TRANSFER AND CHERN CLASSES FOR EXTRASPECIAL
p-GROUPS
DAVID JOHN GREEN AND PHAM ANH MINH
Abstract.In the cohomology ring of an extraspecial p-group, the subring
generated by Chern classes and transfers is studied. This subring is str*
*ictly
larger than the Chern subring, but still not the whole cohomology ring, *
*even
modulo nilradical. A formula is obtained relating Chern classes to trans*
*fers.
Introduction
Methods to determine the cohomology ring of a finite group almost always pre-
suppose that the cohomology of the Sylow p-subgroups is known. Calculating the
cohomology ring of a p-group is however a delicate and difficult task. The extr*
*aspe-
cial p-groups of exponent p are in some sense the minimal difficult cases: mini*
*mal
because their proper quotients are all elementary abelian, and their automorphi*
*sm
groups are very large. For this reason, many papers have been written, investig*
*at-
ing their cohomology. Developments up till 1991 are surveyed in the paper [BC92*
* ].
In particular, M. Tezuka and N. Yagita obtained the prime ideal spectrum of the
cohomology ring.
The usual method to calculate the cohomology of a p-group is to write the
group as an extension, and solve the associated Lyndon-Hochschild-Serre spec-
tral sequence. For the extraspecial p-groups however, such spectral sequences a*
*re
intractable, and one is forced to look for other techniques. Now, standard cons*
*truc-
tions such as transfer (or corestriction) from subgroups and taking Chern class*
*es
of group representations provide us with a large number of cohomology classes. *
*So
many in fact, that for any p-group the classes provided by these two constructi*
*ons
generate a subring that has the same prime ideal spectrum as the cohomology ring
(see [GL96 ]). In this paper, we study this subring in the case of the extraspe*
*cial
p-groups, and ask whether it is the whole cohomology ring.
Actually, these constructions yield very few odd-dimensional classes, and so*
* it is
rather more realistic to ask whether we obtain the whole cohomology ring modulo
nilradical. At least for mod-p cohomology, Proposition 9.4 answers this questi*
*on
too in the negative. For integral cohomology however, the problem remains open,
and the significance of Corollary 8.2 is that this subring is the biggest studi*
*ed to
date in the cohomology ring of an extraspecial p-group.
This last result is proved using cohomology classes which we denote Or;OE. T*
*hey
are constructed in a manner foreshadowed in [Min95 ]. Take a product of Chern
classes for the group of order p2n-1. By inflation and then corestriction, obta*
*in a
cohomology class Or;OEfor the group of order p2n+1. In Theorem 5.2, an elegant
formula is obtained relating the Chern classes and the Or;OE, and in Theorem 7.2
we show that the pth power of any Chern class or any Or;OElies in the subring
____________
Date: 15 November 1996.
1991 Mathematics Subject Classification. Primay 20J06; Secondary 20D15, 55R4*
*0.
The first author was supported by the Deutsche Forschungsgemeinschaft Schwer*
*punktpro-
gramm "Algorithmische Zahlentheorie und Algebra".
The second author held a DAAD fellowship.
1
2 D. J. GREEN AND P. A. MINH
generated by top Chern classes: this is the subring Tezuka and Yagita used to
obtain the spectrum of the cohomology ring.
We are very grateful to Bruno Kahn for interesting discussions; and to Helene
Esnault and Eckhart Viehweg, who arranged for the second author to visit Essen.
1.A relation between Dickson invariants
We shall assume that the reader is familiar with the Dickson invariants: such
familiarity may be acquired by consulting Benson's book [Ben93 ], for example.
Let V be an m-dimension Fp-vector space, and let 0 r m - 1. We shall
write Dr(V) for the Dickson invariant in degree pm - pr in S(V), or just Dm;r i*
*f V
is clear from the context. Recall that, for an indeterminate X, we have the equ*
*ation
m m-1X m-r pr
(1.1) V (V; X) = Xp + (-1) Dr(V)X ;
r=0
where V (V; X) in S(V)[X] is defined by
Y
(1.2) V (V; X) = (X - OE) :
OE2V
The polynomial V (V; X) is called the Mui invariant, as it is the most important
of a family of invariants for subgroups of GL m+1 studied by Mui [Mui75 ]. Note
that V (V; X) is Fp-linear as a function of X. By convention, we define Dm;r = *
*0 if
r < 0, and Dm;m = 1.
