Virtual forms

Virtual forms

Karim Johannes Becher

Abstract Quadratic forms over fields or characteristic different from two are general- ised to virtual forms over arbitrary fields. These objects are related to Milnor's K -theory for fields. The connection is established by a sequence of maps corresponding to Delzant's Stiefel-Whitney classes.

Keywords Milnor K -theory· Quadratic form· Chain equivalence· Witt-Grothendieck group· Pfister form· Stiefel-Whitney class

Mathematics Subject Classification (2000) II E04 . II E81 . 19D45

1 Introduction

Quadratic forms have manifold aspects and they can be generalised in various directions. The aim of this article is to transfer some or the algebraic theory of quadratic forms over fields into a more general context. To this aim virtual forms over an arbitrary field are introduced.

These objects are closely related to Milnor K -theory.

To sketch the idea, let us consider a field F of characteristic different from 2. The algebraic theory of quadratic forms over F, as it may be learned from [6] or [10], starts with a few key observations, mostly going back to Witt's seminal work [II]. First of all, quadratic forms over F can be diagonalised. Witt's Cancellation Law gives information on the interplay between orthogonal sum and isometry. This leads to the definition of the Witt ring W (F) and the Witt-Grothendieck ring W(F). Witt's Chain Equivalence Theorem describes when exactly two given diagonalisations belong to the same quadratic form, and this yields a description of W(F) and of W(F) by means of generators and relations.

This abstract description of the Witt-Grothendieck ring

W

^{(F) will be taken as the}

guiding principle for the definition of a group G (F,

0,

the Grothendieck group of i-forms,

K. J. Becher (181)

Fachbereich Mathematik und Statistik, Universitlit Konstanz. 78457 Konstanz. Germany e-mail: becher@maths.ucd.ie

http://dx.doi.org/10.1007/s00209-009-0529-4

Konstanzer Online-Publikations-System (KOPS) URN: http://nbn-resolving.de/urn:nbn:de:bsz:352-172320

where now F is an arbitrary field and

e

^E N. If F is of characteristic different from 2, then G(F, 2) coincides with the group W(F). The definition of G(F, f.) is based on Milnor K -theory.

In this articlc, first an cquivalcncc rclation on thc sct of finitc scqucnccs of nonzcro c1c- ments of a field is investigated in Sect. 3. This relation, called i-equivalence, leads in Sect. 4 to the definition of virtual forms of degree

e

and the group G (F. 0. In Sect. 5. Pfister rorms are generalised to the new setting. They allow to construct in Sect. 6 a descending sequence of subgroups (G"(F, e) IIEN, destined to replacc thc filtration given by the powers of the fundamental ideal in the Witt-Grothendieck ring. While in general G(F, e) is not a commu- tative group for

e #

2, at least the groups G"(F, e) for n ::: 2 are contained in the center of G (F, e) (6.2). Moreover, if f. is odd, then Gil (F, i) vanishes for n ::: 3 (6.7) and any ele- ment of G I (F, e) has finite order dividing f. (6.10). A sequence of maps from G (F.

n

^{to the}

Milnor K -groups modulo f. of the field F allows the use of f.-forms to investigate the Milnor K -theory of F. These maps generalise Delzant's Stiefel-Whitney classes for quadratic forms from [8]. It turns out that G I (F, £)/ G³ (F, e) can be described as a group extension of the first by the second Milnor K-group modulo

e

(7.10). Kahn [4, Sect. 3] obtains a sufficient condition for the vanishing of higher Milnor K -groups. As an application of the new con- cepts, an alternative proof of this result is given in Sect. 8. Some of the results presented here were announced in [I].

Throughout this article F denotes a field, F x its multiplicative group, and f. a nonnegative integer.

2 Milnor K -theory

We shall blidly go ovt:r the setup or K -tht:Ory for fields introduct:d in l8] and fix some notation.

Let n, e ^E N. Let K,~n F denote the additive abelian group generated by so-called symbols {ai, ... ,all}' with ai, ... ,all E FX, that are subject to the following relations:

(M I) the canonical map { } : (Fx)" ----* K,~n F is Z-multilinear;

(M2) {a" ... , all}

=

0 whenever ai

+

ai+1

=

I in F for some i < n;

(M3)

e·

K,;C) F = O.

Obviously, (M3) is a void condition if f. = O. In fact, K,;O) F is the 'full' nth Milnor K -group, usually denoted by KII F, and K,;£) F is its quotient modulo f.. Note that K

f e)

F can be identified with the group FX / FXf. in additive notation: an element {x} of Kit) F corresponds to the class

x

F X

e

in F x / FX

e.

For

n

= 0 the first two relations are irrclcvant, so K~n F is a cyclic group generated by the empty symbol {}; in view of (M3) we identify

K6e)

^F

⁼

^{Z/£Z. In}

K y)

^F, it follows from (Ml) and (M2) that the equalities {- a, a}

=

0

and {a, b}

=

- {b, a}

=

{b-^I, a} hold for arbitrary a, b E F^X (see [8, Sect. 1]).

(t) (e)

- en n

^{(t) .}

We denote K* F = ffii,:,:oK; F and K* F = i':':O Ki F. With {a I, ... , a,.} . {ar+ I , ... , a,.+s} = {a I ... a,.. a,.+ I , ... , a_r+s }

one has a natural multiplication in both K~e) F and

R !e)

F. Hence, K~n F and

Ri

^e) Fare algebras over

K6 ^e)

^{F = Z/f.Z, and K~f) F is} even a graded algebra.

3 Chain equivalence

We shall introduce and study an equivalence relation on sequences of nonzero elements of a field. The aim is to define virtual forms of degreee by generalising the concept of chain equivalence for quadratic forms, which is involved in Witt's Chain Equivalence Theorem (cf. [6, Chap. I, Sect. 5] or [II, Satz 7]).

We consider finite sequences in FX. We denote by SII (F) the set of all such sequences of length n, and by [ai, ... , all] the sequence with entries ai, ... , all E FX. We use 0 to denote the concatenation of two sequences: for Q' = [a I .... , a,.] E S,. (F) and Q" = [a,.+I ... a,.+s I E Ss(F) we set Q' 0 Q" = [ai, ... , a,., a,.+I .... , a,.+s 1 E S,.+s(F).

For n ::: I we are going to define an equivalence relation on SII (F), which depends on the integer

e ::: o.

