Algebraic K-theory of Special Groups
M. Dickmann
F. Miraglia
November, 2003
Abstract
Following the introduction of an algebraic K-theory of special groups in [6], generalizing
Milnor’s mod 2 K-theory for fields, the aim of this paper is to compute the K-theory of Boolean
algebras, inductive limits, finite products, extensions, SG-sums and (finitely) filtered Boolean
powers of special groups. A parallel theme is the preservation by these constructions of property
[SMC], an analog for the K-theory of special groups of the property “multiplication by l(−1) is
injective” in Milnor’s mod 2 K-theory (see [20]).
2000 Mathematics Subject Classification: 11E81, 11E70, 12D15, 06E99
1
Introduction
In [6] we introduced an algebraic K-theory of special groups. Special groups ([5]) constitute an
axiomatic, first-order approach to quadratic form theory, equivalent to the (non first-order) theories
of abstract Witt rings ([17]) and —in the reduced case— to that of abstract order spaces ([18]),
both due to Marshall. The main (and motivating) class of examples are the special groups arising
from fields of characteristic 6= 2, especially from formally real fields. However, the abovementioned
structures arise naturally in a variety of other contexts, such as semilocal rings, skewfields and
*-domains. In the case of fields of characteristic 6= 2, our K-theory reduces to Milnor’s mod 2
K-theory, [20].
The K-theory of special groups was initially developed for, and instrumental in, solving Lam’s
conjecture ([12], Open Problem B) in the very general form presented in [6]. Further, in [4] we
established for Pythagorean, formally real fields, and in [6] for preorders in an arbitrary formally
real field, the validity of a powerful K-theoretic property —the [SMC] property— which, in the first
case implies Marshall’s signature conjecture, and in the second case the general version of Lam’s
conjecture alluded to above. Indeed, these results follow from the general fact that the [SMC]
property implies Marshall’s signature conjecture for arbitrary reduced special groups ([6], Prop.
4.4).
The category of special groups has a considerably better functorial behavior than that of fields,
and it is possible to perform on special groups certain general constructions with good functorial properties, such as inductive limits, products, profinite limits, extensions, quotients modulo
saturated subgroups, certain kinds of sums, and (finitely) filtered Boolean powers.
The main aim of this paper is to explicitly compute the K-theory of most of these constructions.
We shall prove that our K-theory functor commutes with inductive limits, finite products and the
operation “C” (continuous functions on a Boolean space with values in a discrete special group),
the simplest case of filtered Boolean powers.
For extensions, SG-sums, and filtered Boolean powers in general, the result of applying the
K-theory functor does not have such a straightforward description. Even so, for each of these
constructions we obtain an explicit description of their K-theory in terms of that of its components.
The case of an extension G[∆] of a special group G by a group ∆ of exponent 2 is of particular
1
interest, since it comprises, as is well-known, the example of a special group of a field with a
compatible valuation (see [5], Thm. 1.33, p. 26). In this case we give explicit formulas expressing
the K-theory of the extension G[∆] in terms of that of G and the F2 -dimension of ∆, for finite or
countable ∆ (Theorem 6.11).
We register that in section 6 of [6] we have proved results similar to those described above for
quotients of special groups modulo saturated subgroups.
The only known constructions of special groups for which we have not been able to determine
the K-theory are arbitrary infinite products and projective limits of finite special groups. However,
we do show that the K-theory functor commutes with arbitrary powers of a finite special groups
(Corollary 7.10).
In parallel to the work just summarized, we are also interested in the preservation of the
K-theoretic property [SMC], which is the analog, in the present context, of the injectivity of
the map “multiplication by l(−1)” in Milnor’s mod 2 K-theory of fields. We show that all the
abovementioned constructions of special groups —again, with the possible exception of arbitrary
products and profinite limits— do preserve the property [SMC]. Certain results involving this
property, e.g. Proposition 2.6, have interesting consequences in the case of fields; see Corollary 2.7.
A word is in order concerning certain techniques employed in this paper. We shall use repeatedly
the Boolean hull of a reduced special group, a construction introduced in Chapter 4 of [5]. Its good
functorial properties, together with the simplification which stems from performing calculations in
a Boolean algebra, make it a powerful tool in K-theoretic computations, as illustrated by the proofs
of 2.6 and 2.10.
For the reader’s convenience, we include in section 2 the definition and the statement of some
basic properties of the K-theory of special groups proven in [6], as well as the following observations
on standard notions and notation for special groups.
1.1 List of references for special groups. The basic reference is the monograph [5] where
the notion of special group (SG) is introduced, which also gives a comprehensive account of the
ensuing theory. Extensive summaries may also be found in section 2 of [4] and in section 1 of [6].
For ready reference we note:
a) The basic notions of quadratic form theory over SGs, the notion of SG-morphism, and some
of the basic examples and constructions can be found in Ch. 1, section 1 of [5]; for the notion of
SG-character, see [5], 2.29, p. 46. See also [4], section 2, pp. 245–248, or [6], pp. 152–153.
b) For the definition of the (various) SGs associated to a field, see [5], Ch. 1, section 3.
c) The notion of a saturated subgroup of a SG and quotients thereof is presented and studied in
[5], Ch. 2, sections 2 and 3. For its relation with preorders in the case of SGs of formally real fields,
cf. [6], Lemma 1.4, p. 154. For the notion of formally real SGs see [5], Ch. 3, p. 50. A formally
real SG, G, has a largest reduced quotient, written Gred (see Def. 2.25, p. 44, [5]).
d) Boolean algebras (BA) as reduced special groups (RSG) are dealt with in Ch. 4, section 1, of
[5]. In section 2 of that same chapter the reader will find the construction of the Boolean hull,
BG , of a RSG, G, and the definition of the canonical embedding εG : G −→ BG . The universal
property of the Boolean hull functor, repeatedly used below, is Thm. 4.17(2), p. 71, of [5]. See
also [4], 2.3(c), pp. 248–249. Write ⊥ (bottom) and > (top) for the smallest and largest element
of BA, respectively.
e) Various types of embeddings of SGs − complete, isotropy-reflecting, pure, etc. − are defined
and studied in Ch. 5 of [5].
f) Below we use the fact that any BA, B, has a natural graded ring structure, namely
(F2 , B, B, . . . , B, . . . ),
with B considered as a group under symmetric difference, 4, and graded multiplication given, for
2
n, m ≥ 1, by the meet operation : for x ∈ Bn , y ∈ Bm , set x · y = x ∧ y ∈ Bn+m ; moreover, for
x ∈ Bn = B, 0 · x = ⊥ and 1 · x = x, while multiplication of elements of degree 0 is that in F2 .
Since −1B = > (cf. [5], Def. 4.1, p. 60), multiplication by λ(−1B ) is just the identity map on B:
λ(−1B ) · x = > ∧ x = x.
3
2
The K-theory of a special group
Let G be a special group, written multiplicatively. Let k1 G be G written additively, that is, we
fix an isomorphism
λ : G −→ k1 G,
such that λ(ab) = λ(a) + λ(b), for all a, b ∈ G.
Note that λ(1) is the zero of k1 G and that k1 G has exponent 2, i.e., λ(a) = − λ(a), for a ∈ G.
Define the K-theory of G as the F2 -algebra
k∗ G = (F2 , k1 G, . . . , kn G, . . . )
obtained as the quotient of the graded tensor algebra over F2
(F2 , k1 G, . . . , k1 G ⊗ . . . ⊗ k1 G, . . . ),
|
{z
}
n times
by the ideal generated by {λ(a)λ(ab) : a ∈ DG (1, b)}. Thus, for each n ≥ 2, kn G is the quotient
of the n-fold tensor product k1 G ⊗ . . . ⊗ k1 G over F2 , by the subgroup consisting of finite sums of
elements of the type λ(a1 )λ(a2 ) . . . λ(an ), where for some 1 ≤ i ≤ n − 1 and some b ∈ G we have
ai+1 = ai b and ai ∈ DG (1, b).