Lemma 1.1. Let V be an m-dimensional Fp-vector space, let U be a proper sub-
space of V, and let s be the codimension of U in V. Let Hyp (V; U) denote the s*
*et
of all hyperplanes in V which contain U: this set is clearly nonempty. Then
X s-1
(1.3) V (W; X) = V (U; X)p ;
W2Hyp(V;U)
and it follows that
X s-1
(1.4) Dr-1(W) = Dr-s(U)p for all 1 r m.
W2Hyp(V;U)
Proof. Let T be a complementary subspace of U in V, so dim(T ) = s. The elements
of Hyp(V; U) are those subspaces W of V such that W \T is a hyperplane in T and
W = U (W \ T ). So choosing such a T induces a bijection between Hyp (V; U)
and the projective space PT *of lines in the dual space of T .
Let o1; : :;:os be a basis for T . Observe that V (U; o1), : :,:V (U; os), *
*all ho-
mogeneous of degree pn-s, are algebraically independent over Fp: for the image *
*of
n-s
V (U; oi) in S(V=U) ~=S(T ) is opi .
Denote by __anPs-tuple (1; : :;:s) in Fsp. Given OE 2 PT *, say that __belon*
*gs
to ker(OE) if si=1ioidoes. Using Eqn. (1.2) and the linearity of V (U; -), we*
* obtain
X X Y
(1.5) V (W; X)= V (U; X - t)
W2Hyp(V;U) OE2PTt*2ker(OE)
!
X Y Xs
(1.6) = V (U; X) - iV (U; oi) :
OE_2ker(OE) i=1
Consider the expression (1.6). It is a polynomial in V (U; X), V (U; o1), : :,:*
*V (U; os),
and as such is homogeneous of degree ps-1. Treat V (U; X) as the main vari-
able: then there is no constant term. Moreover, the coefficients are polynomia*
*ls
in V (U; o1), : :,:V (U; os) which are invariant under the natural action of GL*
* s(Fp).
TRANSFER AND CHERN CLASSES FOR EXTRASPECIAL p-GROUPS 3
So the coefficients must be polynomials in the Dickson invariants in these s va*
*ri-
ables: but there are no such polynomials in positive degree less than ps-1.
Therefore the coefficient in (1.6) of V (U; X)j is zero except when j = ps-1*
*._This_
coefficient is the size of PT *, congruent to 1 modulo p. *
*|__|
2.Chern classes and extraspecial p-groups
The integral (group) cohomology of the unitary group U(n) is a polynomial
algebra with n generators. A unitary representation of a finite group G pulls t*
*hese
generators back to the integral cohomology of G; the images of the generators a*
*re
the Chern classes of the representation. We refer the reader to [Tho86 ] for mo*
*re
information about Chern classes and their properties.
Let p be an odd prime. For n 1, denote by P = Pn the extraspecial p-group
of order p2n+1 and exponent p. This fits into a central extension
(2.1) 1 ! Z ! Pn ! En ! 1 ;
where Z = Z(P ) is cyclic of order p, and E = En is elementary abelian of p-
rank 2n. We may identify Z with Fp and view E as a 2n-dimensional Fp-vector
space: then the commutator map on P induces a nondegenerate symplectic bilinear
form E Fp E ! Z.
Ignoring the p2n linear characters, the remaining irreducible characters of *
*P all
have degree pn, and are distinguished by their restrictions to Z. Pick a nontri*
*vial
linear character fl of Z, and define aeflto be the unique irreducible represent*
*ation
of P whose restriction to Z has character pnfl. The other degree pn representat*
*ions
are then j(aefl) for 2 j p - 1.
Since E is the abelianization of P , Chern classes of one-dimensional repres*
*enta-
tions lie in the image of inflation from E to P . The Chern classes of j(aefl*
*) are
scalar multiples of those of aefl.