This relation is introduced lirst ror n = I and n = 2 and then extended in a canonical way to arbitrary n. Given _{a, a'} E FX, we write [a] ~ [a'l if {a) = {a'l holds in

Ki f)

^F, or equivalently, if a' = xta for some x E FX. Given a, b, a'. b' ^E FX, we write [a,b] -S [a',b'] if {{ab) = {a'b'}

{a, b}

=

^{{a', b')}

and

We have thus defined an equivalence relation on SII (F) for n = 1,2. Let now n > 2. We say that the two sequences a

=

[a I ... all] and

f3 =

[bl , .... bll] in SII (F) are simply e-equiva- lent and writeQ' ~

f3

if there exists a positive integer k < n such that [ak. ak+il ~ [bk, bk+il and [ai]

=

^[b;l for I ::: i ::: n with i 1= {k, k

+

I}. This relation is not transitive in general. In order to obtain an equivalence relation ~ we take the transitive closure of ~. We say that Q',

f3

^E SII(F) are chain i-equivalent and write Q'

,.s f3

if

, " e e "

Q' "'" YI "'" Y2 "'" ... "'" Yr "'"

f3

holds for a suitable choice of YI, ... , Yr E SII (F), where r ::: I. For e = 0 we simply speak of simple equivalence and chain equivalence and we write ~ and ""', accordingly; note that simple (resp. chain) equivalence implies simple (resp. chain) i-equivalence for any

e

^{> O.}

As already said, these definitions are motivated by Witt's Chain Equivalence Theorem, which can be reformulated in the following way.

Proposition 3.1 Assume that char(F)

i-

2. Two sequences [al .... , all] and [bl, ... , bll]

in SII (F) are chain 2-equivalent

if

^{and only} if the quadratic forms al

xi + ... +

^{all X~ and}

bl

Xi + .. . +

bllX~ over F are isometric.

Proof For n = 1 the statement is obvious. Let now n = 2. For c, d E FX, the symbols {cd) E

K:

^{2) F and {c, d) E}

Ki

^{2) F are} invariants of the form cX2

+

^dy2 up to isometry.

Hence, if alX2

+

^a2y2 and blX2

+

^b2y2 are isometric, then [al,a2] 1 [bl,b]. Con- versely, assume that [ai, a2]1 [bl, b2]. Since {ai, a2} = {bl, b2) holds in

Ki

^{2) F, the forms}

-a1X2 -a2 y2 +ala2Z2 and - b1X2 - b2y2 +blb2Z2 are isometric, by [3, Theorem 1.8]

and Witt Cancellation (cf. [6, Chap. I, 4.2]). Since further bl b2 = al a2x 2 for some x E F x,

it follows using Witt Cancellation that al X2

+

^{a2y2 and bl X2}

+

^b2y2 are isometric.

So far, it is shown that 2-equivalence for elements in S2 (F) characterises isometry for the corresponding 2-dimensional quadratic diagonal forms over F. The general statement now follows using Witt's Chain Equivalence Theorem (cf. [6, Chap. 1, Sect. 5]), which says that isometry of quadratic diagonal forms in any dimension n ::: 2 is entirely determined by the

isometry of 2-dimensional diagonal subforms. 0

Remark 3.2 Assume that f ::: 2 and that F contains a primitive fth root of unity ~. In this case, the Merkurjev-Suslin Theorem (cf. [7]) implies that the symbols in KY) F are in one- to-one correspondence with the isomorphism classes of cyclic algebras of degree

e

^{over F.}

More precisely, given a, b E F X, the symbol (a, b) can be identified with the symbol algebra (a, b)F.(. By definition, (a, b)F.( is the central simple F-algebra of degree.e generated by two elements u, v subject to the relations £Ie

=

^{a, vf.}

=

^b, and vu

=

~uv.

Hence, under these assumptions on

e

^{and F, chain} f-equivalence can be described in terms of classical algebraic structures not involving K -theory.

Permuting the entries of a sequence does not necessarily yield a sequence that is chain equivalent (modulo e) to the original one. More particularly, for a, b E S2 (F), one has [a,b] J.- [b,a]ifandonlyif2.{a,b}=OinKY)F.

Lemma 3.3 For any a, bE F X, one has [a, b] ~ [b-^I, ab2] ~ [a 2b, a-I].

Proof Note that for any of the three given sequences in S2 (F), the product of both entries is abo Moreover, one easily checks the equalities (a, b)

=

^{{b- I, ab2}}

=

^{{a 2b, a-I} in K2F.}

This together yields the statement. 0

Corollary 3.4

If

^a =

±

I, then [a, bl ~ lb, a]fo,. any b E FX.

Proof Since a-I = a, this is clear from the lemma.

o

A generalisation of the last statement will be obtained in (3.9) below.

Let So (F) denote the singleton set consisting of the empty sequence []. Let further S(F) =

U

^SII(F).

liEN

Hence, (S(F), 0) is the free monoid generated by FX.

We now introduce operations on sequences and study their behaviour with respect to chain equivalence. We say that a map

cP : S(F) ---* S(F)

is compatible with chain €-equivalence if it preserves lengths of sequences and if, for any n E 1':1 and any a. f3 E SII(F), the relation a J.- f3 implies that cP(a) ~ cP(f3). We say that cP as above is compatible if it is compatible with chain i-equivalence for all i E 1':1.

Examples of compatible maps are given by rising the coefficients of a sequence to some power and by reversing the order of the coefficients.

Let a = [ai, .... all] E SII (F). For Z E Z, we write a^Z for the sequence [a_I^Z, ••• ,a,Tl- In particular, a-I

=

[a,I, ... , a,-;-I]. Furthermore, we denote by ii the sequence with the entries of a in reversed order, i.e. ii = [all, ... , ail.

Proposition 3.5 (a) For any Z E Z, the rule a ~ a^Z defines a compatible endomorphism of S(F). In particular, a ~ a-I is a compatible involution on S(F).

(b) The rule a ~ ii defines a compatible involution 011 S(F).

Proof Only the compatibility (with respect to arbitrary f) requires a proof. In view of the definitions, one needs to verify compatibility only for sequences a,

fJ

^{E SII (F)} where n ::'S 2, hence to show for such sequences that (X ~

fJ

implies that (XZ ~ fJz for every Z E Z and ii ~

7f.

^{For n = I} this is obvious.

Let now a. {3 E S2(F) with a ,( {3. Using that, for any u, v E FX and any Z E Z, one has

{UZVZ}=z'{uv} in Kll)F and {uz_{.v Z}=Z2.{u.v}} in KieJF,

one sees that a^Z , ( {3z, for any z E Z. Similarly, using that {v, u} = -{u, v} in Ki^l) F for

any u, v E F^X, one obtains

a ,( if.

