easy to deduce that the cycle homomorphism
induces an isomorphism of ACpm
p with the even dimensional part H
even Cpmp of the coho-mology. We have isomorphisms HC3pm pF Tor Z 1ðH 2 Cp;H 2 mpÞ F Z=pZ; chose a generator s of HC3pm
p (later we will make a canonical choice). We have that s
2¼ 0,
because p is odd, and s has odd degree. The odd-dimensional part HCoddpm
p of the cohomology is isomorphic to the direct sum
L
i; j
TorZ1ðHCip;H
j
mpÞ, with a shift by 1 in degree. Both H
odd Cpmp and L i; j TorZ1ðHCip;H j mpÞ have
natural structures of modules over HCpnH
mp ¼ H
even
Cpmp, and the isomorphism above
is an isomorphism of modules. ButL
i; j
TorZ1ðHi Cp;H
j
mpÞ is easily seen to be a cyclic H
even Cpmp
-module generated by s. From this we obtain the following result. Proposition 5.7.
HCpm
p ¼ Z½x; h; s=ðpx; ph; ps; s
2
Þ:
We are also interested in the action of SL2ðFpÞ on HCpmp. I claim that the class s is
invariant: this is equivalent to the following.
Lemma 5.8. The action of SL2ðFpÞ on HC3pmp is trivial.
This follows, for example, from the construction of Section 11, where we construct a class b A HPGL3 p that maps to a non-zero element of HC3pm
p. It would be logically correct to
postpone the proof to Section 11, as this fact is not used before then; but this does not seem very satisfactory, so we prove it now directly.
Proof. The automorphism group of HC3pm
pF Z=pZ is abelian. On the other hand
the action of SL2ðFpÞ on HC3pmp is the restriction of an action of GL2ðFpÞ, and the
commu-tator subgroup of GL2ðFpÞ is well known to be SL2ðFpÞ. r
Proposition 5.9. The ring of invariantsðHCpmpÞ
SL2ðFpÞ
is generated by q, r and s. Remark 5.10. The group Cp mp is important in the theory of division algebras.
Suppose that K is a field containing k, and E! Spec K is a non-trivial PGLp 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 xp¼ a, yp¼ b, yx ¼ oxy. It is not hard to show that D is
cyclic if and only if E has a reduction of structure group to Cp mp.
One of the main open problems in the theory of division algebra is whether all divi-sion algebras of prime degree are cyclic. Let V be a representation of PGLp over k with a
non-empty open invariant subset U on which PGLpacts freely. Let K be the fraction field
of U=PGLp, E the pullback to Spec K of the PGLp-torsor U! U=G and D the
corre-sponding division algebra; it is well known that D is 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 HPGL p. In fact, consider a non-zero invariant x A HPGL p. Then either x has even degree, so it comes from APGL p, hence it restricts to 0 in V =PGLp for some open invariant subset V L U, or it has odd
de-gree, and then it maps to 0 in AT
PGLp, and it does not map to 0 in A
Cpmp.
This is related with the fact that one can not find such an invariant in e´tale cohomo-logy with Z=pZ coe‰cients (see [6], §22.10).
6. On CpyTGLp
Proposition 6.1. Assume that k¼ C. Then the cycle homomorphism ACpyT
GLp ! H
CpyTGLp
is an isomorphism.
Proof. This is the first illustration of the stratification method: we take a geometri-cally meaningful representation of CpyTGLp and we stratify it.
Denote by V def¼ Apthe standard representation of GLp, restricted to CpyTGLp. We
denote by Vei the Zariski open CpyTGLp-invariant subset consisting of p-tuples of
com-plex numbers such that at most i of them are 0, and by Vi ¼ def
VeinVei1 the smooth locally
closed subvariety of p-tuples consisting of vectors with exactly i coordinates that are 0. Obviously Vep1 ¼ V nf0g and Vp¼ 0.
Lemma 6.2. For each 0 e i e p 1, the cycle homomorphism ACpyT
GLpVi ! H
CpyTGLpVi
is an isomorphism.