Pick an embedding of Fp in Cx : for each elementary abelian p-group A, this
allows us to identify Hom (A; Cx ) with the dual space A*.
The maximal elementary abelian subgroups M of P all have p-rank n + 1, and
are permuted transitively by the automorphism group of P . Each contains Z, and
the map M 7! I = M=Z induces a bijection between the M and the maximal
totally isotropic subspaces I of E. Note that the dual I* is naturally a subspa*
*ce
of M*.
View fl as an element of H2(Z; Z), since this isomorphic to Hom (Z; Cx ). T*
*he
restriction of aeflto M decomposes as the direct sum of all representations who*
*se
restriction to Z has character fl. The first Chern classes of these summands ar*
*e all
the classes in H2(M; Z) whose restriction to Z is fl; pick flto be one of these*
* first
Chern classes, and observe that the kernel in H2(M; Z) ~=M* of restriction to Z
is I*. From the definition of V , the total Chern class of aeflthereforenrestri*
*cts to M
as V (I*; 1 + fl), independent of thenchoicerof fl. Define in 2 H2p (P; Z) and*
*, for
0 r n - 1, define r = n;rin H2(p -p )(Pn; Z) by
(2.2) in = cpn(aefl) n;r= (-1)n-rcpn-pr(aefl) :
Then these are the only non-nilpotent Chern classes of aefl, and
(2.3) ResPMin = V (I*; fl) ResPM(r) = Dr(I*) :
It is useful, and sensible, to define P0 to be Z and i0 to be fl. Also to de*
*fine n;n
to be 1 2 H0(P; Z).
4 D. J. GREEN AND P. A. MINH
3. The new classes
Since every group homomorphism P ! Fp factors through E, we may identify
E* with the Fp-vector space Hom (P; Fp). Each maximal subgroup H P induces a
one-dimensional subspace of Hom (P; Fp), namely the subspace generated by any OE
with ker(OE) = H. Consequently there is a natural bijection between the set of
maximal subgroups of P and the projective space PE*.
We are interested in the corestriction map from maximal subgroups to P . Let
OE 2 PE*, and let H = ker(OE) be the corresponding maximal subgroup of P . Then
H=Z is a (2n - 1)-dimensional Fp-vector space, carrying a symplectic bilinear f*
*orm
with one-dimensional kernel. The centre ZH of H is elementary abelian of p-rank*
* 2,
and ZH =Z is the kernel of the form on H=Z. The maximal elementary abelian
subgroups of H all have p-rank n + 1 and all contain ZH . They are permuted
transitively by the automorphism group of H.
Definition 3.1.For n 1, for 0 r n - 1 and for OE 2 PE*n, the class Or;OEin
H*(Pn; Z) with degree 2(pn - pr) is defined as follows. Set H = ker(OE) and pic*
*k a
rank one subgroup A 6= Z of ZH . Choosing such an A induces a split epimorphism
H ! H=A ~=Pn-1. Set
(3.1) Or;OE= CorPnHInfHPn-1(n-1;rip-1n-1) :
Lemma 3.2. The class Or;OEis well-defined. That is, it does not depend on the
choice of A ZH .
Proof. The inner automorphisms of Pn permute transitively all such subgroups
A ZH , always fixing Z pointwise and therefore sending the aeflfor one Pn-1 to
the aeflfor the other. Since H is normal in Pn and corestriction commutes with_
conjugation, the result follows. |__|
4. Restriction and the new classes
In this section, we study restrictions of the classes Or;OE. We start howeve*
*r with
a preparatory lemma.
Lemma 4.1. Let V be an Fp-vector space, and let U be a hyperplane in V. Pick *
*v in
V \ U. Then the element V (U; v)p-1 of S(V) is invariant under all transformati*
*ons
of V which preserve U. In particular, it is independent of the choice of v.
Proof. Any transformation which preserves U acts on the coset space V=U as mul-
tiplication by some scalar in Fxp. But, from its definition, V (U; v) is invari*
*ant_under
all transformations which preserve U and the coset v + U. |_*
*_|
Definition 4.2.In the situation of Lemma 4.1, we denote by V (U; V) the invaria*
*nt
V (U; v)p-1.