0

Given a sequence a = [al ..... all] E SII(F) and an element c E FX, we denote by c

*

^a the sequence [cf-^Ial , ... , COli all], where ci

=

^(_I)i for i

=

I, ... , n. We call c

*

^a

the conjugate of a by c. Note that conjugation is bijective, but it is in general not an endo- morphism of the monoid S(F). However, (- I)

*

^{a = [-ai, ... , -an],} so conjugation by -1 is an automorphism of S(F). As we shall see now, conjugation by an element of FX is a compatible operation on S(F).

Proposition 3.6 Let a,

f3

E SII (F), c E FX, and f E N. Then a ~

f3 if

and only

if

c

*

a ,( c

* fi.

Proof By the definition of chain e-equivalence, the statement needs to be proven only for n

=

2. Foranya, b, c E FX, we have {c-Ia, cb}

=

^(c- I . -ab}+{a, b) in

K yl

^F. Therefore, if a. b. a', b' E FX are such that [a. b] ~ [a'. b'], i.e. {ab} = {a'b'} in

Kin

^{F and (a, b) =}

{a', b/} in

K yl

^F, then we obtain {c-Ia, cb} = {c-Ia', cb'} and [c-Ia, cb] ~ le-Ia' , cb']. This proves one direction. Replacing c by c-^I yields the converse direction. 0

Fora sequence a

=

[al, ... ,aIlJ E SII(F),weseta*

=

[-a,-;-I, .... -ail] E SII(F).

This defines an involution

* : S(F) --+ S(F), a t----> a*. Corollary 3.7 The involution * : S(F) --+ S(F) is compatible.

Proof The map * is compatible, because it is the combination of three compatible opera- tions on S(F) (which actually commute with each other), namely a f-+ a-I, a f-+

a,

^and

af-+(-I)*a. 0

For a E SII (F) and r EN we put r x a

=

^{a 0 ... 0 a E Srll (F).}

Proposition 3.8 For any a E SII (F) one has a 0 a* ~ n x [I, - I].

Proof It follows from the definition of chain equivalence that [a, -a-I] ~

r

I, - I] holds for any a E F x . Since [1 , - I] commutes up to chain equivalence with any element of S (F)

by (3.4), the statement follows. 0

We now turn our interest to operations on sequences that do not change the equivalence classes of certain sequences.

Proposition 3.9 Let a I, ... ,all E F x be such that a I ... all =

±

I. Then

[

al ... all.

J

~

[

all ^{- I} ,a l ^{- I} , .... all_I^{-I ]} ' Moreovel;

if

ⁿ is odd, then

[al, ... , all] ~ [ail, ... , _{a,-;-I J} ~ [all, al, ... , all-Il.

Proof Using n-I times (3.3) we obtain

[al, ... ,all ] ~ ^{ral, ....} alla~_I'a,~~I]

r ( )

^{2 - I - I ]}

~ all al···all_1 ,a l , ... ,all_l ·

Since by hypothesis (al ... all _I)2 = a,-;2, the first part of the statement follows. This also shows the second equivalence of the second part of the statement, which thus holds without condition on n.

Finally, if n is odd, then applying n times the first part yields [al, .... all] ~ [all, ... , a,-;I].

o

Remark 3.10 Possibly, also the second part of the statement is true for arbitrary n. At least, it can also be proven for even n in case [al . .... all] is chain equivalent to some sequence having one entry equal to ± I.

Proposition 3.11 Let a = [al . ... ,all] E SII (F) with al ... all = ± I. Then for any

f3

E SII/(F) one has a 0

f3

~

f3

⁰ a^C, where F. = (-1)11/.

Proof We may assume that m = I, so that f3 = [c) for some c E FX. Using n times (3.3) we obtain

a 0 f3

=

^{[al, ..}. , all, c] ~ [al, ... , all_I, ca,~, a,-;I]

"

( )2 -I -I] f.J. -I

~ .c al ... all . {II , ... , all = F 0 a ,

because (a I ... all)2 = I, by the hypothesis.

o

4 Virtual forms

With the results of the last section we are now prepared to introduce virtual forms in any degree f ~ 0 over a field in such a way that quadratic forms arc retrieved for

e

^{= 2 in}

characteristic different from 2.

We say that two sequences a, a' ^E S(F) are stably chain f-equivalent if

(X 0 (r x [I, - I)) !v a' 0 (r x [I, -I))

holds for some r E N. We write M(F, e) for the set of equivalence classes in S(F) mod- ulo this equivalence relation. The elements of this set are called virtual forms of degree f over F, or e-forms for short. For a E S(F) we denote by (a) the e-form given by a. If a = [a I, ... , all] then we may write (a I, ... , all) instead of (a). The class of the empty sequence [] is also considered as an element of M(F. f), and it is denoted by 0 and called the trivialform.

It is not clear whether stable chain f-equivalence is actually a coarser equivalence relation than chain f-equivalence.

Question 4.1 For a, (X' E S(F), does (a) = (a') imply that a !v (X'?

We define a (not necessarily commutative) operation -I-on M(F, e). For ex,

fl

E S(F), we set

(ex) -I-(f3) = (ex <> (3).

Using (3.4) it follows that this operation is well-defined and aSSOCl(ltlve. Therefore (M(F, i), -1-) is a monoid with neutral element O. For n E N and rp E M(F. i), we denote by n x rp the n-fold sum rp -I-... -I-rp.

Lemma 4.2 The monoid (M(F.

0 ,

-1-) satisfies the cancellation law.

Proof Given a. a'. f3 ^E S(F) with (a) -I- (f3) = (a') -I- (f3), we claim that (a) = (a').

We may restrict to the case where f3 E SI (F), i.e. (fJ) = (b) for some b E FX. Note that (b) -I-(-b- I)

=

(b, - b- I)

=

(I, -I) by (3.8). Therefore (a) -I- (fn

=

^{(a') -I-}(f3) implies that (ex) -I-(I, - I) = (a') -I-(I, - I). Hence, ex <> ll, - I) and a' <> ll, - 1) are stably chain i-equivalent. The same then holds for a and a', so (a)

=

(a'). This shows that cancellation on the right is possible. The proof of cancellation on the left is analogous. 0

We write IHl for the R-form (1, -I). Note that (a, _a-I)

=

IHl for any a E F^X, by (3.8).

We denote by G (F, e) the set of formal differences rp- nxlHl

where rp E M(F, e) and n EN. By (3.4) thee-form n x IHllies in the center of M(F, e) for any n E N. Therefore, the operation -I-extends naturally to a monoid operation on G (F, e).

Since M(F,e) satisfies the cancellation law, the same is true for G(F, e), and M(F. e) can be seen as a submonoid of G (F,

0

by identifying an e-form rp with the difference rp - 0 x 1Hl.

Proposition 4.3 G(F. e) is a group.