It is readily verified that for all n ≥ 1 and all η ∈ kn G, η + η = 0, that is, kn G is a group of
exponent 2. Note that for all a, b ∈ G, λ(a)λ(ab) = λ(a)2 + λ(a)λ(b). Thus,
a ∈ DG (1, b) ⇒ In k2 G, λ(a)λ(ab) = 0 (equivalently, λ(a)2 = λ(a)λ(b)).
The next result shows that some of the basic relations of classical mod 2 K-theory hold in k∗ G.
Proposition 2.1 (Prop. 2.1 and Cor. 2.2, [6]) Let G be a SG and let a, b, a1 , . . . , an be elements
in G. Let σ be a permutation of {1, . . . , n}.
a) In k2 G, λ(a)2 = λ(a)λ(− 1). Hence, in km G, λ(a)m = λ(a)λ(− 1)m−1 , m ≥ 2.
b) In k2 G, λ(a)λ(− a) = 0.
c) In kn G, λ(a1 )λ(a2 ) . . . λ(an ) = λ(aσ(1) )λ(aσ(2) ) . . . λ(aσ(n) ).
d) For n ≥ 1 and ξ ∈ kn G, we have ξ 2 = λ(− 1 )n ξ.
e) (Thm. 2.10, [6]) If G is a RSG and x, y, a1 , . . . , an are elements of G, then
x ∈ DG (1, y) and λ(y)λ(a1 ) . . . λ(an ) = 0 imply λ(x)λ(a1 ) . . . λ(an ) = 0.
3
An element a of a special group G induces a graded ring homomorphism of degree 1,
ω a = {ω an : n ≥ 0} : k∗ G −→ k∗ G,
where ω an : kn G −→ kn+1 G is multiplication by λ(a); write ω = {ω n : n ≥ 0} for the graded
homomorphism ω −1 = {ω −1
n : n ≥ 0} associated with multiplication by λ(− 1 ).
The following result, relates the K-theory of a reduced special group with its Boolean hull.
Theorem 2.2 (Thm. 2.7, [6]) Let G be a reduced special group and let εG : G −→ BG be its
Boolean hull. Then, there is a graded ring homomorphism (εG )∗ = ε∗ : k∗ G −→ BG , defined on
generators by the rule
εn (λ(a1 )λ(a2 ) . . . λ(an )) = εG (a1 ) ∧ εG (a2 ) ∧ . . . ∧ εG (an ).
Moreover,
3
a) For all n ≥ 1, λ(− 1 )n = > (− 1 of BG ) and λ(1)n = ⊥ (1 of BG ).
b) Let BG (n) be the subgroup of BG generated (with respect to symmetric difference) by
{εG (a1 ) ∧ . . . ∧ εG (an ) : {a1 , . . . , an } ⊆ G}. Then, BG (n) = Im εn .
3
ε
If G is a formally real special group, let Gred −→ BGred be the Boolean hull of Gred and let
π
ε
0
εG = ε0 : G −→ BGred be the composition G −→ Gred −→ BGred , with π the canonical quotient
map.
Corollary 2.3 (Cor. 2.8, [6]) Let G be a formally real special group.
a) There is a graded ring homomorphism π ∗ : k∗ G −→ k∗ Gred , sending λ(a1 ) . . . λ(an ) to
λ(π(a1 )) . . . λ(π(an )).
b) There is a graded ring homomorphism (ε0G )∗ = ε0∗ = {ε0n : n ≥ 1} : k∗ G −→ BGred , defined on
generators by the rule
ε0n (λ(a1 ) . . . λ(an )) = ε0 (a1 ) ∧ . . . ∧ ε0 (an ).
In particular, for all n ≥ 0, ε0n (λ(− 1 )n ) = > in BGred . Moreover, for η ∈ kn G and ζ ∈ km G
(1) ε0n+m (λ(− 1 )m η + λ(− 1 )n ζ) = ε0n (η) 4 ε0m (ζ)
3
(2) ε0n+m (ηζ) = ε0n (η) ∧ ε0m (ζ).
Definition 2.4 If G is a SG, recall that XG is the space of characters of G and I(G) is the ideal
of classes of even dimensional forms under Witt-equivalence, the fundamental ideal of G.
a) A reduced special group G is [MC] if it satisfies
For all n ≥ 1 and all forms ϕ over G,
[MC]
∀ σ ∈ XG , σ(ϕ) ≡ 0 mod 2n implies
ϕ ∈ I n G.
b) A reduced special group is [SMC] if it satisfies
[SMC]
For all n ≥ 1, multiplication by λ(− 1 ) is an injection from kn G into kn+1 G.
3
In Proposition 4.4 of [6] it is shown that every [SMC] group is [MC]. Property [SMC] generalizes
Marshall’s problem in [16], Open question 2, p. 575 − originally stated for abstract order spaces
− whether every reduced special group satisfies [MC]. In [4] and [6], respectively, the authors have
shown that the special groups associated to a Pythagorean field and to a preorder on any formally
real field are [SMC]. Hence, we shall be interested in the preservation of this property by the
constructions to be discussed in later sections.
A useful criterion for a reduced special group to be [SMC] is given in
Proposition 2.5 (Prop. 6.6, [6]) Let G be a reduced special group. With notation as in 2.2, the
following conditions are equivalent :
(1) G satisfies [SMC];
(2) For all n ≥ 1, εn : kn G −→ BG is an injection.
Hence, if G satisfies [SMC], εn is an isomorphism between kn G and the subgroup BG (n) of BG , for
all n ≥ 1.
3
As an application of the preceding results we give a criterion for the graded ring homomorphism
induced by a SG-morphism to be a monomorphism, i.e., injective in all degrees.
Proposition 2.6 Let G be a formally real SG and f : H −→ G be a complete embedding. If H
is [SMC], then f∗ is a graded ring monomorphism, that is, for all n ≥ 0, fn is injective.
Proof : Recall that a [SMC] group is necessarily reduced; see Lemma 6.2 in [6].
4
Define a SG-morphism fred : H −→ Gred by fred = π ◦ f , where π : G −→ Gred = G/Sat(G)
is the canonical quotient map (cf. [5], Definition 2.25, p. 44).
Claim. fred is a complete embedding.
Proof of Claim. We must show that is ϕ, ψ are forms over H,
fred ? ϕ ≡Gred fred ? ψ
⇒
ϕ ≡H ψ .
The antecedent of this implication amounts to π ? (f ? ϕ) ≡Gred π ? (f ? ψ ). Since Sat(G)
S
= k≥0 DG (2k h 1 i), by Proposition 2.21 and 2.28 of [5], there is an integer k ≥ 0 so that
(f ? ϕ) ⊗ 2k h 1 i ≡G (f ? ψ ) ⊗ 2k h 1 i, or equivalently,
f ? (ϕ ⊗ 2k h 1 i) ≡G f ? (ψ ⊗ 2k h 1 i).
Since f is a complete embedding, it follows that ϕ ⊗ 2k h 1 i ≡H ψ ⊗ 2k h 1 i and, H being reduced,
this in turn implies ϕ ≡H ψ ([5], Proposition 1.6(e)), establishing the Claim.
Next we show that for all n ≥ 1 the square
fn kn G
kn H
ε0n = (ε0G )n
(εH )n = εn
?
?
- BG
red
BH
B(fred )
is commutative, where the maps εn and ε0n are defined in Theorem 2.2 and Corollary 2.3, and
B(·) is the morphism associated to the Boolean hull functor ([5], Chapter 4). Indeed, let η =
λ(a1 ) . . . λ(an ) (a1 , . . . , an ∈ H) be a generator of kn H. Then,
(
= εGred (π(f (a1 ))) ∧ . . . ∧ εGred (π(f (an )));
ε0n ◦ fn (η)
B(fred )◦ εn (η) = (B(fred ) ◦ εn )(a1 ) ∧ . . . ∧ (B(fred ) ◦ εn )(an ),
as B(fred ) is a Boolean morphism. By Theorem 4.17(2) in [5]
B(fred ) ◦ εH = εGred ◦ fred = εGred ◦ π ◦ f ,
proving the commutativity of the diagram above.