Lemma 4.3. Let V be an Fp-vector space, and let U be a subspace with codimens*
*ion
two in V. Then X
(4.1) V (U; W) = 0 ;
W
where the sum is over all hyperplanes W in V which themselves contain U as a
hyperplane.
Proof. Let v; w be a basis for a subspace of V complementary to U. Then
X X
(4.2) V (U; W)= V (U; v + w)p-1
W [:]2FpP
X p-1
(4.3) = (V (U; v) + V (U; w)) :
[:]
TRANSFER AND CHERN CLASSES FOR EXTRASPECIAL p-GROUPS 5
This is an invariant of GL 2(Fp) acting on the rank 2 polynomial algebra genera*
*ted
by V (U; v) and V (U; w). Its degree in the generators is however p - 1, and th*
*ere_
are no invariants other than zero in this degree. |*
*__|
We now investigate the image of Or;OEunder restriction to each maximal eleme*
*n-
tary abelian M P . If M is contained in H = ker(OE), then ZH is contained in M.
Define IOEto be the quotient M=ZH . Then I*OEis a hyperplane in I*, which is it*
*self
a hyperplane in M*. Concretely, I*OEis the annihilator in I* of ZH =Z and also *
*the
annihilator in M* of ZH .
Proposition 4.4. With the above notation, we have
( * * *
(4.4) ResM (Or;OE) = -Dr(IOE)V (IOE; I )if M ker(OE)
0 otherwise.
Proof. Since Or;OEwas defined as a corestriction, we use the Mackey formula to
determine its restriction to M. Both M and H are normal in P , and corestriction
to M from any proper subgroup is the zero map, at least in positive degree. This
proves the result when M is not contained in H. So we may now assume that M
H.
As in the definition of Or;OE, choose a cyclic subgroup A of ZH such that ZH*
* =
A x Z. Then M=A is a maximal elementary abelian subgroup of H=A ~=Pn-1.
Note that (M=A)* is a hyperplane in M*, and itself contains I*OEas a hyperplane:
namely, the annihilator of Z. Then
X p-1
(4.5) ResM (Or;OE)= g* ResHMInfHPn-1n-1;rin-1
g2Pn=H
X
(4.6) = Dr(I*OE)V (I*OE; g*(M=A)*) :
g2Pn=H
Now, there were p possible choices for A, each of which yields a different (M=A*
*)*.
The possible (M=A)* are exactly those hyperplanes in M* which contain I*OEbut
are not equal to I*. There are permuted faithfully and transitively by Pn=H. Th*
*e_
result therefore follows from Eqn. (4.6) by Lemma 4.3. |_*
*_|
5.Describing Chern classes in terms of corestrictions
The Chern classes n;rrestrict to Z as zero. A theorem of Carlson [Eve91,
x10.2] says that any such class has some power which is a sum of corestrictions
from proper subgroups. In this section we shall derive a formula for n;rin terms
of the corestrictions Or;OEand the image of inflation from E. First however, we*
* recall
a well-known fact about Dickson invariants.
Lemma 5.1. Let V be an Fp-vector space, and U a hyperplane in V. Then
(5.1) V (V; X) = V (U; X)p - V (U; X)V (U; V) ;
and so, for 0 r n - 1,
(5.2) Dr(V) = Dr-1(U)p + Dr(U)V (U; V) :
Here, recall that D-1(U) is zero. |___|
Let M0 be a maximal elementary abelian subgroup of P , and denote by J0 the
quotient P=M0, itself elementary abelian of rank n. Then J*0is the annihilator *
*in
E* of M0. In addition, we may view J*0as a subspace of H2(E; Z).
6 D. J. GREEN AND P. A. MINH
Theorem 5.2. For all 0 r n - 1, the degree 2(pn - pr) class r;M0in H*(P; Z)
defined by
X
(5.3) r;M0= n;r- InfPEDr(J*0) + Or;OE
OE2PJ*0
is nilpotent.