Proof We have to show that, for n E N and a E S(F), the difference (ex) - n x IHl has an inverse in G (F, i). In fact, if m is the length of the sequence a, then the inverse is given by (a*) -I-(n - m) x 1Hl, because (a) -I- (a*)

=

m x IHl by (3.8). 0

Example 4.4 Let us assume that char(F)

I-

2 and consider the case

e

^{= 2. By (3.1) and}

Witt's Cancellation Theorem, 2-forms over F correspond bijectively to isometry classes of regular quadratic forms over F. In particular, Question (4.1) has a positive answer in this case and G(F, 2) is isomorphic to the Witt-Grothendieck group W(F).

We call G(F, R) the Grothendieck group ofe-forms over F. Note that this group is abelian if and only if 2 . Ki

e )

F

=

0 (for example, if

e =

2).

Lemma 4.5 For any a, bE F^X, we have (a. b)

=

(b- I, ab²)

=

(a²b, a-I).

Proof This is immediate from (3.3).

o

We next prove a universal property for the group G(F, i).

Lemma 4.6 Let (G. 0) be a group and g : F x --* G a map. Assume that the following holdforanya,b,a',b' E F^X:

(i)

if

{a} = (bl in K~e) F, then g(a) = g(b) in G;

(ii)

if

lab}

=

{a'b'} in Kif.) F and (a, b)

=

{a', b'} in KY) F, then g(a) 0 g(b)

=

g(a') 0

g(b') in G.

Then there is a unique group homomorphism r : G(F, £) ~ G such that r«(a) = g(a) for every a E FX.

Proof The rule [at, ... , all] ~ g(at) 0 . . . 0 g(all ) defines a monoid homomorphism S(F) ~ G. In view of the two conditions and since G is a group, it is obvious that the image of a sequence a E S(F) under this homomorphism depends on a only up to stable chain £-equivalence. This yields a homomorphism M(F, £) ~ G that maps the I:'-form (at, "" all) to g(at) 0 . .. 0 g(all ). Since the group G(F, e) is generated by M(F, e) and since G is a group, it is elear that this map extends to a map

r :

^G (F, £) ---+ G with the

desired property. The uniqueness of

r

is obvious. 0

Remark 4.7 The lemma shows that (G(F, t), +) is equal to the group defined by generators and relations in the following way. G(F, e) is generated by elements (a) with a E FX, and the defining relations arc:

(VI) (a)

=

(b) if {a}={b} in KitlF;

(V2) (a)

+

(b)

=

(a')

+

(b') if {{ab}

=

{a'b'}

{a, b}

=

{ai, b'}

We will frequently use these rules without particular mention.

Let ep be an £-form over F, say ep = (a J , ••• , a,.). We shall refer to r as the rank of ep and denote it by rk(ep). We further put dJ (ep) = {at· .. a,.} E Kit) F and call this the determinant of rp. This yields two group homomorphisms

rk : G(F,t) ~

z,

dJ : G(F,£) ~ K J (C) F.

One can apply (4.6) to check that they are actually well-defined. Note that the group G (F, £) is generated by the I:'-forms of rank one.

For an /:'-form ep given by ep

=

^{(a) with a E} S(F), we use the notations ep*

=

^(a*),

epz = (a^Z) for Z E Z, in particular rp-J = (a-J), and further c * rp = (c

*

^{a) for c E FX. By}

(3.5)-(3.7), these operations on £-forms are well-defined.

Proposition 4.8 For any £-form ep over F, we have:

(a) ep

+

rp* = rk(ep) x 1HI;

* { c

*

ep*

(b) (c*ep) = (c-I)*ep*

if

rk(ep) is even, x

if

^{k( )} . dd for any c ^E F . / r ep /s 0 ,

Proof Part (a) follows from (3.8), and (b) is easily checked.

o

We may extend the conjugation operation to elements of G(F, .1:'): for a E FX, rp E M(F, £), and m EN, we put

c

*

(ep - m x 1HI) = c

*

rp - m x 1HI.

We denote the inverse of an element ~ E G (F, t) by -~. (This should not be confused with the element (-I)*~.) 1ft ~ E G(F . .1:'), then ~ -~ is meantto be ~ +(

-no

Furthermore, with ~ E G(F, f) and n EN, we write n x ~ for the n-fold sum ~

+ ... +

~.

Proposition 4.9 Let ep be an I:'-form over F such that dl (ep) = 0 or dl (ep) = {- I}. Then rp = ep-I. Moreover,

if

^aI, ... , all E F X are such that ep = (al, ... , all)' then also ep

=

^{(all.al, ... ,all_I).}

Proof Let a), ... , all E F^X be such that cp

=

(a), ... , all)' The hypothesis on d) (cp) says that a) ... all = ±c for some c E Fxt. Since e-forms remain unchanged if one entry of a representing sequence is multiplied by an element of F x t, we may assume that c = I, whence a) ... all

= ±

I. If n is odd, then the claims now follow directly from (3.9). If n is even, then we apply (3.9) to cp .1 (1) instead and use that (I) lies in the center of G (F, i).

o Corollary 4.10 The kernel of d) : G(F, e) ~ Kit) F lies in the center ofG(F,

0.

Proof Let rp and VI be i-forms over F with d) (rp)

=

O. Then cp

+

VI

=

VI

+

cpO' with &

=

±I by (3.11) and cpi:

=

^cp by (4.9). Hence cp and

1/1

commute. 0

Corollary 4.11 Let cp be an e-form of even rank and trivial determinant over F. Then c²

*

^cp

=

cp for any c E F x .

Proof Applying (4.9) to cp and to c

*

cp yields

c*cp= (c*cp)-) =c-) *cp-) =c-) *cp,

whence c²

*

^cp = cp.

o

For any [-form cp over F, let

° {m x JH[

+

(c) if rk(cp)

=

^2m

+

I, cp = m x JH[

+

(I, c) if rk(cp) = 2m +2, where c E F^X is chosen such that d) (cp)

=

d) (cpO).

Proposition 4.12 Let ~ E G(F, e). To have ~ = cp - cpo for some e-form cp over F, it is necessary and sufficient that rk(O = 0 and d) (i;) = O.

Proof We write ~

=

rp - m x JH[ with i{J E M(F, i) and mEN. If rk(O

=

0 and d) (0

=

0, then rk(cp)

=

2m and d) (cp)

=

^{d) (m} x JH[)

=

((_I)IIl), so that cpo

=

^m x JH[ and ~

=

^{cp _ cpo.}

Conversely, for any i-form cp over F one has rk(cp - cpO) = 0 and d) (cp - cpO) = d) «(p) -

d) (cpO) = O. 0

Given x E JR, let LxJ denote the integral part of x, that is, the unique integer such that LxJ .::: x < LxJ

+

I.