Proof. First of all, assume that i¼ 0. Then the action of CpyTGLp on V0is
transi-tive, and the stabilizer ofð1; . . . ; 1Þ A V0ðkÞ is Cp; hence we have a commutative diagram
ACpyT GLpðV0Þ ! H CpyTGLpðV0Þ ? ? ? y ? ? ? y ACp ! HCp
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 CpyTGLp on Vi expresses Vi as a
disjoint union of open orbits W1; . . . ;Wr, where r ¼ def 1
p p i
, and the stabilizer of a point of each Wj is an i-dimensional torus Tj; hence we get a commutative diagram
ACpyT GLpðViÞ ! H CpyTGLpðViÞ ? ? ? y ? ? ? y Lr h¼1 ATj ! L r h¼1 HTj
where the columns and the bottom row are isomorphisms. r Lemma 6.3. For each 0 e i e p 1, the cycle homomorphism
ACpyT
GLpVei ! H
CpyTGLpVei
is an isomorphism.
Proof. We proceed by induction on i. When i¼ 0 we have Ve0¼ V0, and the thesis
follows from the previous lemma. For the inductive step, we have a commutative diagram with exact rows
ACpyT GLpðViÞ ! A CpyTGLpðVeiÞ ! A CpyTGLpðVei1Þ ! 0 ? ? ? y ? ? ? y ? ? ? y1 HCpyT GLpðViÞ ! 3 HCpyT GLpðVeiÞ ! 2 HCpyT GLpðVei1Þ;
by inductive hypothesis, the arrow marked with 1 is an isomorphism, hence the arrow marked with 2 is surjective. However, the bottom row of the diagram extends to a Gysin exact sequence HCpyT GLpðViÞ ! H CpyTGLpðVeiÞ ! H CpyTGLpðVei1Þ ! H CpyTGLpðViÞ !
showing that the arrow marked with 3 is 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 desired. r
Let us proceed with the proof of the theorem. For each i we have a commutative diagram with exact rows
ACpyT GLp ! cpðV Þ ACpyT GLp ! A CpyTGLpðV nf0gÞ ! 0 ? ? ? y ? ? ? y ? ? ? y HCpyT GLp ! cpðV Þ HCpyT GLp ! H CpyTGLpðV nf0gÞ:
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 HCipyT
GLpðV nf0gÞ ¼ 0, hence the multiplication homomorphism
HCi2ppyT
GLp !
cpðV Þ
HCipyT
GLp
is an isomorphism. From this we deduce, by induction on i, that HCipyT
GLp ¼ 0 for all odd i.
When i is even, one proceeds similarly by induction on i, with a straightforward dia-gram chasing in the diadia-gram above. r
Let us compute the Chow ring of the classifying space of CpyTGLp. The Weyl group
Spacts on ATGLp ¼ Z½x1; . . . ; xp by permuting the xi’s. Consider the action of Cpon ATGLp:
the group permutes the monomials, and the only monomials that are left fixed are the ones of the form spr ¼ xr
1. . . x r
p, while on the others the action of Cp is free. We will call the
monomials that are not powers of sp free monomials. Then ATGLp splits as a direct sum
Z½sp l M, where M is the free ZCp-module generated by the free monomials. Hence the
ring of invariantsðATGLpÞ Cp
is a direct sum Z½sp l MCp, and MCp is a free abelian group
on the generators P
s A Cp
sm, where m is a free monomial.
We will denote by x A AC1p the first Chern class of the character Cp! Gmobtained by
sending the generatorð1; . . . ; pÞ of Cp to the fixed generator o of mp, and also its pullback
to CpyTGLp through the projection CpyTGLp ! Cp.
We will also use the subgroup mpL TGLp of matrices of the form zIp, where z A mp.
The Chow ring Amp is of the form Z½h=ðphÞ, where h is the first Chern class of the 1-dimensional representation given by the embedding mp,! Gm. The action of Cp on mp is
trivial, so there is a copy of Cp mpin CpyTGLp; the Chow ring A
Cpmp is Z½x; h=ðpx; phÞ.
Here are the facts about ACpyT
GLp that we are going to need.
Proposition 6.4. (a) The image of the restriction homomorphism ðATGLpÞ
Cp
! Amp ¼ Z½h=ðphÞ is the subring generated by hp, which is the image of s
p. The kernel is the subgroup of
ðATGLpÞ
Cp generated by the P
s A Cp
sm, where m is a free monomial, and by psp.