Since fred is a complete embedding of RSGs, B(fred ) is injective (Theorem 5.2, [5]). For n ≥ 0
and η ∈ kn H, suppose that fn (η) = 0 in kn G. Then,
B(fred )(εn (η)) = ε0n (fn (η)) = ⊥.
The injectivity of B(fred ) implies that εn (η) = ⊥ in BH . Now, the fact that H is [SMC] and 2.5
guarantee that η = 0 in kn H, as needed.
3
In the proof of Theorem 6.9 of [6] we show that any reduced quotient of the special group of
a formally real field is [SMC]. In particular, the special group of a Pythagorean field is [SMC] (cf.
also Cor. 6.5, [4]). Moreover, by Theorem 2.5 in [6], the Milnor mod 2 K-theory of a field F of
characteristic 6= 2 is isomorphic to the K-theory of the special group of F . Taking into account
that the characters of the special group associated to a formally real field L are exactly the orders
of L, Lemmas 1.3 and 1.4 in [6], Proposition 5.3 of [5] and Proposition 2.6 yield :
Corollary 2.7 Let F ⊆ K be formally real fields.
a) If F is Pythagorean and all orders of F extend to K, then k∗ F is a graded subring of k∗ K, that
is, the graded ring homomorphism induced by the canonical inclusion ι : F −→ K is injective in
all degrees.
b) Let T be a preorder on F and assume that all orders of F extending T extend to K. Then,
5
k∗ GT (F ) is a graded subring of k∗ GT 0 (K), where GT (F ) is the RSG associated to the preorder T
of F , and T 0 = Σ (T ·K 2 ) is the preorder of K generated by T .
In particular, if T = Σ F 2 ,
c) If all orders of F extend to K, then k∗ Gred (F ) is a graded subring of k∗ K.
3
Example 2.8 If K is the Pythagorean closure of F , or the order closure of F ([3]), then k∗ Gred (F )
is a graded subring of k∗ K.
3
We end this section with a description of the K-theory of Boolean algebras and fans. In the
case of Boolean algebras, we state the pertinent result, but its proof − although it may be obtained
directly −, will follow from the more general Theorem 7.9 (Corollary 7.11). Keeping in mind that
Id
the Boolean hull of a BA, B, is the identity map, B −→ B, we have:
Theorem 2.9 If B is a Boolean algebra, the homomorphism Id∗ : k∗ B −→ B of 2.2 is an
isomorphism, that is, for each n ≥ 1, the map Idn defined on generators by
λ(a1 )λ(a2 ) . . . λ(an ) 7→ a1 ∧ a2 ∧ . . . ∧ an ,
is a group isomorphism from kn B onto B. In particular, B is a [SMC]-group.
3
A description of the K-theory of a fan was given, under a different terminology, by Elman and
Lam ([8], Thm. 5.13.(2)). Recall from Example 1.7 and Lemma 1.8 in [5] that a special group G
is a fan if 1 6= − 1 and the representation relation in G verifies
(
G
if a = − 1
D(1, a) =
{1, a} otherwise.
Fans are reduced special groups; cf. [13] or [5].
Proposition 2.10 Let G be a fan and let T ∪ {− 1 } be a basis for G over F2 . If n ≥ 1 is an
integer, write 2Tn for the family of subsets of T of cardinal at most n. For α ∈ T , let c(α) be the
cardinal of α. Then, for each n ≥ 1, kn G is the group of exponent two freely generated by the set
Q
β n = {λ(−1)n−c(α) a∈α λ(a) : α ∈ 2Tn }.
Proof : We first show that every element of kn G is a sum of elements of β n . It is enough to verify
this for a generator η = λ(c1 ) · · · λ(cn ). Since T ∪Q
{− 1 } is a basis for G, for each
P 1 ≤ i ≤ in there is
i , and so λ(c ) =
a finite subset Ti of T ∪ {− 1 } such that ci =
d
i
di ∈Ti
di ∈Ti λ(d ). Hence,
P
P
1
n
η = λ(c1 ) . . . λ(cn ) =
d1 ∈T1 λ(d ) · · ·
dn ∈Tn λ(d )
P
=
hx1 ,...,xn i∈Πn Ti λ(x1 ) . . . λ(xn ).
i=1
In each term in this last sum, let {a1 , . . . , ar } be the set of distinct elements of T among the xi ;
then 2.1(d) yields
λ(x1 ) · · · λ(xn ) = λ(−1)n−r λ(a1 ) · · · λ(ar )
and η is a sum of elements in β n . It remains to see that
P β n is linearly independent over F2 .
T
Let F 6= ∅ be a finite subset of 2n and assume that
α∈F ξα = 0 in kn G, where for α ∈
F , ξα = λ(−1)n−c(α) Πa∈α λ(a). Let F 0 be the set of non-empty elements in F . Recalling that
εn (λ(−1)) = >, by Theorem 2.2 we obtain
(
V
4α∈F 0 a∈α εG (a)
if ∅ 6∈ F
P
⊥ =
ε
(ξ
)
=
(I)
n
α
V
α∈F
> 4 4α∈F 0 a∈α εG (a) if ∅ ∈ F
holds in the Boolean hull BG of G. It follows from (I) that in BG we have
6
(
4α∈F 0
V
a∈α
εG (a) =
⊥
if ∅ 6∈ F
>
if ∅ ∈ F .
(II)
By Proposition 5.19.(c) in [5], BG is the free Boolean algebra having {εG (a) : a ∈ T } as a set of
free generators. Hence, relation (II) cannot hold true in BG . Since G is [SMC], εn is injective and
so no sum of elements of β n can be zero, as desired.
3
3
qSG-morphisms
We introduce a new concept of morphism of special groups that will be useful in the sequel.
f
Definition 3.1 Let G, H be special groups. A group homomorphism G −→ H is a quasimorphism of special groups (qSG-morphism) if
For all a, b ∈ G, a ∈ DG (1, b)
implies
f (a) ∈ DH (1, f (b)).
3
A qSG-morphism f is a SG-morphism iff f (− 1 ) = − 1 .
Remark 3.2 Let G, H be special groups.
a) The constant 1-valued map from G to H is a qSG-morphism; it is a SG-morphism iff 1 = − 1
in H.
b) In general, there is no canonical SG-embedding of G into G × H (the category of special groups
does not have direct sums). There are many non-canonical complete embeddings, as is shown by
Example 5.23 in [5]. However, the map a ∈ G 7→ (a, 1) ∈ G × H is a qSG-morphism. The same
phenomenon takes place for an arbitrary set of indices in an infinite product.
3
f
Lemma 3.3 A qSG-morphism, G −→ H, induces a graded ring homomorphism
f∗ = {fn : n ≥ 0} : k∗ G −→ k∗ H,
where f0 = IdF2 and, for n ≥ 1, fn is defined on generators by
fn (λ(a1 ) . . . λ(an )) = λ(f (a1 )) . . . λ(f (an )).
f
g
Moreover, for qSG-morphisms G −→ H and H −→ K,
a) (g ◦ f )∗ = g∗ ◦ f∗ and Id∗ = Id.
b) For all a ∈ G, the following diagram is commutative :
kn G
ω an kn+1 G
fn
fn+1
?
kn H
ω fna
?
- kn+1 H
c) f is a SG-morphism iff for all n ≥ 0, the diagram in (b) is commutative, when the horizontal
arrows are both multiplication by λ(− 1 ).
Proof : Because f is a qSG-morphism, it is easily seen that for each n ≥ 1 :
(i) The map (k1 G)n −→ kn H defined by (a1 , . . . , an ) 7→ λ(f (a1 )) . . . λ(f (an )), is multilinear;
N
(ii) This map induces a homomorphism from ni=1 k1 G to k1 H, whose kernel contains the ideal
generated by {λ(a)λ(ab) : a ∈ DG (1, b)}.