Proof. By a theorem of Quillen [Eve91, Cor. 8.3.4], a class in the mod-p cohomo*
*logy
ring of a finite group is nilpotent if and only if the restriction to every ele*
*mentary
abelian p-subgroup is. Now, a class in integral cohomology reduces to zero in m*
*od-
p cohomology only if it is in the ideal generated by p. But in positive degree,*
* all
such classes are nilpotent. Hence it suffices to prove that the restriction to *
*every
elementary abelian subgroup is zero.
Let M be a maximal elementary abelian subgroup of P . As before, define I =
M=Z, and IOE= M=(Z(kerOE)) for OE 2 PE*. Define S to be the annihilator in J*0of
M, recalling that J*0may be viewed as a space of homomorphisms P ! Fp. Then
S = {OE 2 E* | M; M0 2 ker(OE)}.
If S = {0}, then ResM (Or;OE) = 0 for every OE 2 J*0, and both n;rand InfPED*
*r(J*0)
restrict to M as Dr(I*). So ResM (r;M0) = 0 as required.
We may therefore assume that the Fp-vector space S has positive dimension s,
which implies that the number of elements of the projective space PS is congrue*
*nt
to one modulo p. Denote by R the subspace ResM J*0of I*, and observe that S is
the kernel of the quotient map J*0! R. Now, S contains by Proposition 4.4 the
set of those OE 2 J*0such that ResM (Or;OE) is nonzero. Hence
0 1
X X
(5.4) ResM @ Or;OEA= - Dr(I*OE)V (I*OE; I*) by Proposition 4.4
OE2PJ*0 OE2PS
X
(5.5) = Dr-1(I*OE)p - Dr(I*) by Lemma 5.1
OE2PS
s *
(5.6) = Dr-s(R)p - Dr(I ) by Lemma 1.1
P *
(5.7) = ResM InfEDr(J0) - n;r :
Therefore ResM (r;M0) = 0, as desired. |___|
6. Taking pth powers of Chern classes
In the next section we shall show that the pthpowers of the new classes Or;O*
*Elie
in the image of inflation from E, at least modulo nilradical. In preparation fo*
*r this,
we shall here prove the same result for the pthpowers of the Chern classes n;r.
Recall that E carries a nondegenerate symplectic form, say (; ). Pick a symp*
*lectic
basis A1; : :;:An, B1; : :;:Bn for E: so Ai? Aj, Bi? Bj and (Ai; Bj) = ffiij. L*
*et
ff1; : :;:ffn, fi1; : :;:fin be the corresponding dual basis for E*, which we r*
*ecall may
be identified with H2(E; Z).
Definition 6.1.For r 1, define z(r)nin S(E*) by
Xn r r
(6.1) z(i)n= (ffifipi- ffpifii) :
i=1
By the work of Carlisle and Kropholler on symplectic invariants (see [Ben93 ,
x8.3]), the z(i)nare invariant under symplectic transformations of E. In partic*
*ular,
this means that they do not depend upon the choice of symplectic basis. Tezuka
and Yagita proved that, at least for p odd, z(1)n; : :;:z(n)nis a regular seque*
*nce
in S(E*); and that the ideal they generate contains every z(i)nand is the inter*
*section
TRANSFER AND CHERN CLASSES FOR EXTRASPECIAL p-GROUPS 7
with S(E*) of the kernel of inflation from H*(E; Z) to H*(P; Z). (See [BC92 ,
Prop. 8.2 and x10].)
Lemma 6.2. Let x1; : :;:xm be a regular sequence in a commutative ring R, and
suppose that elements a1; : :;:am of R satisfy
Xm
(6.2) aixi= 0 :
i=1
Then each ai lies in the ideal generated by x1; : :;:^xi; : :;:xm .
Proof. We note that the case m = 1 is trivial, and proceed by induction on m. T*
*he
product am xm lies in thePideal generated by x1; : :;:xm-1 , and therefore so d*
*oes am
by regularity:Psay am = m-1i=1bixi. Defining a0i= ai+ bixm for 1 i m - 1, we
have m-1i=1a0ixi = 0. Thus we have reduced to the case of m - 1, and the_resu*
*lt_
follows by induction. |__|
Corollary 6.3. Suppose that f 2 S(E*) lies inPthe kernel of InfPE. Then there e*
*xist
elements f1; : :;:fn of S(E*) such that f = ni=1fiz(i)n, and the images InfPE*
*(fi)_
of the fi under inflation depend only on f, not on the choice of the fi. *
* |__|
Proposition 6.4. There are unique classes g0; : :;:gn-1 in H*(P; Z) such that
there exist f0; : :;:fn-1 in S(E*) which satisfy both gi= InfPE(fi) and
n-1X
(6.3) z(n+1)n+ (-1)n-iz(i+1)nfi= 0 :
i=0
Each gi has degree 2(pn+1 - pi+1), and pn;r- gr is nilpotent for all 0 r n - *
*1.