(b) The ring homomorphism ACpyTGLp ! ðA
TGLpÞ
Cp
induced by the embedding TGLp ,! CpyTGLp is surjective, and admits a canonical splitting f :ðA
TGLpÞ
Cp ! A
CpyTGLp,
which is a ring homomorphism. (c) As an algebra over ACpyT
GLp, the ring A
CpyTGLp is generated by the element x, while
the ideal of relations is generated by the following: px¼ 0, and fðuÞx ¼ 0 for all u in the ker-nel of the ring homomorphism AT
GLp ! A
mp induced by the embedding mp,! TGLp.
(d) The ring homomorphism ACpyT
GLp ! A
TGLp A
Cpmp induced by the embeddings
TGLp ,! CpyTGLp and Cp mp,! CpyTGLp
is injective.
(e) The restriction homomorphism ACpyT
GLp ! ðA
TGLpÞ
Cp
sends the kernel of ACpyT
GLp ! A
Cpmp bijectively onto the kernel of A
TGLp ! Amp.
Remark 6.5. The proofs of these statements could be made somewhat shorter by making use of Totaro’s results on the Chow ring of the classifying space of a wreath prod-uct ([19], Section 9). However I find the present approach more transparent; it also has the merit of generalizing to other situations of interest.
Proof. Let us prove part (a). All the xi in ATGLp map to h in A
mp, so sp maps to h
p,
and all the P
s A Cp
sm map to phdeg m¼ 0.
Let us prove (b). First of all let us construct the splitting f :ðAT
GLpÞ
Cp ! A
CpyTGLp as
a homomorphism of abelian groups. The groupðATGLpÞ
Cp is free over the powers of s
pand
the P
s A Cp
sm.
The restriction of the canonical representation V of CpyTGLp to the maximal torus
TGLp splits a direct sum of 1-dimensional representations with first Chern characters
x1; . . . ; xp; hence the ith Chern class ciðV Þ A ACipyTGLp restricts to siAðA
i
TGLpÞ. We define
the splitting by the rules (a) fðsr
pÞ ¼ cpðV ÞrAACpyTGLp for each r > 0, and
(b) f P s A Cp sm ¼ trTGLp CpyTGLpm A A
CpyTGLp for each free monomial m.
Notice that the transfer in the second part of the definition only depends on the orbit of m; hence f is well defined.
We need to check that f is a ring homomorphism, by taking two basis elements u and v and showing that fðuvÞ equals fðuÞfðvÞ. This is clear when both u and v are powers of sp.
Consider the product sr p P s A Cp sm¼ P s A Cp sðsr pmÞ; we have
f spr P s A Cp sm ¼ f P s A Cp sðsprmÞ ¼ trTGLp CpyTGLpðs r pmÞ ðbecause s r
pm is still a free monomialÞ
¼ cpðV Þrtr TGLp
CpyTGLpðmÞ ðby the projection formulaÞ
¼ fðsr pÞf P s A Cp sm :
Now the hardest case. Notice that if m is any monomial, not necessarily free, we have the equality f P s A Cp sm ¼ trTGLp CpyTGLpm:
When m is free this holds by definition, whereas when m¼ sr
pwe have f P s A Cp spr ¼ pfðsr pÞ ¼ pcpðV Þr ¼ trTGLp CpyTGLpres CpyTGLp TGLp cpðV Þ r ¼ trTGLp CpyTGLps r p:
Take two free monomials m and n. We have f P s A Cp sm P s A Cp sn ¼ f P s; t A Cp sm tn ¼ f P s; t A Cp sm stn ¼ P t A Cp f P s A Cp sðm tnÞ ¼ P t A Cp trCTpGLpyT GLpðm tnÞ ¼ trTGLp CpyTGLp m P t A Cp tn ¼ trTGLp CpyTGLpðm res CpyTGLp TGLp tr TGLp CpyTGLpnÞ ¼ trCTpGLpyTGLpðmÞ tr TGLp CpyTGLpðnÞ ¼ f P s A Cp sm f P s A Cp sn
as claimed. This ends the proof of part (b).