7
Then, f∗ is the graded ring homomorphism determined by this construction. The functoriality of
the association f 7→ f∗ is clear, as are the other assertions in the statement.
3
In the case of fields, every graded ring homomorphism in K-theory arises from a qSG-morphism
of the corresponding special groups, as remarked by the referee of a previous version of this paper.
In fact, this is true under far more general circumstances, as shown by Proposition 3.6 below.
Following [19] (p. 421), we have
Definition 3.4 A SG, G, is AP(3) if the Arason-Pfister Theorem holds for forms in I 3 (G): for
every form ϕ over G,
ϕ ∈ I 3 (G) and dim(ϕ) < 8
imply
ϕ is hyperbolic.
3
Thus, SGs of fields and RSGs are AP(3) and, indeed, AP(n) for all n (in an obvious sense); see
[11], Ch. X, pp. 289-292, and [5], Theorem 7.31, p. 171, respectively. We observe
Lemma 3.5 Let G be a SG. For a, b ∈ G, consider the statements :
(1) a ∈ DG (1, b);
(2) λ(a)λ(−b) = 0 (in k2 G);
(3) λ(a)λ(ab) = 0 (in k2 G).
Then, (1) ⇒ (2) ⇔ (3). If G is AP(3), then (2) ⇒ (1).
Proof : (1) ⇒ (3) is by the definition of k2 G. For the equivalence between (2) and (3), just observe
that λ(a)λ(ab) = λ(a)λ(−b). Indeed, λ(a)2 = λ(a)λ(−1) (Proposition 2.1(a)) yields :
λ(a)λ(ab) = λ(a)[λ(a) + λ(b)] = λ(a)2 + λ(a)λ(b) = λ(a)λ(−1) + λ(a)λ(b)
= λ(a)[λ(−1) + λ(b)] = λ(a)λ(−b).
Now assume that G is AP(3) and that (2) holds. By Theorem 4.1 in [6], there is a canonical
group homomorphism, s2 : k2 G −→ I 2 (G)/I 3 (G), defined on generators by
s2 (λ(x)λ(y)) = (h 1, −x i ⊗ h 1, −y i)/I 3 (G).
The assumption λ(a) λ(ab) = 0 then yields h 1, −a i ⊗ h 1, b i ∈ I 3 (G), and so AP(3) implies that
this form is hyperbolic in G; by Prop. 2.2(k) in [5] (p. 34), we obtain a ∈ DG (1, b).
3
Proposition 3.6 Let G, H be SGs and β = {β n : n ≥ 0} : k∗ G −→ k∗ H be a graded ring
homomorphism, such that β 0 = IdF2 . If H is AP(3), there is a qSG-morphism, f : G −→ H, such
that β = f∗ (as in Lemma 3.3).
Proof : Note that there are no conditions on G. Recall that for n, m ≥ 0, η ∈ kn G and ζ ∈ km G,
β n+m (ηζ) = β n (η)β m (ζ).
(I)
Let f : G −→ H be the unique group homomorphism such that β 1 ◦ λG = λH ◦ f , where
λG : G −→ k1 G is the isomorphism changing multiplication in G into addition in k1 G (similarly
for H).
G
f
- H
λG
λH
?
?
- k1 H
k1 G
β1
To verify that f is a qSG-morphism, we must show :
a ∈ DG (1, b)
⇒
f (a) ∈ DH (1, f (b)).
8
(II)
For x, y ∈ G, (I) and the commutative diagram (II) yield (with λ = λG ) :
β 2 (λ(x)λ(y)) = β 1 (λ(x))β 1 (λ(y)) = λH (f (x))λH (f (y)).
(III)
If a ∈ DG (1, b), then λ(a)λ(ab) = 0 and so (III) implies λH (f (a))λH (f (a)f (b)) = 0. Since H is
AP(3), Lemma 3.5 gives f (a) ∈ DH (1, f (b)), as required. If f∗ = {fn : n ≥ 0} is the graded
ring morphism induced by f as in Lemma 3.3, it is clear from the hypothesis, the definition of f1
and the commutative diagram (II) that f0 = β 0 and f1 = β 1 . It now follows readily from (I) that
β = f∗ , concluding the proof.
3
4
The K-theory of inductive limits
In Chapter 5 of [5] there is a detailed discussion of embeddings of special groups. We recall two
f
types which will be important in what follows. A SG-morphism G −→ H is
∗ an embedding if for all a, b ∈ G, h 1, a i ≡G h 1, b i iff h 1, f (a) i ≡H h 1, f (b) i.
∗ a complete embedding if for all forms ϕ, ψ over G, of the same dimension
ϕ ≡G ψ
iff
f ? ϕ ≡H f ? ψ .
It is easily verified that an embedding must be injective. If the inclusion map of a subgroup G
into H is complete, we say that G is a complete subgroup of H.
Recall that a partially ordered set (I, ≤) is right-directed if for all i, j ∈ I, there is k ∈ I
such that i, j ≤ k.
Let G = (Gi ; {fij : i ≤ j}) be an inductive system of special groups and SG-morphisms over
the right-directed poset I. Let lim G = (G; {fi : i ∈ I}) be the inductive limit of the system G.
−→
If ai ∈ Gi is such that fi (ai ) = a ∈ G, ai is said to be a representative of a in Gi . Observe that if
ai ∈ Gi is a representative of a ∈ G, then so is fij (ai ), for all j ≥ i. If ϕ is a form over G, a form θ
over Gi , of the same dimension, is a representative of ϕ if all entries of θ are representatives in Gi
of the corresponding entries in ϕ. The isometry relation in G is characterized by
For forms ϕ, ψ of the same dimension over G,
There are i ∈ I and representatives θ of ϕ and χ of
ϕ ≡G ψ
iff
ψ in Gi , such that θ ≡Gi χ.
Lemma 4.1 (Prop. 3.1, [7]) With notation as above
a) G is a special group, which is reduced if {i ∈ I : Gi is reduced} is cofinal in I.
b) If for S
all i ≤ j, fij is injective, then all fi ’s are injective. If all fij ’s are embeddings, then
I n (G) = i∈I I n (Gi ), for all n ≥ 1.
c) If each fij is a complete embedding, then every fi : Gi −→ G is a complete embedding.
3
Remark 4.2 Any inductive system of SGs − even if the transition maps fij are not injective −
induces an inductive system of groups h I n (Gi ) : i ∈ I i with homomorphisms induced by the fij
in an obvious way. An argument similar to that proving 4.1(b) shows that I n (G) is the inductive
limit of the I n (Gi ).
3
For lack of a convenient reference (and with due apologies to the reader), we present a
4.3 Summary of inductive limits of graded rings. Let T = (Ti ; {τ ij : i ≤ j in I}) be
an inductive system of graded rings and graded ring homomorphisms (of degree 0) over the rightdirected poset I. Write Tin for the group of degree n of Ti and τ nij for the nth -component of τ ij .
The inductive system T gives rise to a family T n , n ≥ 1, of inductive systems of Abelian groups,
9
defined by taking as nodes the group of degree n of each Ti and, as arrows, the nth -component of
the homomorphism τ ij .
Lemma 4.4 Let T be an inductive system of graded rings and let {T n : n ≥ 1} be the associated
family of inductive systems of Abelian groups.
a) For n ≥ 1, let Sn = lim T n ; then the sequence S = (Sn )n≥1 has a natural structure of graded
−→
ring, and for each i ∈ I there is a graded ring homomorphism τ i : Ti −→ S making the following
diagram commutative for i ≤ j :
τ ij Tj
Ti
τj
A
A
τ iA
A
A
AU
S
b) lim T = (S; {τ i : i ∈ I}).