Proof. Existence and uniqueness of the gifollows from Corollary 6.3. We once ag*
*ain
demonstrate nilpotence by proving that restriction to every maximal elementary
abelian subgroup M is zero.
Now, each z(j)nis a symplectic invariant. Hence, for any symplectic transfo*
*r-
mation oe of E, Corollary 6.3 says that Eqn. (6.3) still holds if each fi is re*
*placed
by oe*(fi). Consequently, oe* fixes each gi. As any linear transformation of a *
*max-
imal totally isotropic subspace I may be extended to a symplectic transformation
of E, it follows that each ResM (gi) is a polynomial in the Dickson invariants *
*for I*.
Since the symplectic transformations of E permute the I transitively, ResM (gi)*
* is
the same polynomial for each M. Finally, we see by comparing degrees that for
each i there is a scalar i2 Fp such that ResM (gi) = iDi(I*)p for all M.
To establish the result from here, we have to prove that each i is 1. First *
*we
shall prove that
n+1 n-1X n-i pi+1 * 2
(6.4) OEp + (-1) OE gi= 0 for all OE 2 E H (P; Z).
i=0
Since the gi are invariants for the symplectic group, it suffices to prove this*
* for
one nonzero OE, say fi1. Recall that the inflationjof z(j)nis zero, and observ*
*e that
differentiating z(j)nwith respect to ff1 yields fip1. Consequently, differentia*
*ting both
sides of Eqn. (6.3) with respect to ff1 and then inflating yields Eqn. (6.4) wi*
*th OE
replaced by fi1. This establishes Eqn. (6.4).
Now restrict to any one maximal elementary abelian M P and take pthroots
to obtain
n n-1X n-i pi * p *
(6.5) OEp + (-1) OE iDi(I ) = 0 for all OE 2 I .
i=0
8 D. J. GREEN AND P. A. MINH
This equation also holds for all OE 2 I* with each ireplaced by 1. Therefore ea*
*ch i
must be 1, or else taking the difference of the two left hand sides would yield*
* a_
polynomial of degree less than pn, with too many roots in an integral domain. *
*|__|
7. Taking pth powers of the new classes
P n-1 (i)
There are h1; : :;:hn-1 in S(E*n-1) such that z(n)n-1= i=1 hizn-1, by Coro*
*l-
lary 6.3. Now, S(E*n-1) embeds in S(E*n), and z(i)n= z(i)n-1+ z01(i). Here z0*
*1(i)
signifies z(i)1as a function of ffn; fin, that is ffnfipin- ffpinfin. So we have
n-1X n-1X
(7.1) z(n)n= z01(n)- hiz01(i)+ hiz(i)n:
i=1 i=1
As z01(i)is divisible by fin, there is a unique j 2 S(E*) such that
n-1X
(7.2) z(n)n= finj + hiz(i)n:
i=1
We may consider j to be an element of H2pn(E; Z).
Lemma 7.1. For every OE 2 PE*, the cohomology class Opn-1;OElies in the image
of InfPEmodulo nilpotent elements.
Proof. As the symplectic group permutes PE* transitively, it is enough to prove
the lemma for one OE. We shall show that Opn-1;fin- InfPE(j) is nilpotent. Wr*
*ite
O for On-1;fin.
Once more, we prove that the restriction of Op - InfPE(j) to every maximal
elementary abelian M P is zero. The restriction of Op to each M is known by
Proposition 4.4. If the image of fin in I* is nonzero, then it follows from Eqn*
*. (7.2)
that ResM InfP(j) is zero. Henceforth we may assume that M lies in the kernel
of fin, which is to say that M contains An and lies in En-1x. Then ResM Op =
-V (I*fin; ffn)p by Proposition 4.4.
