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Published by | universitat_regensburg |
Published | 01 January 2003 |
Reads | 14 |
Language | English |
Exrait
Cohomological Invariants
for Higher Degree Forms
Dissertation zur Erlangung des Doktorgrades
der Naturwissenschaften (Dr. rer. nat.)
der Mathematischen Fakult at der Universit at Regensburg
vorgelegt von
Christopher Rupprecht
aus Philadelphia
2003Das Promotionsgesuch wurde am 30.1.2003 eingereicht.
Die Dissertation wurde von Prof. Dr. Uwe Jannsen angeleitet.
Den Prufungsaussc huss bildeten
Prof. Dr. Harald Garcke (Vorsitzender),
Prof. Dr. Uwe Jannsen (erster Gutachter),
Prof. Dr. Manfred Knebusch (zweiter Gutachter),
Prof. Dr. Gun ter Tamme.
Das Promotionskolloquium fand am 2.5.2003 statt.Contents
0 Introduction 2
1 The Witt-Grothendieck Ring of r-Forms 11
2 Multilinear and Homogeneous r-forms 17
3 The Center of r-Forms, Separable r-Forms 20
4 Cohomological Classi cation of Separable r-Forms 25
5 Invariants of Degree 2 34
6 The Generalized Leibniz Formula 42
7 Discriminants 46
8 Zeta Functions of Separable r-Forms over Finite Fields 52
9 Hyperbolic r-Forms and the Witt Ring 58
10 Introduction
The motivation for this work is to generalize a concept from the theory of quadratic
forms to higher degree forms. Let us rst recall some de nitions for quadratic forms
(For a detailed exposition, see e.g. [28]):
LetK be a eld of characteristic = 2. A quadratic form overK is a pair (V;b),
consisting of a nite-dimensional K-vector spaceV and a symmetric bilinear form
b : V V ! K. The set of isomorphism classes of non-degenerate quadratic
forms overK with direct sum and tensor product is a semiring, which embeds into
^a commutative K-algebra W (K), the Witt-Grothendieck ring of quadratic forms
^over K. The 2-dimensional quadratic form h =h1; 1i2 W (K) is called the
^hyperbolic plane, and the ideal H W (K) generated by h is the ideal of hyper-
^bolic forms. The quotient ring W (K) = W (K)=H is the Witt ring of quadratic
forms over K. The structure of this ring is the principal object of study in the
theory of quadratic forms.
^The dimension map dim :W (K)! , (V;b)7! dim (V ) induces a homomor-K
phism e : W (K)! =2, called the dimension index. Let I = I(K) W (K)0
be its kernel, called the fundamental ideal of the Witt ring. The ltration of the
Witt ring by the powers of the fundamental ideal relates the Witt ring of quadratic
forms to Milnor K-Theory and Galois cohomology of the eld K as follows:
MLetK (K) be the n-th Milnor K-group of the eld K, de ned by Milnor in [25].n
M n n+1In this article, Milnor also gives a a surjections :K (K)!I =I , which mapsn n
a product l(a )l(a ) to the class of the n-fold P ster form1 n
(hai h 1i) (hai h 1i). Milnor’s conjecture that s is an isomorphism was1 n n
proved by Orlov, Vishik and Voevodsky in [26].
M rFor r 2, we have K (K)=r K =K , and in [31], Tate shows that=1
r 1the Kummer isomorphism K =K ! H (K; ) extends to a homomorphismr
M n
nh : K (K)! H (K; ) via the cup product. In ([19], p.608), Kato conjec-n;r n r
tures that h is bijective. In the case r = 2, this had been conjectured earliern;r
by Milnor and by Bloch. The conjecture was proved by Voevodsky in the case
mthat r = 2 is a power of 2 (cf. [17]). Hence we obtain commutative diagrams of
abelian groups and isomorphisms
hn
nM nK (K)=2 H (K; )n 2
M
M p
p
M
M p
p
M
p
M
p
M
p
M ps en M n
p
M p
p
n n+1I =I :
For n = 0; 1; 2, the morphism e has the following interpretation in terms ofn
quadratic forms: For n = 0, this is the dimension index e , which was de ned0
above.
The morphism e is de ned as follows: For a quadratic form( V;b), the class1
2of the determinant det(V;b) in K =K is an invariant for its isomorphism class.
2
/&6&ZZ77/dim(b)
b c
2The discriminant of (V;b) is de ned as d(V;b) := ( 1) det(V;b) (cf. [28],
2 1Def. 2.2.1). The discriminant gives a morphism d : I! K =K H (K; ),= 2
2and e is the induced map on I=I .1
The morphism e is given by the Cli ord invariant, which maps a quadratic2
form to the class of its Cli ord algebra in the Brauer group. This class has degree
22 2 2, so that the image of e lies in Br(K) = H (K; ) = H (K; ) (cf. [22],2 2 2 2
Chap. 5.3).