−→
Proof : a) For n ≥ 1, write (Sn ; {σ in : i ∈ I}) for lim T n . Thus, for i ≤ j in I, the diagram below
−→
left is commutative :
τ nij Tjn
Tin
A
A
A
A
hinA
σ jn
σ inA
A
AA
U
Sn
σ in -
Tin
A
AAU
Sn
gn
Hn
Clearly, Sn is an Abelian group. To define multiplication in S = (Sn )n≥1 , for a ∈ Sn and b ∈ Sm ,
select representatives x ∈ Tin of a, and y ∈ Tjm of b, for suitable i, j ∈ I; now pick k in I such that
i, j ≤ k and define
m (y)).
a · b = σ k(n+m) (τ nik (x) · τjk
(I)
It is straightforward that (I) is independent of representatives and that S = (Sn )n≥1 is a graded
ring with this operation. Set τ i = (σ in )n≥1 ; for x ∈ Tin and y ∈ Tim , x is a representative of σ in (x)
and y is a representative of σ im (y). Thus, it follows immediately from (I) that
σ i(n+m) (x · y) = σ in (x) · σ im (y),
(II)
and so τ i is a graded ring homomorphism from Ti to S. The commutativity of the diagram in the
statement is clear.
h
i
H such that
b) Let (H; {hi : i ∈ I}) be a graded ring and graded ring homomorphisms Ti −→
hj ◦ τ ij = hi , for all i ≤ j in I. Since Sn = lim T n , there is a unique group homomorphism,
−→
gn : Sn −→ Hn , such that the diagram above right is commutative for all i ∈ I. To end the proof,
it is enough to verify that g = (gn )n≥1 is a graded ring homomorphism from S to H. For a ∈ Sn ,
b ∈ Sm , pick representatives x ∈ Tin of a and y ∈ Tim of b. Then, (II) yields
gn+m (a · b) = gn+m (σ i(n+m) (x · y)) = hi(n+m) (x · y) = hin (x) · him (y)
= gn (σ in (x)) · gm (σ im (y)) = gn (a) · gm (b).
3
An inductive system of SGs, G = (Gi ; {fij : i ≤ j ∈ I}), gives rise to an inductive system of
graded rings with nodes k∗ Gi and arrows (fij )∗ : k∗ Gi −→ k∗ Gj , for i ≤ j in I.
10
Theorem 4.5 Let G = (Gi ; {fij : i ≤ j in I}) be an inductive system of special groups over the
right-directed poset I and let (G; {fi : i ∈ I}) = lim G. Then, k∗ G ≈ lim k∗ Gi .
−→
−→
Proof : Set lim k∗ Gi = (M ; {µi : i ∈ I}); by functoriality (Lemma 3.3), we have graded ring
−→
homomorphisms fi ∗ : k∗ Gi −→ k∗ G such that for all i ≤ j in I, the diagram on the left is
commutative :
fij ∗-
k∗ Gi
fj ∗
A
A
fi ∗ A
A
AA
U
µi M
k∗ Gi
k∗ Gj
µ
A
A
fi ∗ A
A
AAU
k∗ G
k∗ G
µ
Therefore, there is a unique graded ring homomorphism M −→ k∗ G making the diagram on the
right commutative for all i ∈ I. We show that µ is an isomorphism by constructing a graded ring
homomorphism θ : k∗ G −→ M such that θ ◦ µ = IdM and µ ◦ θ = Idk∗ G . For later use, we give
an explicit description of µ, whose verification is left to the reader :
P
For n ≥ 1 and a ∈ Mn , let η ∈ kn Gi be a representative of a, η = pj=1 λ(aj1 )λ(aj2 ) . . . λ(ajn ),
with ajl ∈ Gi . Then,
P
µn (a) = pj=1 λ(fi (aj1 ))λ(fi (aj2 )) . . . λ(fi (ajn )).
(I)
Fix n ≥ 1 and consider the map gn : (k1 G)n −→ Mn given as follows :
For h λ(u1 ), λ(u2 ), . . . , λ(un ) i ∈ (k1 G)n , select i ∈ I and representatives xj ∈ Gi of uj for
1 ≤ j ≤ n, to define
gn (h λ(u1 ), λ(u2 ), . . . , λ(un ) i) = µin (λ(x1 )λ(x2 ) . . . λ(xn )).
(II)
It is straightforward that
Nng is independent of representatives and multilinear. Thus, gn induces
a homomorphism hn :
j=1 k1 G −→ Mn , taking λ(u1 ). . . λ(un ) to µin (λ(x1 )λ(x2 ) . . . λ(xn )) in
Mn . That h = (hn )n≥1 is a graded ring homomorphism follows directly from the fact that each
µi = (µin )n≥1 is a graded ring homomorphism.
Nn
For n ≥ 2, let λ(u)λ(uv)λ(u3 ) . . . λ(un ) be a generator in
j=1 k1 G, with u ∈ DG (1, v).
Choose i ∈ I and representatives x ∈ Gi of u, y ∈ Gi of v and xj ∈ Gi of uj , 3 ≤ j ≤ n, such that
x ∈ DGi (1, y). Note that xy represents uv. Since λ(x)λ(xy)λ(x3 ) . . . λ(xn ) = 0 in kn Gi , (II) yields
hn (λ(u)λ(uv)λ(u3 ) . . . λ(un )) = µin (λ(x)λ(xy)λ(x3 ) . . . λ(xn )) = 0,
and h factors through k∗ G to yield a graded ring homomorphism θ : k∗ G −→ M which, for each n
≥ 1, sends λ(u1 ) . . . λ(un ) in kn G to µin (λ(x1 )λ(x2 ) . . . λ(xn )), where xj is a representative of uj
in Gi .
P
For a ∈ Mn , let η = pj=1 λ(aj1 ) . . . λ(ajn ) represent a in kn Gi . Then, (I) yields
Pp
P
θ(µ(a)) = θ( pj=1 λ(fi (aj1 )) . . . λ(fi (ajn )) =
(III)
j=1 θ(λ(fi (aj1 )) . . . λ(fi (ajn ))).
Since ajl is a representative of fi (ajl ) in Gi , 1 ≤ j ≤ p, 1 ≤ l ≤ n, (II) implies
θ(λ(fi (aj1 )) . . . λ(fi (ajn ))) = µin (λ(aj1 ) . . . λ(ajn )),
and so, from (III) comes
Pp
Pp
θ(µ(a)) =
j=1 θ(λ(fi (aj1 )) . . . λ(fi (ajn ))) =
j=1 µin (λ(aj1 ) . . . λ(ajn ))
P
p
= µin
= µin (η) = a,
j=1 λ(aj1 ) . . . λ(ajn )
verifying that θ ◦ µ is the identity on M . Now let ζ = λ(u1 ) . . . λ(un ) be a generator in kn G. Then,
with notation as above, (I) yields
11
µ(θ(ζ)) = µ(µin (λ(x1 ) . . . λ(xn ))) = λ(fi (x1 )) . . . λ(fi (xn )) = λ(u1 ) . . . λ(un ) = ζ,
3
and µ ◦ θ is the identity on k∗ G, as needed.
Corollary 4.6 The inductive limit of an inductive system of [SMC]-groups is [SMC].
Proof : Let G = h Gi ; {fij : i ≤ j in I} i be an inductive system of [SMC]-groups; thus, if − 1 i is
the distinguished element of Gi , multiplication by λ(− 1 i ) is an injection from kn Gi into kn+1 Gi ,
for all n ≥ 1. Let G = lim G. It is easily checked that the map λ(− 1 ) · − : kn G −→ kn+1 G is the
−→
inductive limit of the maps λ(− 1 i ) · −. But it is well-known that the inductive limit of injective
maps is injective.
3
Remark 4.7 A slight modification of the argument proving 4.5 and 4.6 will establish that if
G = (Gi ; {fij : i ≤ j in I}) is an inductive system of SGs such that {i ∈ I : Gi is [SMC]} is cofinal
in I, then lim G is a [SMC]-group.