For parts (c) and (d), notice the following fact: since the restriction of x to ATGLp
is 0, from the projection formula it follows that x trCTGLppyT
GLpðmÞ ¼ 0 A A
CpyTGLp for any
m A AT
GLp; hence we get that f
P
s A Cp
m
x¼ 0 A ACpyTGLp, as claimed. Thus, the relations
of the statement of the proposition hold true. Denote by AþT
GLp the ideal of A
TGLp generated by homogeneous elements of positive
degree. Then the image of ðAþTGLpÞ
Cp in A
CpyTGLp via f maps to 0 under the restriction
homomorphism resCCppyTGLp :ACpyT
GLp ! A
Cp. In fact, the image of A
þ
TGLp is generated by
elements of the form trCTGLppyT
GLpm, where m A A
TGLp is a monomial of positive degree, and
by positive powers cpðV Þr of the top Chern class of V . The fact that the restriction of
trTGLp
CpyTGLpm 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; x; 2x; . . . ;ðp 1Þx, so the restriction of V has trivial top Chern class. Lemma 6.6. The kernel of the restriction homomorphism
resCCppyTGLp :ACpyT
GLp ! A
Cp
consists of the sum of the image of ðAþTGLpÞ
Cp in A
CpyTGLp via f, and of the ideal
cpðV ÞL ACpyTGLp.
Proof. Consider the hyperplane Hi in the canonical representation V ¼ Ap defined
by the vanishing of the ithcoordinate. Denote by H ¼ Sp i¼1
HiL V the union. If V0¼ V nH
we have an exact sequence ACpyT GLpðHÞ ! A CpyTGLpðV Þ ! A CpyTGLpðV0Þ ! 0: We identify ACpyT GLpðV Þ with A
CpyTGLp via the pullback A
CpyTGLp ! A
CpyTGLpðV Þ, which
is an isomorphism. The action of TGLp on V0is free and transitive, and the stabilizer of the
point ð1; 1; . . . ; 1Þ is CpL CpyTGLp. Hence we have an isomorphism of A
CpyTGLpðV0Þ
with ACp, and the pullback ACpyT
GLpðV Þ ! A
CpyTGLpðV0Þ is identified with the
restric-tion homomorphism ACpyT
GLp ! A
Cp. So the kernel of this restriction is the image of
ACpyT
GLpðHÞ.
Denote by ~HH the disjoint union ‘
p i¼1
Hit f0g of the Hi with the origin f0g L V . I
claim that the proper pushforward ACpyT
GLpð ~HHÞ ! A
CpyTGLpðHÞ is surjective. This follows
from Proposition 4.5: we need to check that every CpyTGLp-invariant closed subvariety
of H is the birational image of a CpyTGLp-invariant subvariety of ~HH. Denote by W a
CpyTGLp-invariant closed subvariety of H. If W ¼ f0g we are done. Otherwise it is easy
to see that W will be the union of p TGLp-invariant irreducible components W1; . . . ; Wp,
such that each Wi is contained in Hi. Then the disjoint union ‘ p i¼1 WiL ‘ p i¼1 HiL ~HH is
CpyTGLp-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 ACpyT
GLpðf0gÞ ! A
and ACpyT GLp ‘p i¼1 Hi ! ACpyT GLpðV Þ: After identifying ACpyT GLpðV Þ with A
CpyTGLp, the first pushforward is just
multiplica-tion by cpðV Þ, so its image is the ideal
cpðV Þ
L ACpyT
GLp.