Independently from the proof of the Milnor conjecture, it was shown forn = 3
by Arason in [1] and for n = 4 by Jacob and Rost in [15] that the map e com-n
pleting the diagram is well de ned.
Now let r > 2 be an integer. One observes that, while the upper part of the
diagram has a degree r analogue, the lower part has not:
hn;rM 1
nK (K)=r H (K; )n r
N
p
N
N p
N p
p
N
p
N
p
N p
N
p
N
ps ? e ?n;r N p n;r
p
n n+1I =I ?
This raises the following questions:
Is there a degree r analogue of the Witt-Grothendieck ring?
Can we give cohomological invariants for higher degree forms generalizing
the maps e in the diagram above?n
Can we give a degree r analogue of the hyperbolic plane or the hyperbolic
ideal and de ne a Witt ring of higher degree forms?
Can we give degree r P ster forms generalizing the maps s in the diagramn
above?
Forms of degree r. Let K be a eld such that (char( K);r!) = 1, i.e. such
that char(K) = 0 or char(K) > r. An r-form over K is a pair (V; ), consist-
ing of a nite-dimensional K-vector space V and a symmetric multilinear map
:VV!K, de ned on the r-fold product of V .
The condition (char(K);r!) = 1 on the characteristic ofK allows us to identify
r-forms with homogeneous forms of degree r over K as follows: Let (V; ) be an
r-form overK, and letfv ;:::;vg be aK-basis ofV . Then there is a homogeneous1 n
form f =f 2K[x ;:::;x ] such that 1 n
n nP P
f(x ;:::;x ) = ( xv;:::; xv ).1 n i i i i
i=1 i=1
Just as in the case of quadratic and bilinear forms, we obtain a bijective cor-
respondence between isomorphism classes of symmetric multilinear r-forms and
homogeneous forms of degree r. In times it will be convenient to switch from one
3
/&/77&viewpoint to the other. We will speak of multilinear and homogeneousr-forms, or
simply of r-forms if there is no ambiguity.
Regularity. A quadratic form on V is called non-degenerate if the induced
linear mapV!V has full rank. A quadratic form is if and only
if it is non-singular, meaning that it describes a non-singular quadric. For forms
of degree r> 2, there is more than one analogue of this de nition:
De nition. Let r 2 and let 1 k < r be an integer. An r-form (V; ) over
K is called k-regular, if, for every non-zero k-tuple (v ;:::;v ) of vectors in V ,1 k
the (r k)-form (V; ) given by (v ;:::;v ) := ( v ;:::;v ) is(v ;:::;v ) (v ;:::;v ) k+1 r 1 r1 1k k
non-zero. A 1-regular r-form is also called regular.
Anr-form overK is non-singular, meaning that it describes a non-singular hy-
persurface in projective space, if and only if it is (r 1)-regular over the separable
closure K.
The Witt-Grothendieck ring of r-forms. The starting point for this work
is the article [10], in which Harrison introduces a ring ofr-forms. He shows that the
set of isomorphism classes of regular r-forms over K with direct sum and tensor
product is a commutative semiring over K, which embeds into a commutative
^K-algebra W (K), called the Witt-Grothendieck ring of r-forms.r
Although the de nition of thek ring of r-forms is the same
for r = 2 and r > 2, the obtained rings have quite di erent properties. This is
illustrated by the following observations:
Consider the generators in the Witt-Grothendieck ring. Every quadratic form
is isomorphic to a diagonal form, and therefore the Witt-Grothendieck ring of
quadratic forms is generated by 1-dimensional forms. In particular, the Witt-
Grothendieck ring of quadratic forms over a nite eld is nitely generated.
Forms of degree r > 2 are not always diagonal. We call an r-form indecom-
posable if it has no non-trivial sum decomposition. Over any eld, there are
indecomposable r-forms of dimension > 1. If K is a nite eld, then there are in-
decomposable r-forms of arbitrary dimension over K, and the Witt-Grothendieck
ring of r-forms over K is not nitely generated.
Now consider the relations in the Witt-Grothendieck ring. Witt’s Theorem
gives a cancellation rule for quadratic forms, which allows the construction of the
Witt-Grothendieck group. For r > 2 one obtains a stronger result: The decom-
position of an r-form into indecomposable r-forms is unique. Thus, the Witt-
Grothendieck group of degree r > 2 is a free abelian group, having much less
relations than in the quadratic case.