3
−→
5
The K-theory of finite products
If S, T are graded F2 -algebras, with S0 = T0 = F2 , their direct sum, S ⊕ T , is the sequence
of groups
(S ⊕ T )0 = F2
and
(S ⊕ T )n = Sn ⊕ Tn , n ≥ 1,
with multiplication defined by the relation (x, y) · (u, v) = (xu, yv). The operation of F2 in each
Sn ⊕ Tn , n ≥ 1, is the usual operation as a F2 -module.
The main result of this section is
Q
Theorem 5.1 Let G1 , . . . , Gm L
be special groups and let P = m
i=1 Gi . Then, there is a graded
m
ring isomorphism, γ : k∗ P −→ i=1 k∗ Gi , defined on generators by the rule
γ n (λ(a1 )λ(a2 ) . . . λ(an )) = h λ(π1 (a1 )) . . . λ(π1 (an )), . . . , λ(πm (a1 )) . . . λ(πm (an )) i,
where πi : P −→ Gi is the canonical projection, 1 ≤ i ≤ m. Moreover,
Lγ takes multiplication by
λ(−1, −1, . . . − 1) in k∗ P to componentwise multiplication by λ(−1) in m
i=1 k∗ Gi .
The basic observation needed for the proof of Theorem 5.1 is stated in
Lemma 5.2 Let {Gi : i ∈ I} be special groups and let P =
[a = 1] = {i ∈ I : ai = 1}. Then, for all a, b ∈ P ,
Q
i∈I
Gi . For a = h ai i ∈ P , write
If I is the disjoint union of [a = 1] and [b = 1], then λ(a)λ(b) = 0 in k2 P .
Proof : Suppose a, b ∈ P are such that {[a = 1], [b = 1]} is a partition of I. Define c ∈ P by
(
ai if i ∈ [b = 1]
ci =
bi if i ∈ [a = 1]
It is readily verified that for all i ∈ I, ai ∈ DGi (1, ci ); thus, a ∈ DP (1, c). Since ac = b, the
definition of k2 P yields λ(a)λ(b) = λ(a)λ(ac) = 0.
3
Proof of Theorem 5.1. It is enough to verify the result for m = 2. Let G, H be special groups
and set P = G × H. Recall that − 1 in P is (− 1 , − 1 ) and that isometry is defined componentwise.
Let π G , π H be the canonical projections. Let αG : G −→ P be defined by a 7→ (a, 1); αH is defined
similarly. The projections are SG-morphisms, while αG and αH are qSG-morphisms. Since
π G ◦ αG = IdG and π H ◦ αH = IdH ,
Lemma 3.3(a) implies that
12
H
H
π∗G ◦ αG
∗ = Idk∗ G and π∗ ◦ α∗ = Idk∗ H .
(I)
For n = 0, γ 0 is the identity on F2 . For n = 1, it is clear that γ 1 (λ(a, b)) = (λ(a), λ(b)) is an
isomorphism between k1 P and k1 G ⊕ k1 H.
For n ≥ 2, consider the map fn : (k1 P )n −→ kn G ⊕ kn H, given by
fn (λ(a1 , b1 ), . . . , λ(an , bn )) = (λ(a1 ) . . . λ(an ), λ(b1 ). . . λ(bn )).
Nn
Since fn is multilinear, it induces a homomorphism gn :
i=1 k1 P −→ kn G ⊕ kn H. For a generator
ζ = λ(a1 , b1 ) . . . λ(a, b)λ(ac, bd) . . . λ(an , bn ), such that (a, b) ∈ DP (1, (c, d)), we have, recalling
that a ∈ DG (1, c) and b ∈ DH (1, d),
gn (ζ) = (λ(a1 ) . . . λ(a)λ(ac) . . . λ(an ), λ(b1 ), . . . λ(b)λ(bd) . . . λ(bn )) = 0.
Therefore, gn induces a homomorphism γ n : kn P −→ kn G ⊕ kn H, that sends a generator
λ(a1 , b1 ) . . . λ(an , bn ) to the pair (λ(a1 ) . . . λ(an ), λ(b1 ) . . . λ(bn )).
It is clear that γ n is surjective; moreover, since product in k∗ G ⊕ k∗ H is defined coordinatewise,
γ = {γ n : n ≥ 0} is a graded algebra homomorphism. To show that γ is an isomorphism, it remains
to verify that γ n is injective for all n ≥ 0. This is clear for n = 0, 1. For n ≥ 2, we first prove
Fact 1. For n ≥ 2, every element in kn P can written as a sum of the form
λ(a1 , 1)λ(a2 , 1) . . . λ(an , 1) + λ(1, b1 )λ(1, b2 ) . . . λ(1, bn ),
with (a1 , . . . , an ) ∈ Gn and (b1 , . . . , bn ) ∈ H n .
Proof : It is enough to verify the statement for a generator ζ = λ(a1 , b1 ) . . . λ(an , bn ). Because
(ai , bi ) = (ai , 1)(1, bi ), 1 ≤ i ≤ n, we may write
ζ = [λ(a1 , 1) + λ(1, b1 )][λ(a2 , 1) + λ(1, b2 )] . . . [λ(an , 1) + λ(1, bn )].
(II)
Note that λ(ai , 1)λ(1, bj ) = 0, for all 1 ≤ i, j ≤ n. This follows from Lemma 5.2 if both ai and bj
6= 1; if one of them is 1, we have a factor λ(1, 1) and the product is 0. This observation and (II)
immediately imply that ζ has the desired form.
For (a1 , . . . , an ) ∈ Gn and (b1 , . . . , bn ) ∈ H n , Lemma 3.3 yields
 G

 αn (λ(a1 ) . . . λ(an )) = λ(a1 , 1) . . . λ(an , 1)
and

 H
αn (λ(b1 ) . . . λ(bn )) = λ(1, b1 ) . . . λ(1, bn ).
(III)
H
If η ∈ kn P , write γ n (η) = (η G , η H ); then, since αG
n and αn are homomorphisms, Fact 1 and (III)
yield
H
η = αG
n (η G ) + αn (η H ).
(IV)
Thus, if γ n (η) = 0, that is, η G = 0 and η H = 0, then (IV) guarantees that η = 0, and γ n is indeed
an isomorphism.
P
If η ∈ kn P , we have η = m
j=1 λ(aj1 , 1) . . . λ(ajn , 1) + λ(1, bj1 ) . . . λ(1, bjn ); thus, Lemma 5.2
yields
λ(− 1 , − 1 )η = [λ(− 1 , 1) + λ(1, − 1 )]η
Pm
=
j=1 λ(− 1 , 1)λ(aj1 , 1) . . . λ(an , 1) + λ(1, − 1 )λ(1, bj1 ) . . . λ(1, bjn ),
and so multiplication by λ(− 1 , − 1 ) in k∗ P is taken by γ to multiplication by (λ(− 1 ), λ(− 1 )) in
k∗ G ⊕ k∗ H, ending the proof.
3
With respect to infinite products we pose
5.4 Problem. How is the K-theory of an infinite product of special groups related to the Ktheory of its components ?
3
An immediate consequence of Theorem 5.1 is
13
3
Corollary 5.5 The finite product of [SMC]-groups is [SMC].
We shall now apply the preceding results to compute the K-theory of the SG-sum of special
groups (Definition 5.27, [5]). Recall (1.1.(e)) that Z2 = {± 1} is a reduced special group, in fact,
just the Boolean algebra {⊥, >} under a different guise; by Theorem 2.9, Z2 is a [SMC]-group.
In Remark 3.2 we registered that, in general, there is no canonical SG-embedding of G into
G × H. It is a different story if we introduce a factor Z2 in our considerations.
Let I Q
⊆ J be finite
Q
sets and Gj , j ∈ J, be formally real special groups. Set GJ = j∈J Gj and GI = i∈I Gi . Let
πJI : GJ −→ GI be the canonical projection (forget coordinates outside I). Let ∗ denote the last
(Z2 )-coordinate in the product GI × Z2 .