Notice that the disjoint union ‘
p i¼1
Hiis canonically isomorphic, as a CpyTGLp-scheme,
toðCpyTGLpÞ
TGLp H
1; hence there is a canonical isomorphism
ACpyTGLp ‘p i¼1 Hi F AT GLpðH1Þ: The pushforward AT GLpðH1Þ ! A
CpyTGLpðV Þ is the composite of the proper pushforward
AT
GLpðH1Þ ! A
TGLpðV Þ, followed by the transfer homomorphism
AT GLpðV Þ ! A CpyTGLpðV Þ: After identifying AT GLpðH1Þ and A TGLpðV Þ with A TGLp, A CpyTGLpðV Þ with A CpyTGLp, we see
that this implies that the image of ACpyT
GLp ‘p i¼1 Hi in ACpyT GLpðV Þ ¼ A CpyTGLp is the
image of the idealðx1Þ L ATGLp under the transfer map ATGLp ! ACpyTGLp. So each element
of the image of ACpyTGLp ‘p i¼1 Hi
can be written as a linear combination with integer coe‰cients of transfers of monomials of positive degree: and this completes the proof of the
lemma. r
Now we show that ACpyT
GLp is generated, as an algebra over ðA
TGLpÞ
Cp, by the
single element x. Take an element a of ACpyT
GLp of degree d, and write its image in
ACp ¼ Z½x=ðpxÞ in the form mxd, where m is an integer. Then a mxdAA
CpyTGLp maps
to 0 in ACp, so according to Lemma 6.6 it is of the form bþ spg, where b is inðA
TGLpÞ
Cp
and g A ACdpp . The proof is concluded by induction on d.
Now we prove that the relations indicated generate the ideal of relations, and, simul-taneously, part (d).
Take an element a A ACdpyT
GLp; using the given relations, we can write a in the form
a0þ a1xþ a2x2þ , where a0AðATdGLpÞ
Cp, while for each i > 0 the element a
i is of the
form dispr, where 0 e die p 1, and rp ¼ d i, when p divides d i, and 0 otherwise.
Assume that the image of a in AT
GLp A Cpmp is 0. The image of a in A TGLp is a0, hence a0¼ 0.
Lemma 6.7. The restriction of fðspÞ ¼ cpðV Þ to ACpmp ¼ Z½x; hðpx; phÞ equals
hp hxp1.
Proof. The restriction of V to Cp mpdecomposes as a direct sum of 1-dimensional
representations with first Chern classes h; h x; h 2x; . . . ; h ðp 1Þx, and hðh xÞðh 2xÞ . . .h ðp 1Þx¼ hp hxp1: r
Since x and hp hxp1 are algebraically independent in the polynomial ring F p½x; h,
it follows that all the ai are 0. This finishes the proof of (c) and (d).
Finally, let us prove part (e).
Injectivity follows immediately from part (d). To show that the restriction homo-morphism is surjective, it is su‰cient to show that if u is in the kernel of the homomor-phismðATGLpÞ
Cp
! Amp, then fðuÞ is in the kernel of ACpyTGLp ! A
Cpmp. Each element of ðAT GLpÞ Cp of the form P s A Cp
sm goes to 0 in ACp, while spgoes to hp; hence u is a linear
com-bination of elements of the form P
s A Cp
m and psr
p. So fðuÞ is a linear combination of elements
of ACpyT
GLp of the form pcpðV Þ
r
and trCTGLppyT
GLpm; from the following lemma we see that all
these elements of ACpm
p are 0.
Lemma 6.8. If u is an element of positive degree in AT
GLp, the restriction of tr
TGLp CpyTGLpu
to ACpm
p is 0.
Proof. The double coset space ðCp mpÞnðCpyTGLpÞ=TGLp consists of a single
point andðCp mpÞ X TGLp ¼ mp, so we have
resCCppmyTGLp p tr TGLp CpyTGLpu¼ tr mp Cpmpres TGLp mp u:
However, I claim that the transfer homomorphism trmp Cpmp :A mp ! A Cpmp
is 0 in positive degree. In fact, the restriction homomorphism resCpmp mp :A Cpmp ! A mp
is surjective, because the embedding mp,! Cp mpis split by the projection Cp mp! mp.
It follows immediately, again from Mackey’s formula, that the composition trmp
Cpmpres
Cpmp
mp
is multiplication by p; and all classes in ACpm
p in positive degree are p-torsion.
r
This concludes the proof of Proposition 6.4. r
Remark 6.9. When k¼ C, Propositions 6.1 and 6.4 give a description of the coho-mology HCpyT
GLp. This can be proved directly, by studying the Hochschild-Serre spectral
sequence E2ij ¼ HiðC p;H j TGLpÞ ) H iþj CpyTGLp:
7. On CpyTPGLp
In this section we study the Chow ring of the classifying space of the group CpyTPGLp. Here is our main result. Consider the subgroup mpL TPGLp defined, as in the
Introduction, by the formula z7! ½z; z2; . . . ;zp1;1. This defines a homomorphism of rings ATPGLp ! Amp.