Proposition 4.13 Letcp bean i-form over F and let m = L~(rk(cp)-l)J. There exist t-forms rJI, ... , rJlIl over F, each of rank 4 and with trivial determinant, such that

cp - cpo = (rJI - 2 x JH[)

+ . .. +

(rJlIl - 2 x JH[).

Proof We proceed by induction on m. If m

=

0, then cp

=

cpo and thus the statement holds trivially. Let now m > O. We may write cp = cp'

+

(x. y, z) for certain x, y, Z E F x and an i-form cp'. We put fJ

=

^{(x, y, Z, (xyz)-)) and}

1/1 =

^cp'

+

(-xyz). Since d) (fJ)

=

0, we obtain cp

+

JH[

=

cpl

+

fJ

+

(-xyz)

=

VI

+

lJ by (4.10). Since rk(V/)

=

rk(cp) - 2 and d) (cp) = d)

(1/1 +

JH[), we have cpo =

1/10 +

JH[. Thus using (4.10) yields

cp - cpo =

1/1 - 1/1 0 +

^{rJ -} 2 x JH[.

Now we apply the induction hypothesis to

1/1. o

5 Pfister forms

The discovery of the peculiar properties of quadratic forms that are tensor products of 2-dimensional quadratic forms was a breakthrough in quadratic form thcory over lields. If such a quadratic form in addition represents I, it is called a Pfisler form. We generalise this concept to the setting of virtual forms.

For a E F^X, we put ((a)) = (I, - a) and call this a I-fold Pfisterform (ofdegree /!). Given n > I and ai, ... ,all ^E F^X, we define recursively

(((I], ... , all)) = rp

+

^(al

*

^{cp)* = cp}

+

^al

*

^{cp* where cp}

=

((a2, ... , all)), and we call ((ai, ... , all)) an n-fo/d Pfisterform. This is an l-form of rank 2". Note that, for a ^E F^X, we have ((a))

=

(I)

+

(a

*

^{(1) *.}

LemmaS.1 Foranya,b E FX, we have ((a,b)) = (I, -a, -b, (ab)-I).

Proof We compute ((a, b)) = (I, -b)

+

(a-I, -ab)* = (I, -b, (ab)-I, -a) and then use

(4.9). 0

Question 5.2 Assume that a, b, c, d E F^X are such that {a, b}

=

{c, d} in

Kit)

F. Does it follow that thee-forms ((a, b)) and ((c, d)) coincide?

In fact, the converse holds, as we shall see in (7.4). For

e

= 2

:p

char(F), a positive answer to Question (5.2) is given by [3, Theorem 1.8.]. We will see in (7.13) that the answer is also positive whenever l is odd.

Proposition 5.3 Assume that a, b, c E F^X are such that {a, b}

((a, b)) = ((b, c)).

{b, c} in K~Cl F. Then

Proof Let a, b, c E F^X be such that {a, b}

=

{b, c}. By (4.9) and (4.5) we have (-a, -b, (ab)-I)

=

(-b, (ab)-I, -a)

=

(-b, -a-I, ab- I ). With (5.1) we conclude that

«(a, b))

=

«b, a-I )).

Using now that {a, b} = {b, e} = - {c, b} we obtain

{-a-I. b-Ia}

=

{-a, b}

=

-{-e, b}

=

{-c, b- I }

=

{-c, (be)-I).

Therefore (-a-I, b-Ia)

=

(-c, (bc)-I) and thus «b, a-I))

=

«b, c)).

o

Corollary 5.4 If a, b E F x are such that {a, b} = 0, then ((a, b)) = 2 x 1HI.

Proof This follows from (5.3), because {b, I}

=

^{{I, I} = O.}

o

Remark 5.5 Let a, b, c, d E F^X be such that {a, b}

=

{c, d} in

K?l

F. It is not known whether in this situation one can always find a chain of elements ai, ... ,all E F^X with al =a, a2 = b, all_1 = c, all = d and such that {ai-I, ai} = (ai, ai+d for I < i < n. For l = 2 and char(F)

:p

2 this is possible and one can achieve this with n = 5 [6, Chap. Y, Sect. 4]. If

e

^{= 3 and if F} contains a primitive 3rd root of unity (in particular char(F)

:p

^3),

then an analogous statement where n = 7 follows from [9, Corollary 2.2].

6 Subgroups

We are going to define a sequence of subgroups (G" (F. f))"EN of G (F, e) general ising the powers of the fundamental ideal in the Witt-Grothendieck ring of quadratic forms.

We put GO(F, £) = G(F, l). Forn > 0, we define G"(F, e) to be the group generated by thedifferences~ -c*~ where~ E G,,-I (F, e) and c E FX. Clearly G"(F,

0

is a subgroup of G,,-I (F. l).

Proposition 6.1 G I (F, f.) is equal to the kernel of rk : G (F, e) ----* Z.

Proof It is clear that every element of G I (F, l) has trivial rank. On the other hand, the kernel of rk is generated by the differences (a) - (I)

=

(a) - a

*

^{(a) with a E F x,} and these belong

to G^I (F, £). 0

Proposition 6.2 The groups G" (F, e) with n ~ 2 lie in the center of G (F, e).

Proof For ~ E GI(F,f) and c E FX, we have rk(~)

=

0, thus dl(c

* 0 =

dl(~) and dl (~-c*O = 0. By (4.10) the kernel ofdl : G(F, l) ---+ K~n F is contained in the center

of G(F.

0,

so the statement follows. 0

Lemma 6.3 Foranya,b,c E FX, one has

(a, b, c, (abc)-I) = ((-a, - b))

+

((-ab, -c)) - ((-I, -ab)). Proof We compute

(a. b, c, (abc)-I)

+

((-I, - ab))

=

^(1. a, b, (ab)-I)

+

(I, ab, c, (abc)-I)

=

((-a, -b))

+

((-ab, -c))

o

Theorem 6.4 G²(F, f) is equal to the kernel of dl : G¹ (F,

0

---+ K~t) F. Furthermore, G² (F, f) consists of all the differences cp - cpo with cp E M (F, e), and it is generated by the differences ((a, b)) - 2 x IHl with a, b E FX.

Proof We denote by H the kernel of dl : G ^I (F, f) ----* K~t) F. By (4.12), it consists of the differences cp - cpo with cp E M (F, f). By (4.13), H is generated by the differences /~ - 2 x IHl where tJ is an e-form over F of rank 4 and trivial determinant. Note that any such form can be written as ()

=

(x, y,

z,

(xYZ)-I) with x, y,

z

E FX, and then

() - 2 x IHl

=

«((-x, -y)) - 2 x 1Hl)

+

«((-xy, -z)) - 2 x 1Hl) - «((-I, -xy)) - 2 x 1Hl), by (6.3). Therefore H is already generated by the differences ((a, b)) - 2 x IHl with a, b E FX.