Lemma 5.6 Notation as above, the map αIJ : GI × Z2 −→ GJ × Z2 defined, for s ∈ GI × Z2
and j ∈ J ∪ {∗}, by
(
s(j) if j ∈ I
αIJ (s)(j) =
s(∗) if j ∈
6 I,
is a complete embedding. Moreover, (πJI × IdZ2 ) ◦ αIJ is the identity in GI × Z2 .
Proof : By Lemma 5.17 in [5], a SG-morphism having a retract is a complete embedding; hence,
the second assertion implies the first. Checking that αIJ and (πJI × IdZ2 ) are SG-morphisms with
the stated property is straightforward.
3
The SG-sum of special groups is defined as follows :
L∗
Definition 5.7 (Definition 5.27,
[5])
If
G
,
i
∈
I,
is
a
family
of
special
groups,
its
SG-sum,
i
i∈I
Q
Gi , is the following subgroup of i∈I Gi :
Q
L∗
i∈I Gi : For some cofinite J ⊆ I, x|J is either constantly 1 or − 1 },
i∈I Gi = {x ∈
Q
L
where x|J denotes the restriction of x to J. Isometry in ∗i∈I Gi is that induced by i∈I Gi and
− 1 = (−1Gi : i ∈ I).
3
Note that if I is finite, then SG-sum and product coincide. Henceforth, we assume that I is
infinite. We emphasize that the SG-sum is not the coproduct in the category of special groups; the
latter, in general, does not have coproducts (Prop. 10.11, [5]).
A slight modification of the proof given in Example 5.28
[4], will
L in [5], using Lemma 2.5 in Q
show that if Gi , i ∈ I, is a family of formally real groups, ∗i∈I Gi is a pure subgroup of i∈I Gi
(see Definition 5.25, [5]).
LetQGi , i ∈ I, be a family of formally real special groups. For each finite subset A of I, let
GA = i∈A Gi × Z2 . If A ⊆ B are finite subsets of I, Lemma 5.6 yields a complete embedding
αAB : GA −→ GB . Since the set of finite subsets of I, P f in (I), is right-directed under inclusion,
G = (GA ; {αAB : A ⊆ B in P f in (I)})
is a complete inductive system of formally real special groups. We now prove
Theorem 5.8 Let Gi , i ∈ I, be an infinite family of formally real SGs. Write S =
With notation as above, we have S = lim G. Moreover,
−→
a) k∗ S = lim KG, where KG is the inductive system of K-theory rings associated to G.
−→
b) The SG-sum of [SMC]-groups is [SMC].
Proof : To show that S = lim G, for each finite A ⊆ I, define αA : GA −→ S by :
−→
(
s(i) if i ∈ A
αA (s)(i) =
s(∗) if i ∈
6 A.
14
L∗
i∈I
Gi .
The proof of 5.6 can easily be adapted to show that
S αA is a complete embedding. It is clear that
A ⊆ B implies αA = αB ◦ αAB , as well as that S = A∈Pf in (I) Im αA . Since all morphisms involved
are injective, we must have S = lim G. Now item (a) follows directly from Theorem 4.5, while item
−→
(b) is a consequence of Theorem 2.9, Corollary 5.5 and Theorem 4.5.
3
6
The K-theory of extensions
The definition of extension appears in Example 1.10 in [5]. Since we shall use a slightly
different notational convention, we recall the relevant definitions.
Let G be a special group and T a group of exponent two, written additively (0 is the neutral of
T ). The extension of G by T , G[T ], is defined as follows :
1. The domain of G[T ] is G × T . Thus, 1 = (1, 0) and (g, δ) · (g 0 , δ 0 ) = (gg 0 , δ + δ 0 ).
2. The distinguished element − 1 in G[T ] is (− 1 , 0).
3. For a, b ∈ G and s, t ∈ T , the representation relation in G[T ] is defined by


 b = − 1 and t = 0 or
s = t = 0 and a ∈ DG (1, b) or
[ext]
(a, s) ∈ DG[T ] (1, (b, t))
iff


t 6= 0 and ((a, s) = (1, 0) or (a, s) = (b, t)).
In [13] there is a proof that G[T ] is a SG, which is reduced iff G is reduced. For reduced SGs,
a proof of this, in the dual language of abstract order spaces, appears in [1], Prop. IV.2.13, p. 93.
ι
π
We have two SG-morphisms, G −→ G[T ] and G[T ] −→ G, given by ι(a) = (a, 0) and π(a, t) = a.
It is clear that π ◦ ι = IdG ; hence ι is a complete embedding.
We shall write 2 = {0, 1} for the additive group of the field of two elements F2 .
Proposition 6.1 Let G be a special group and S, T be groups of exponent two.
a) The map (a, (s, t)) 7→ ((a, s), t) is a SG-isomorphism between G[S ⊕ T ] and G[S][T ].
b) If T has (finite) dimension n ≥ 1 over F2 , then G[T ] is isomorphic to G [2][2] . . . [2].
| {z }
n times
c) If S ⊆ T , let A be a subgroup of T such that T = S ⊕ A. Then, the maps
(
ιST : G[S] −→ G[T ], ιST (x, s) = (x, (s, 0))
πST : G[T ] −→ G[S], πST (x, (s, a)) = (x, s)
are SG-morphisms such that πST ◦ ιST = IdG[S] . Hence, ιST is a complete embedding.
d) Let F in(T ) be the right-directed poset (under inclusion) of finite subgroups of T and let
T = (G[S]; {ιSS 0 : S ⊆ S 0 in F in(T )}) be the complete inductive system of SGs in item (c).
Then, G[T ] = lim T .
−→
Proof : a) It is clear that the given map, say f , is bijective. The verification that
For all η ∈ G[S ⊕ T ], f DG[S⊕T ] (1, η) = DG[S][T ] (1, f (η)),
is a straightforward application of the definition of representation given above ([ext]). Item (b) is
a direct consequence of (a), while (c) is clear.
d) For S ∈ F in(T ), let ιS : G[S] −→ G[T ] be the complete embedding
S associated in (c) to S ⊆ T .
It is clear that for S ⊆ S 0 in F in(T ), ιS = ιS 0 ◦ ιSS 0 . Since G[T ] = S∈F in(T ) Im ιS and all maps
involved are injective, we must have G[T ] = lim T , as asserted.
3
−→
Proposition 6.1 and Theorem 4.5 imply
15
Corollary 6.2 Let G be a special group and T a group of exponent 2. Then, k∗ G[T ] is the union
of the K-theory rings of the extensions of G by the finite subgroups of T .
Proof : If S ⊆ S 0 are finite subgroups of T , 6.1.(c) implies that in K-theory we have
(πSS 0 )∗ ◦ (ιSS 0 )∗ = Idk∗ G[S]
and so the connecting maps (ιSS 0 )∗ are injective graded ring homomorphisms, as needed.
3
Remark 6.3 Clearly, any exact sequence
f
g
0 −→ A −→ B −→ C −→ 0
of groups of exponent 2 is split, since they are vector spaces over F2 . This also the case if our
groups have a distinguished element − 1 6= 1 and the morphisms f , g send − 1 to − 1 . In the
specific case where A, B and C are K-theoretic groups, we shall use the adjective “split” in the
following sense :
Definition 6.4 Let G, H be SGs, let C be a group of exponent 2 and let n ≥ 1 be an integer. We
say that an exact sequence
f
g
0 −→ kn G −→ kn H −→ C −→ 0
where f , g are group homomorphisms, is split if there is a SG-morphism, h : H −→ G, so that
hn ◦ f = Idkn G , where hn is induced by h as in Lemma 3.3.