Proposition 7.1. (a) The image of the restriction homomorphism ðATPGLpÞ
Cp
! Amp¼ Z½h=ðphÞ
is the subring generated by hp.
(b) The ring homomorphism ACpyT
PGLp ! ðA
TPGLpÞ
Cp induced by the embedding
TPGLp ,! CpyTPGLp is surjective, and admits a canonical splitting
f :ðAT
PGLpÞ
Cp ! A
CpyTPGLp;
which is a ring homomorphism. (c) As an algebra over ðATPGLpÞ
Cp
, the ring ACpyTPGLp is generated by the element x,
while the ideal of relations is generated by the following: px¼ 0, and fðuÞx ¼ 0 for all u in the kernel of the ring homomorphism AT
PGLp ! A
mp induced by the embedding mp,! TPGLp.
(d) The ring homomorphisms ACpyT PGLp ! A TPGLp A Cpmp and HCpyT PGLp ! H TPGLp HCpmp
induced by the embeddings
TPGLp ,! CpyTPGLp and Cp mp,! CpyTPGLp
are injective.
(e) The restriction homomorphism ACpyT
PGLp ! ðA
TPGLpÞ
Cp sends the kernel of
ACpyT
PGLp ! A
Cpmp bijectively onto the kernel of A
TPGLp ! A mp.
Proof. One of the main ideas in the paper is to exploit the fact, already used in [21] and rediscovered in [20], that there is an isomorphism of tori
F : TPGLpF TSLp
defined by
Fðt1; . . . ; tpÞ ¼ ½t1=tp; t2=t1; t2=t2; . . . ; tp1=tp2; tp=tp1:
This isomorphism is not Sp-equivariant, but it is Cp-equivariant; therefore it induces an
isomorphism
F : CpyTPGLpF CpyTSLp:
The composite of the embedding mp,! TPGLp with the isomorphism F is the
embed-ding mp,! TSLp defined by z7! ½z; z; . . . ; z.
Now, take an open subset U of a representation of CpyTGLp on which CpyTGLp
acts freely. The projection U =CpyTSLp ! U=CpyTGLp is a Gm-torsor, coming from the
determinant det : CpyTGLp ! Gm of the canonical representation V of TGLp. Lemma 4.1
implies that there is an exact sequence ACpyTGLp !
c1ðV Þ
ACpyTGLp ! A
CpyTSLp ! 0
and a ring isomorphism ACpyT
SLpF A CpyTGLp= c1ðV Þ . Consider the splitting f :ðAT
GLpÞ
Cp
! ACpyT
GLp constructed in the previous section. I
claim that c1ðV Þ coincides with fðs1Þ ¼ tr TGLp
CpyTGLpx1. To prove this it is enough, according
to Proposition 6.4 (d), to show that these two classes coincide after restriction to AT
GLp and
to ACpm
p. The restrictions of both classes to A
TGLp coincide with x1þ þ xp.
The action of Cp mp on V splits as a direct sum of 1-dimensional representations
with characters hþ x; h þ 2x; . . . ; h þ ðp 1Þx; h, so the restriction of c1ðV Þ to ACpmp is
hþ x þ h þ 2x þ þ h þ ðp 1Þx þ h ¼ ph þpðp 1Þ
2 x¼ 0:
So we need to show that the restriction of trCTpGLpyT
GLpx1to A
Cpmp is also 0. This is a
partic-ular case of Lemma 6.8.
There is also an exact sequence 0! ATGLp ! s1 AT GLp ! A TSLp ! 0; so AT SLp is the quotient A TGLp=ðs1Þ.