Separable r-forms. Another di erence between the quadratic and the degree
r> 2 case comes from the following de nition given by Harrison:
4De nition. Let r > 2, and let (V; ) be an r-form over K. Let the center of
(V; ), written Cent (V; ), denote the set of K-endomorphisms ’2 End (V )K K
such that
( ’v ;v ;v ;:::;v ) = ( v ;’v ;v ;:::;v )1 2 3 r 1 2 3 r
for all v ;:::;v 2 V . The center is a commutative K-algebra. The r-form1 r
(V; ) over K is called separable if its center is a separable K-algebra such that
dim (Cent(V; )) = dim (V ).K K
sep^ ^Harrison shows that separable r-forms generate a subring W (K) W (K)rr
in the Witt-Grothendieck ring ofr-forms, and he gives the following classi cation
of separable r-forms:
Let L=K be a nite separable eld extension, let tr : L! K be the traceL/K
map, and letb2L . We considerL as aK-vector space with the multilinear map
tr hbi :LL!K ; (l ;:::;l )7! tr (bl l ).L/K r 1 r L/K 1 r
Then (L; tr hbi ) is an indecomposable separabler-form overK and every inde-L/K r
composable separable r-form over K is isomorphic to an r-form (L; tr hbi ) forL/K r
some L and b.
Cohomological classi cation of separable r-forms. In the theory of
quadratic forms, Weil descent is used to classify quadratic forms by Galois co-
homology: Since every quadratic form is diagonal, all forms of the same
dimension overK are isomorphic over a separable closureK. Therefore the set of
1quadratic forms of dimensionn overK is bijective to the cohomology setH (K; O )n
by Weil descent.
Forr> 2, it is not true that allr-forms become isomorphic to a diagonal form
over the separable closure. However, restricting attention to those who do so, we
obtain a subring in the Witt-Grothendieck ring, and we nd that this is the ring
of separable r-forms. This leads to a cohomological classi cation for separable
r-forms as follows:
The automorphism group of the diagonal r-form over K is the wreath product
S s of the symmetric group S and the group of r-th roots of unity in K.n r n r
nThe wreath product is the set S with the semidirect product induced byn r
nS -action on . Using Weil descent, we obtain a classi cation of separablen r
1r-forms of dimension n over K by the cohomology set H (K;S s ). The corre-n r
spondence between this classi cation and Harrison’s classi cation by trace forms
is explicitly computed.
Cohomological invariants for separable r-forms. Consider the classi -
1cation of quadratic forms by cohomology sets H (K; O ), which was describedn
before. In these terms, the determinant of quadratic forms, which is closely
related to the map e in the diagram above, is equal to the cohomology map1
1 1H (K; O )! H (K; ). In the same way, we obtain invariants for separablen 2
r-forms from the cohomological classi cation:
5Consider the projection from the wreath product S s to the symmetricn r
1group S . The cohomology set H (K;S ) classi es isomorphism classes of sepa-n n
rable K-algebras of dimension n, and we nd that the induced cohomology map
1 1H (K;S s )! H (K;S ) maps a separable r-form to the isomorphism classn r n
of its center. Concatenation with the sign homomorphism S ! gives an in-n 2
1 1variant map H (K;S s )! H (K; ), which maps a separable r-form to then r 2
2 1determinant of the bilinear trace form of its center in K =K =H (K; ).2
Next, consider the permanent morphism
nQ
per :S s ! ; (; ( ;:::; ))7! :n r r 1 n i
i=1
The induced cohomology map gives a rst degree cohomological invariant for sep-
arable r-forms
sep 1^per :W (K)!H (K; ).rr
Finally, consider the determinant morphism
nQ
det :S s !K ; (; ( ;:::; ))7! sgn() :n r 1 n i
i=1
The image of the determinant is equal to if r is even, and equal to if r isr 2r
odd. Hence we obtain another rst degree cohomological invariant for separable
r-forms
1H (K; ) evenrsep^det :W (K)! if r is .r 1H (K; ) odd2r
All these invariants are given explicitly in terms of separable trace forms.
Cohomological invariants of degree 2. We interpreted the determinant
of quadratic forms as a cohomology map of degree 1, induced by thet
morphism det : O ! . Its kernel is the special orthogonal group SO , hencen 2 n
1the cohomology set H (K; SO ) classi es quadratic forms of dimension n andn
determinant 1. The simply-connected covering of SO is the spin groupn
0! ! Spin ! SO ! 0,2 nn
and the induced long exact sequence of Galois cohomology gives a map
1 2 :H (K; SO )!H (K; ),n 2
which is related to the morphism e in the diagram above (cf. [21],x2.4).2
Starting from the cohomological classi cation for separable r-forms, we want to
construct second degree invariants in this way. For this purpose, let
(i)SO 2 S s , (i = 1; 2; 3) denote the kernel of the permanent, the determi-n rr;n
nant, and the sign respectively. In the case that r = 2; 3 is a prime number, we
give a classi cation for central extensions of Galois modules
(i)0! ! Spin ! SO ! 0.r n;r n;r
6
6
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