Differentiating Eqn. (7.2) with respect to fin, then inflating to P and fina*
*lly
restricting to M, we have
n P n-1X P pi
(7.3) -ffpn = ResM Inf (j) - ResM Inf (hi)ffn :
i=1
By Proposition 6.4 (for Pn-1) and Eqn. (1.1), the choice of the hi implies that
n n-1X E pi * p
(7.4) ffpn - ResM Inf (hi)ffn = V (Ifin; ffn) ;
i=1
whence the result follows. |___|
Theorem 7.2. p_For all 0 r n - 1 and all OE 2 PE*, the class Opr;OElies in
Im (InfPE) + 0.
Proof. By a result of Evens [Eve68], Steenrod operations commute with corestric-
tion. Therefore, at least in mod-p cohomology, taking pth powers also commutes
with corestriction. Hence Lemma 7.1 and Proposition 6.4 (applied to Pn-1)_yield_
the result. |__|
TRANSFER AND CHERN CLASSES FOR EXTRASPECIAL p-GROUPS 9
8.Linear independence of corestrictions
We have been looking at three subrings of the cohomology of the extraspecial*
* p-
groups. Firstly the subring generated by top Chern classes, then the Chern subr*
*ing,
and then the subring generated by top Chern classes and the corestrictions Or;O*
*E.
In [Gre96], it is shown that top Chern classes generate a strictly smaller subr*
*ing
than the Chern subring, even modulo nilradical. Theorem 5.2 demonstrates that
(at least modulo nilradical) the Chern subring is contained in the subring gene*
*rated
by top Chern classes and the Or;OE. This containment is now shown to be strict.
Proposition 8.1. Let E00be a nondegenerate, codimension 2 subspace of E. Ex-
actly p + 1 elements OE 2 PE* satisfy E00 ker(OE). When np=_2, no non-trivial
Fp-linear combination of these On-1;OElies in Im(InfPE) + 0.
Corollary 8.2. Assume n = 2. Modulo nilradical, the Chern subring is strictly
contained in the subring generated by top Chern classes and the Or;OE.
Proof of Corollary.Modulo nilradical, the degree 2(p2 - p) part of the Chern su*
*b- __
ring consists of the image of inflation together with one extra class, 1. *
* |__|
Let E0 be the orthogonal complement of E00. Pick a symplectic basis for E su*
*ch
that E0has basis A1; B1 and E00has basis A2; B2. Note that each maximal isotrop*
*ic
subspace I of E satisfies either I = (I \ E0) (I \ E00) or (I \ E0) = (I \ E00*
*) = 0.
Lemma 8.3. Consider the x 2 S(E*) whose restriction to S(I*) is zero for every
maximal totally isotropic I E satisfying (I \ E0) = (I \ E00) = 0. These x form
the ideal generated by z(1)2and fl2 = D1(E0*) - D1(E00*).
Proof. Recall that z(1)2; z(2)2form a regular sequence in S(E*), and generate t*
*he joint
kernel of restriction to every S(I*). Observe that (ff1fip1- ffp1fi1)fl2 2 z(2)*
*2+ (z(1)2).
Moreover, fl2 has zero restriction to S(I*) if and only if I satisfies (I \ E0)*
* =
(I \ E00) = 0, whereas (ff1fip1- ffp1fi1) restricts to zero if and only if I sa*
*tisfies
I = (I \ E0) (I \ E00). So the ideal in question is the ideal of x such that *
* __
(ff1fip1- ffp1fi1)x has zero restriction to every S(I*). *
* |__|
Proof of Proposition 8.1.Let OE 2 PE* have E00 ker(OE), and let H be the corre-
sponding maximal subgroup of P . For any maximal elementary abelian M H,
have that I = M=Z satisfies I = (I \ E0) (I \ E00). By Proposition 4.4 it foll*
*ows
that ResM (O1;OE) = 0 for all such OE and all M satisfying (I \ E0) = (I \ E00)*
* = 0.