Lemma 7.2. If G is a subgroup of Sp, the projectionðATGLpÞ G ! ðATSLpÞ G induces an isomorphism ðAT SLpÞ G =ðs1Þ F ðATGLpÞ G :
Proof. This is equivalent to saying that the exact sequence above stays exact after taking G-invariants; but we have that H1ðG; AT
GLpÞ ¼ 0, because A
TGLp is a torsion-free
permutation module under G. r
Part (a) comes from the surjectivity of the restriction homomorphism ðATGLpÞ
Cp
! ðATSLpÞ Cp
We construct the splittingðATSLpÞ Cp
! ACpyTSLp by taking the splitting
ðAT
GLpÞ
Cp ! A
CpyTGLp
constructed in the previous section, tensoring it withðATGLpÞ Cp =ðs1Þ over ðATGLpÞ Cp, to get a ring homomorphism ðATGLpÞ Cp =ðs1Þ ! ACpyTGLp=ðs1Þ
and using the isomorphisms
ðATSLpÞ CpF ðATGLpÞ Cp= ðs1Þ and ACpyT GLp=ðs1Þ F A CpyTSLp
constructed above. This proves part (b). Part (c) follows from Proposition 6.4 (c). To prove part (e) consider the diagram of restriction homomorphisms
ACpyTGLp ! A CpyTSLp ! A Cpmp ? ? ? y ? ? ? y ? ? ? y ðATGLpÞ Cp ! ðATSLpÞ Cp ! Amp:
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 ACpyT
SLp that maps to 0 in A Cpmp and in ðAT SLpÞ Cp. Let v be an element of A
CpyTGLp mapping to u. Since the kernel of the
homomor-phismðAT GLpÞ Cp ! ðAT SLpÞ Cp
is generated by s1, we can write the image of v inðAT
GLpÞ
Cp
as s1w for some w AðATGLpÞ
Cp. Then the element v
fðs1wÞ maps to 0 in Amp and in ðATGLpÞ Cp; hence, by Proposition 6.4 (d), it is 0. So v ¼ fðs1ÞfðwÞ maps to 0 in ðATSLpÞ Cp, as claimed.
Let us prove part (d). The statement on Chow rings is an immediate consequence of part (e).
For the cohomology, we argue as follows. We have a long exact sequence ! HCi1pyTSLp HCi2pyT GLp !c 1ðV Þ HCipyT GLp ! H i CpyTSLp HCi1pyT GLp ! : q q
By Proposition 6.1, the cycle homomorphism ACpyTGLp ! H
CpyTGLp is an
isomor-phism. Hence, for each i we have a commutative diagram with exact rows ACi1pyT GLp ! A i CpyTGLp ! A i CpyTSLp ! 0 ? ? ? y ? ? ? y ? ? ? y HC2i2pyT GLp ! H 2i CpyTGLp ! H 2i CpyTSLp ! H 2i1 CpyTGLp ¼ 0
in which the first two columns are isomorphisms. This implies that the third column is also an isomorphism: so the cycle homomorphism ACpyT
SLp ! H
even
CpyTSLp is an isomorphism.
Therefore the homomorphism HCevenpyT
SLp ! H
even TSLp H
even
Cpmp is injective.
When i is odd, we have an exact sequence 0¼ HCipyTGLp ! H i CpyTSLp ! q HCipyT GLp ! c1ðV Þ HCiþ2pyT GLp;
hence the boundary homomorphism q : HCoddpyT
SLpH
even
CpyTGLp yields an isomorphism of
HCoddpyT
SLp with the annihilator of the element c1ðV Þ of H
even
CpyTGLp ¼ A
CpyTGLp. From the
description of the ring ACpyT
SLp in (c), it is easy to conclude that this annihilator is the ideal
generated by x.
Consider a free action of CpyTGLp on an open subscheme U of a representation.
The diagram of embeddings
Cp mp ! Cp Gm ? ? ? y ? ? ? y CpyTSLp ! CpyTGLp
induces a cartesian diagram
U=Cp mp ! U=Cpy Gm ? ? ? y ? ? ? y U =CpyTSLp ! U=CpyTGLp
in which the rows are principal Gm-bundles, and the columns are Gm-equivariants. This in
turn induces a commutative diagram HCoddpyT SLp ! q HCevenpyT GLp ? ? ? y ? ? ? y HCoddpm p ! H even CpGm
in which the top row is injective, and has as its image the idealðxÞ L HCevenpyTGLp as we have
just seen. Furthermore, every element of ðxÞ L HCevenpyT
GLp maps to 0 in H