Since ((a, b))- 2xlHl

=

(I, - b)-a*(l. - b), it follows that H is contained in G²(F,

0.

On the other hand, it is clearthat G² (F, f) is contained in H, the kernel of d I : G I (F, e) ----* Kit) F.

o

Corollary 6.5 For any n ~ I, the group G" (F, e) is generated by the differences 7r -2,,-1 x IHl where 7r is an n~fold Pfister form.

Proof For n

=

1 this is easy to see and for n

=

2 the statement is contained in (6.4). We proceed by induction on n. Let n > 2. By definition, G" (F, f) is generated by the elements

~ - c

*

~ with ~ ^E G,,-I (F. ^f) and c ^E FX. We want to show that ~ - c

*

~ is a sum of

elements of the form ^{Jr: -} 2"-¹ X 1HI where ^Jr: is an n-fold Pfister form. Applying the induction hypothesis to ~ and using that C"-¹ (F, e) is commutative by (6.2), we can restrict to the case where $ = p - 2"-² X 1HI for some (n -I )-fold Pfister form p. We obtain

~ - c

*

~

=

p - c

*

^p

=

p

+

c

*

^{p* - 2"-1 X 1HI,}

and since p

+

c

*

p* is an n-fold Pfister form, this finishes the proof. o Question 6.6 Do we have n~o C" (F, e) = O?

In the case where

e

^{= 2}

ⁱ⁼

char(F), the Arason-Ptister Hauptsatz gives a positive answer to this question (cf. [6, Chap. X, Sect. 5]). There are other cases where the answer is positive, in fact for rather simple reasons.

Theorem 6.7 Assume that f is odd or that FX ₌ Fx2. Then C³(F, f) =

o.

Proof By hypothesis, F X / F xe is 2-divisible. Therefore C³(F, e) is generated by elements

~ - c2

*

~ with ^C E F X and ~ E C 2(F,

0 .

Now, for ~ E C 2(F,

0

there is rp E M(F, e) and m EN such that dl (rp)

=

0, rk(rp)

=

2m, and ~

=

rp - m x 1HI. Using (4.11), we obtain for any C E F^X that

and this finishes the proof.

o

Proposition 6.8 The commutator subgroup ofC(F, t) is generated by the elements of the shape ((a 2, b)) - ((I, I)) with a, b E FX.

Proof Let ~,I; E C (F,

0 .

We want to compute the commutator of ~ and 1;. We choose a, b E F^X such that dl

(0

= {a} and dl

(n

= (b) in Kit) F. Using (4.10), it follows that

~

==

^(a) and I;

==

(b) modulo the center of C(F,

0.

This yields

~

+

^{I; -} ~ -I; = (a)

+

(b) - (a) - (b)

=

(a, b, _a-I, - b- I ) - ((I, I)).

Furthermore, we have

(a, b, _a-I, - b- I )

=

(a, -a, a-2b, - b-I)

=

(I. _a 2, a- 2b, -b-I)

=

((a 2, b)).

Therefore ~

+

I; - ~ -I; = ((a², b)) - ((1, I)), and the statement follows.

o

Corollary 6.9 Iff is odd, then C 2(F, f) is the commutator subgroup ofC(F,

0.

Proof If

e

is odd, then the generators of C² (F, 0 given by (6.5) can all be written as

2 x 1+1

((a, b)) - ((I, I)) (for given c, b E F ,one may set a = c^T in order to have ((c, b)) =

«a

^{2,b»). 0}

Theorem 6.10

If

- 1 E FX e, then

e

^{x ~ = 0 for any ~ E C 1 (F,}

0 .

Proof Assuming that - I E Fxe , we have (-x)

=

(x) for any x E FX.

Let d E FX. By induction on i one obtains the equalities (di - 1, d) = (di , 1) and i x «(d) - (I)

=

(d^{i) -} (1) for all i ::: 1. In particular, for ~

=

(d) - (1) we have thate x $ = O.

Let a, b E F^X and i :::: I. Using (5.3) and (4.10), we compute ((ai, b»

+

((a, b» = ((b, (a i b)-I»

+

((a, b»

=

(1. I, b, a- i b- I , ai, a, b, (ab)-l)

=

(I, I, b, a-ib- I , a i + 1, I, b, (ab)-l)

=

(b,a-ib-I,I,ai+1,b,I,I, (ab)-I)

=

(b, a-i b-1,l, a i + 1, b, (ai+1b)-I, ai+1b, (ab)-I)

=

(b, a-ib- 1)

+

((ai+l, b»

+

(ai+1b, (ab)-I)

=

((a i +^l, b»

+

(b, a-i b- 1, a i+ 1 b, (ab)-I).

Since (a- i b- 1, ai+1b) = (a- i b- 1, a) = (a, b)

=

^{(b-1, ab)} in Kin F, we have further that (a-ib- I, ai+1b)

=

(b- 1, ab). Therefore we obtain

((ai, b»

+

((a, b»

=

((a i+ 1 b»

+

(b, b- 1, ab, (ab)-l) = ((a i + 1, b»

+

((I, I».

By induction on i we deduce that

i x «((a, b» - ((I, I») = ((ai, b» - ((I, 1».

Since ((at, b» = ((1, I», iffollows for ~ = «((a, b» - (( I, 1 ») thatf x ~ = O.

Since G²(F,f.) is an abelian group, generated by elements «((a, b» - ((I, I») with a, b E FX, we obtain that f. x ~ = 0 holds for any ~ E G²(F, f.).

Let now ~ ^E G¹ (F, f). Let d E F X be such that dl

(0

= (d) in Kif) F and

S

= ~ - (d)

+

(I) ^E G²(F,

0.

By what we have shown,

e

x

S

= 0 and

e

x «(d) - (I» = O.

Since

s

lies in the center of G(F, 0, it follows that £ x ~

=

£ x (I;

+

(d) - (I»

=

O. 0 Let R(t) F denote the subgroup of F X consisting of the elements x E F X satisfying (x, y) = 0 in Kyl F for any y E FX. Note that F^xt C R(C) F. Let ^f?(e) F denote the cor- responding subgroup of Kif.) F, consisting of the symbols (r) with r E R(e) F. In the case where char(F)

i=

2, the group ^R(2l F was introduced by Kaplansky [5] and called 'the radical of F' and denoted by R(F).

Proposition 6.11

If

£ is odd, then the center of G (F, £) is equal to the pre image of

f?(t)

F

d d . (e)

un er 1. G(F, C) ---+ Kl F.