Invoking Lemma 8.3 and comparing degrees, the only element of Im(InfPE) that
comes into consideration is fl2 itself.
For no two of these OE do the corresponding maximal subgroups H share a
common maximal elementary abelian subgroup. Applying Proposition 4.4 again,
these O1;OEare therefore Fp-linearly independent. Moreover, no nonzero restrict*
*ion __
of a O1;OEis a scalar multiple of the restriction of fl2. *
* |__|
Remark 8.4. Proposition 8.1 and Corollary 8.2 can be proved in the same way for
general n 2, using the results of [Gre96].
9.Chern classes and transfer not sufficient
Calculating cohomology rings of p-groups would be easier if there were a lis*
*t of
constructions, such as transfer and Chern classes, which together always yielde*
*d a
set of generators for the cohomology ring. Being able to construct the cohomolo*
*gy
ring modulo_nilradical would be an important first step. This gave rise to the *
*so-
called Ch -conjecture, related to a construction of Moselle [Mos89 ]. We provid*
*e a
counterexample to that conjecture.
10 D. J. GREEN AND P. A. MINH
___
Definition 9.1.For a finite group G and a prime p, the subring Ch(G) of H*(G; F*
*p)
is defined recursively as the subring_generated by Chern classes for G, togethe*
*r with
the images under corestriction of Ch(H) for proper subgroups H of G.
Remark 9.2. For p = 2, one should use Stiefel-Whitney rather than Chern classes
to get the largest possible subring.
___
Remark 9.3. Inclusion of Ch (G) in H*(G; Fp) always induces an isomorphism of
varieties: see [GL96 ].
Proposition_9.4.pLet_G be the extraspecial 3-group ofporder_27 and exponent 3.
Then Ch (G)= 0 is strictly contained in H*(G; Fp)= 0.
Proof. Take a symplectic basis A; B for E = E1. We may view A; B as elements
of P = P1. We shall consider restrictions to the four cyclic subgroups generat*
*ed
by A, AB, AB2, B respectively. For each of these four cyclic groups K, denote
by the class in H1(K; Fp) ~=Hom (K; Fp) taking the generator to 1. Let x 2 H2
be the Bockstein of .
For p-groups H1 < H2, recall that CorH2H1ResH2H1is zero in mod-p cohomology.
In particular, CorH2H1is zero whenever H2 is elementary abelian. Using the Mack*
*ey
formula, it follows that ResPKCorPHis zero for any H < P and for K any of the f*
*our
cyclic subgroups.
Let ff; fi be the dual basis for E*. These may be viewed as elements of H1(P*
*; Fp).
Let a; b 2 H2 be their Bocksteins. Then a; b are also the first Chern classes o*
*f the
corresponding representations, and all first Chern classes are in their span. N*
*ote
that the restrictions of a to the four cyclics are x; x; x; 0 respectively, whe*
*reas the
restrictions of b are 0; x; -x; x respectively.
We refer the reader to [Lea92] for the fact that fffi = 0 in H2(P; Fp), and *
*for
an account of Massey triple products in this context. Define Y to be the Massey
product 2 H2(P; Fp). By naturality, this has zero restriction to t*
*he cyclics
generated by A; B, whereas its restriction_to_the subgroup generated by AB is *
*__
<; ; >. This is -x. Hence Y lies outside Ch(P ), even modulo nilradical. |*
*__|
Remark 9.5. We conclude from the proof of Proposition 9.4 that any list of con-
structions that was always guaranteed to provide enough generators modulo nil-
radical for mod-p cohomology would have to include Massey products.
Remark 9.6._Strictly, Proposition 9.4 is only a counterexample to the mod-p_ver*
*sion
of the Ch -conjecture. One may however define in the same way a subring Ch (G)
of H*(G; Z), and it is an open question whether containment modulo nilradical is
strict in this case.
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Inst. f. Exp. Math., Ellernstr. 29, D-45326 Essen, Germany
E-mail address: david@exp-math.uni-essen.de
Fachbereich 6 Mathematik, Universit"at Essen, D-45117 Essen, Germany
Current address: Department of Mathematics, University of Hue, Dai hoc Tong *
*hop Hue, Hue,
Vietnam