Proof Let ~ E G(F, e). Let d E F X be such that dl

(0

= (d). Then ~

==

(d) modulo the center of G (F, f). Thus ~ is in the center of G (F, £) if and only if the identity (d, e) = (e, d) holds for every e ^E FX. But for any e ^E F X we have the equivalences

(d, e)

=

(e, d) <=? (d, e) = (e, d) <=? 2· (d, e) = 0 <=? (d, e) = O.

Therefore ~ belongs to the center of G (F, £) if and only if d ^E R(f) F.

o

7 Delzant classes

Stiefel-Whitney classes of quadratic forms were defined by Delzant [2] and later adapted by Milnor [8] in such a way that these maps take their values in K -groups instead of Galois cohomology groups. Here they will be generalised to what shall be named Delzant classes.

They are designed as a tool to perform computations in Milnor K -theory.

We denote by U (F, £) the multiplicative group of the ring

Ki

^e) F and by U I (F, £) the subgroup consisting of the elements with constant term equal to I (in Kcil'.) F = Z/ £Z). By (4.6), there is a unique group homomorphism

d: C(F,f.) ~ UI(F,e)

that maps (x) to I + (x), for any x E FX. We call d the Delzant homomorphism (modulo

0.

For any n E N let _dll : C(F, e) ~ K,\1'.l F be the composition of d with the projection from

Kie)

F to the nth component K,\fJ F. In other terms, for ~ ^E C(F, e) we write

00

d(O = Ldll(O 11=0

with _dll

(0

E K,\e) F (n EN); we call _dll

(0

the 11th Delzant class of ~. Note that do C(F, e) ~ Kci

e )

F is the constant map I and dl : C(F, £) ---* K~e) F is the determinant homomorphism defined earlier.

The interest of these maps in the study of K -groups lies in the fact that K,\fl F is generated by the image of _dll : C F, e) ---* K,\ fI F. Indeed K,\ t) F (n ::: I) is generated by symbols and any symbol (ai, ... ,all) is equal to the nth Delzant class of the i-form (ai, ... ,all)' Proposition 7.1 For any ep ^E M (F, C) and n > rk(ep), one has dn (rp) = O.

PIVof Let ep be an £-form of rank rover F. With ai, ... , a,. E F X such that rp = (ai, ... , a,.), we see thatd(rp) = (I + (ad) ... (I + (a,.)) lies in ffi;'=oK?) F, whencedll (rp) = 0 for n > r.

o

Proposition 7.2 For any ~I, ... ,~k E C(F, e) one has

dll(~I+"'+~d= L (dil(~I) .. ·dik(~k)).

iI, .... ik ~~.o

il+'+h=1I

Proof This is straightforward from the definitions of d and d_n (n ::: I).

Lemma 7.3 Leta,b E FX. Thend«(I, I))-«a,b)))

=

1- (a,b).

o

PIVofOne has d«(a,b)))

=

1+(a,b)+(- I, -I}andd«(1,I)))

=

1+ (- 1, -1)

=

(I - (a,b)) · d«(a,b))). The statement follows, because d : C(F,£) ~ UI(F,£) is a

group homomorphism. o

Proposition 7.4 The restricted map d2 : C2(F, e) ---* Ky) F is a surjective homomor- phism. Foranya,b E FX, it maps «a, b)) - «1.1)) to (a, b).

Proof For any ~,~ E C (F, e) we obtain from (7.2) that

d2(~ +

n

= d2(~) + dl (~)dl

(n

+ d2U;)·

The term dl (Odl

(n

vanishes if any of the elements ~ and ~ belongs to C 2(F, e). There- fore d2 : C 2(F, £) ---* Kie) F is a homomorphism. It follows from (7.3) that d2«(a, b)) -

«(1, I))) = (a, b) for any a, b E FX. Using (6.5), the surjectivity is now obvious. 0

Corollary 7.S One has C 2(F, £) = 0 ifand only if KJe) F = 0, and in this case C(F,e) is commutative and C^I (F, £) is isomofphic to FX/Fxe.

Proof If C²(F, e)

=

^{0, then} Kif) F

=

^{0 by} (7.4). Conversely, assume that Kin F vanishes.

Then (5.4) and (6.4) together imply that

C

²(F, t) = O. In this casedl : C l (F. e) ~

Ki n

F is an isomorphism, whence Cl _{(F. f)} ~

K: n

F ~ FX/Fxe, and using (6.8) it follows that

C(F. l) is commutative. 0

The next aim is to show that C³(F, t) is equal to the kernel of the homomorphism

d2 : C²(F,e) ~ Kif.) F.

Lemma 7.6 Let a, b, c ^E FX. Then

((a, b, c)) = ((a, c))

+

((b, c)) - ((ab, c))

+

2 x IHL Proof Since ((a, b, c)) = ((b, c)) +a

*

((b, c))*, the statement follows from

a

*

((b, c))*

=

(-a-I be, ac- I , (ab)-I, -a) (4J) (a-Ie. -(ab)c-I, (ab)- I. -a) (4J) (-a, a-Ie, -(ab)c-I, (ab)- I)

(-a, a-Ie)

+

^{(-c-I , c)}

+

(-(ab)c- I, (ab)-I) -1HI (l, - a, a-Ie. - c- I )

+

(c, - (ab)c- I, (ab)-I , - 1) - 2 x 1HI ((c-^I. a))

+

((c-^I ,ab))* - 2 x 1HI

((a, c)) - ((c-^I ,ab))

+

2 x 1HI ((a, c)) - ((ab, c))

+

2 x 1HI.

Corollary 7.7 For any x, y, Z E FX, the following congruences modulo C³(F, l) hold:

((xy, z))

+

2 x 1HI

==

((x, z))

+

((y, z)) ((x, yz))

+

2 x 1HI

==

((x, y))

+

((x, z))

o

Proof The first conglUence immediately follows from (7.6). For the second, one uses that

((x, yz)) = ((yz, [ I)) by (5.3). 0

Lemma 7.8 For any a, b, c E FX and n ~ lone has

d" «((a, b, c)) - 4 x 1HI) = {

b

^{_l}"-3 . (a, b, c)}

^{if 41}

^n,

otherwise.

Proof For any n ~ I, let s" denote the term on the right hand side in the claimed equality. It is easily checked that

( I

+ I:

^{s,,). (1 - {ab.c}}

⁺

^{(- l,a,b,c)) = I - {ab,c}.}

,,=1

Using (7.6) and (7.3) we obtain that

d «((a, b, c)) - 4 x 1HI) = (1 - {a, c))-I . (1 - {b. c))-I . (1 - (ab, c))

=

^{(1 - (ab, c)}

+

{- I. a, b, c})-I . (1 - lab, c)).

Therefore d«((a, b, c)) - 4 x 1HI)

=

^(I

+

L~I s,,).

o

Theorem 7.9 C³ (F, f.) is the kernel of d2 :

c

² (F, f.) ~ Kit) F.

Proof By (7.4), the restriction of d2 to C²(F, f.) is a homomorphism. It follows from (6.5) and (7.8) that d2 is trivial on C³(F,0. Hence we obtain a homomorphism d2 : C2(F, e)/C³(F,f.) ~ KiflF, which maps the difference ((a, b)) - ((1. 1)) to the symbol (a. bl.

Now we consider the pairing F^X x F^X ~ C²(F. e)/c³(F, e) that associates to a pair (a, b) the class of ((a, b)) - ((1, 1 )). Since ((a, b)) = (( 1, J)) whenever (a. bl = 0, this induces a homomorphism Kit) F ~ ^{C2(F, f.)/C3(F, f.)} that is inverse to d2. Thus d2 is

an isomorphism. 0

Remark 7.10 Combining (6.4) and (7.9), one obtains an exact sequence

- I

o ~

KylF

(~

C^I(F,e)/C³(F.e)

~

KltlF

~

O.

This group extension of Kin F by Kit) F corresponds to the trivial operation of Kin F on Kif) F together with the 2-cocycle Kif) F x Kif) F ~ Kii') F given by the multiplication in K!fl F.

Corollary 7.11

If

- I ^E F^X t, then the kernel of d : C ^I (F, e) ~ v I (F, e) is equal to

c

^{3(F, f.).}

Proof The kernel of d : C ^I (F, f.) ~ V ^I (F, f.) is contained in the kernel of d2 : C2(F,e) ~ Kit)F, whence in C³(F, f.), by (7.9). Assume now that - J ^E F^xf,

i.e. (-J

I

= 0 in Kit) F. Since (-I

I

= 0 in Kin F, (7.8) yields that d(ll' - 4 x 1Hl) = I for any 3-fold Pfister form ll' over F. Using (6.5), this implies that d is trivial on C³ (F. e). 0 Corollary 7.12 The homomorphism d : C ^I (F, e) ~ V ^I (F, e) is injective in each of the following cases:

(1)

e

^{is odd;}

(2) F^X = F^x2 and

e

^> 0;

(3) char(F) = 2 and F is perfect.

Proof In any of these cases we have - J ^E F x

e ,

so that the previous corollary applies. Hence the kernel of d : C^I (F, e) ~ VI (F, f.) is equal to C³(F. f.). The statement then follows,

because C³(F, f.) = 0 by (6.7). 0

Remark 7.13 If d : C I (F. e) ~ V ^I (F. f.) is injective, then we have immediately a positive answer to Question (5.2).

Proposition 7.14 The homomorphism d : C ^I (F, e) ~ V I (F. f.) is surjective if and only ifKjiJ F = O.

Proof Given a symbol ^(J in K~n F, it is easy to see that 1

+

^(J lies in the image of d if and only if (J = O. This shows that the condition in the statement is necessary.

Assume now that KjeJ F = O. The elements of V ^I (F, e) then are of the shape I

+

^ex.

+

^f3

with ex. E KiflFandf3 E Kie)F.Letsuchex.andf3begiven.Thereisanelement~ ^E C²(F,f.) such that

f3

= d2(n Let dE F^X be such that ex. = (dl in Kif) F. For ~ = ~

+

(d) - (1) E C^I (F, e), we obtain

den = d(~) . d«(d) - (1) = (1

+

f3) . (I

+

ex.) = 1

+

ex.

+

f3,

since

fi .

ex. E Kje) F = O. Therefore the condition is sufficient.

o

Question 7.15

If Kyl

F = 0, is then d :

e

^l (F, e) ---+

U

^l (F, £) an isomorphism? We finish this section with a result. on correlations between the different Dclzanl classes of a form. It generalises [8, Remark 3.4].

Theorem 7.16 Let n be a positive integer. Let n I > ... > nk be the decreasing sequence of 2-powers such that n

=

^{n I}

+ .. . +

nk. Then for any i-form ep over F one has the equality

d" (ep)

=

d"l (ep) .. . d" k (ep ) .

Proof We put d;~(ep)

=

d"l (ep) ... d"k (ep) and want to prove the equality d,~(ep)

=

d" (cp). If rk(cp) = 1 this is trivial. We proceed by induction on rk(cp). Assume that rk(ep) > I and write cp = ljI

+

(x). For any m :::: 1, we have by the induction hypothesis

d,;, (ljI) = dill (ljI).

and we know further from (7.1) and (7.2) that

We compute

dm (cp)

=

^dill (ljI)

+

dlll_1 ("if I) . (xl·

d,,(cp)

=

d,,(V/)

+

d,,_1 (ljI) . (xl

= d,;(V/)

+

d'~_1 (ljI) . {xl

= d';-"k (1/1) . (d"k (ljI)

+

_d,7k-1 (ljI) . (xl)

= d,7-"k (ljI). (d"k(ljI)

+

d"k-I (ljI) . (xl)

= d,7-"k (VI) . d"k (cp)

= d"l (VI)'" d"k_1 (1/1) . d"k (ep).

To finish the proof, we shall show for 1

:s

^{i < k that}

d"j (VI) . d"k (cp) = d"j (ep) . dllk (cp).

Since ni is a 2-power greater than nk, the element ~ = d';j _ I (ljI) . {x

I

in K~O F is a multiple of dllk-I (1/1) . {xl as well as of _{dllk (VI)}. Using that (2 = {- I 1111 . ( holds for any ( E

K,\[-l F

and m :::: 0, it follows that

d';~_1 (ljI) . {xl' ~ = (- ll"k . ~ = d"k(ljI) ·~·

Since d,7k-1 (1/1) = d"k-I (1/1), we obtain that

d"k (cp) . ~ = (d"k (VI)

+

dllk-I (VI) . (xl) . ~ = O. Since further ~ = d"j _ I (ljI) . (x), we conclude that

d"j (cp) . d"k (cp) = (dllj (ljI)

+

^d"j_1 (ljI) . (x

I) .

dllk (cp) = d"j (ljI) . d"k (cp), which is what we claimed above.

8 Vanishing of higher K -groups

o

To illustrate how virtual forms and Delzant classes can be used for computations in Milnor K -theory, we give an alternative proof of a result in [4, Sect. 3] relating the 'symbol length' of KJfl F to the vanishing of higher K -groups modulo i under the hypothesis that - I E F x e.