3
Since the K-theory of a finite extension is the iteration of extensions by 2 (Proposition 6.1.(b)),
Corollary 6.2 implies that in order to determine the K-theory on an extension it is sufficient to
characterize the K-theory of G[2]. The main result of this section is
Theorem 6.5 Let G be a special group. For each n ≥ 1, there is a split exact sequence
0
- kn G
ιn kn G[2]
δn kn−1 G
-0
δn kn−1 G
-0
such that the following diagram is commutative
0
- kn G
ιn kn G[2]
ωn
0
?
- kn+1 G
ωn
ιn+1 - ?
kn+1 G[2]
ω n−1
δ n+1 -
?
kn G
- 0
where the vertical arrows are multiplication by λ(−1).
We fix a special group G and let ι : G −→ G[2], π : G[2] −→ G be the canonical embedding
and projection, respectively. Since π ◦ ι = IdG , in K-theory we have π∗ ◦ ι∗ = Idk∗ G . Thus,
For all n ≥ 0, πn ◦ ιn = Idkn G .
(I)
In particular, ιn is injective for all n ≥ 0. This is precisely the horizontal arrow ιn in the diagrams
of Theorem 6.5. Because ι is a SG-morphism, Lemma 3.3(c) guarantees that the left square in the
two-row diagram of Theorem 6.5 is commutative.
We now turn to the construction of δ n . We shall use the additive structure of 2, as well as the
multiplicative structure of F2 .
16
For an integer n ≥ 1, let Ln be the family of subsets of [1, n] of cardinal ≤ n − 1; write c(p) for
the cardinal of p ∈ Ln . If p = (p1 , . . . , pc(p) ) ∈ Ln , a = (a1 , . . . , an ) ∈ Gn and x = (x1 , . . . , xn ) ∈
2n , set
Q
xp = i6∈p xi (in F2 ) and λ(a, p) = λ(ap1 )λ(ap2 ) . . . λ(apc(p) ) (in kc(p) G),
with the empty product equal to 1 ∈ F2 for the term in the right-hand side.
Proposition 6.6 With hypothesis as in Theorem 6.5, for each n ≥ 1, there is a surjective homomorphism δ n : kn G[2] −→ kn−1 G, such that
P
a) If ζ = λ(a1 , x1 )λ(a2 , x2 ) . . . λ(an , xn ), then δ n (ζ) = p∈Ln xp λ(−1)n−c(p)−1 λ(a, p).
b) For all β ∈ kn−1 G, δ n (λ(1, 1)ιn−1 (β)) = β.
Proof : If n = 1, δ 1 (λ(a, x)) = x, which is clearly surjective. From here on, assume n ≥ 2.
For p ∈ Ln , define fp : (k1 G[2])n −→ kc(p) G by
fp (λ(a1 , x1 ), λ(a2 , x2 ), . . . , λ(an , xn )) = xp λ(a, p).
To see that fp is multilinear, write u = (λ(a1 , x1 ), λ(a2 , x2 ), . . . , λ(an , xn )), v for the result of
substituting λ(c, z) for the ith coordinate of u, and w for the n-tuple consisting of adding λ(c, z) to
the ith -coordinate of u. Thus, the ith -coordinate of w is λ(cai , z + xi ). We have two possibilities :
i) i ∈ p : In this case the value of xp is the same for u, v and w. Thus,
fp (w) = xp (λ(ap1 ) . . . [λ(c) + λ(ai )] . . . λ(apc(p) ))
= xp λ(ap1 ) . . . λ(c) . . . λ(apc(p) ) + xp λ(ap1 ) . . . λ(ai ) . . . λ(apc(p) ) = fp (u) + fp (v).
ii) i 6∈ p : In this case the value λ(a, p) is the same for u, v and w. On the other hand, if t is the
product of the xj , for j 6∈ p and distinct from i, we have

for u

 txi
tz
for v
xp =


t(xi + z) for w.
Consequently, fp (w) = t(xi + z)λ(a, p) = fp (u) + fp (v).
Nn
Therefore, fp induces a homomorphism gp :
i=1 k1 G[2] −→ kc(p) G, such that
gp (λ(a1 , x1 )λ(a2 , x2 ) . . . λ(an , xn )) = xp λ(a, p).
Nn
It follows that we have a homomorphism f :
i=1 k1 G[2] −→ kn−1 G, given by
P
n−c(p)−1 λ(a, p).
f (λ(a1 , x1 )λ(a2 , x2 ) . . . λ(an , xn )) =
p∈Ln xp λ(−1)
(I)
To prove f surjective, it is enough to show that any generator η = λ(c1 ). . . λ(cn−1 ) ∈ kn−1 G is in
the image of f . It is straightforward that λ(c1 , 0)λ(c2 , 0) . . . λ(cn−1 , 0)λ(1,1) is taken to η by f .
Thus, for all β ∈ kn−1 G, we have
f (λ(1, 1)ιn−1 (β)) = β.
(*)
To show that f factors through kn G[2], let
α = λ(a1 , x1 ) . . . λ(ai , xi )λ(bai , z + xi ) . . . λ(an , xn )
be a generator such that (ai , xi ) ∈ DG[2] (1, (b, z)). It must be verified that f (α) = 0. We discuss
three cases, corresponding to the definition of representation (see [ext], page 15).
1. b = − 1 and z = 0 : Thus, α = λ(a1 , x1 ) . . . λ(ai , xi )λ(−ai , xi ) . . . λ(an , xn ). Define
Q = {p ∈ Ln : p ∩ {i, i + 1} = ∅};
Pi = {p ∈ Ln : p = {i} ∪ q, for some q ∈ Q};
Pi+1 = {p ∈ Ln : p = q ∪ {i + 1}, for some q ∈ Q};
P = {p ∈ Ln : {i, i + 1} ⊆ p}.
Note that Ln is the disjoint union of these sets. If p ∈ P , then λ(a, p) contains λ(ai )λ(−ai )
as a factor and so the terms in f (α) corresponding to these p’s are 0. Setting, for q ∈ Q,
17
tq =
Q
{xj : j 6∈ q and j 6= i, i + 1}, we may write :
 P
n−c(q)−2 λ(a )λ(a, q)
i

q∈Q xi tq λ(−1)




+

 P
n−c(q)−2 λ(−a )λ(a, q)
f (α) =
i
q∈Q xi tq λ(−1)




+


P

2
n−c(q)−1 λ(a, q)
q∈Q xi tq λ(−1)
(terms corresponding to Pi )
(terms corresponding to Pi+1 )
(terms corresponding to Q).
Consequently,
f (α) =
P
q∈Q
[xi λ(ai ) + xi λ(−ai ) + xi 2 λ(− 1 )] tq λ(−1)n−c(q)−2 λ(a, q).
But xi λ(ai ) + xi λ(−ai ) + xi 2 λ(− 1 ) = xi λ(− 1 ) + xi 2 λ(− 1 ) = 0, because x2 = x in F2 . Thus,
f (α) = 0, as required.
2. xi = z = 0 and ai ∈ DG (1, b) : Here α = λ(a1 , x1 ) . . . λ(ai , 0)λ(bai , 0) . . . λ(an , xn ). Let Q, P ,
Pi , Pi+1 stand for the same sets as in case 1. If p 6∈ P , then the term corresponding to p in f (α) is
zero, because xi = xi+1 = 0. For p ∈ P , λ(a, p) contains λ(ai )λ(bai ) as factor, and so it must be
zero. Thus, f (α) = 0, as desired.
3. z = 1 : Suppose that ai = 1 and xi = 0 and α = λ(a1 , x1 ) . . . λ(1, 0)λ(b , 1) . . . λ(an , xn ). With
notation as above, if p ∈ Pi , then the term corresponding to p in f (α) is zero because it has a factor
λ(1); if p 6∈ Pi , the term corresponding to it is also zero, because xi = 0. Thus, f (α) = 0. If ai =
b and xi = 1, a similar reasoning, with Pi+1 in place of Pi shows that f (α) = 0.
It follows directly from the above considerations that f factors through kn G[2] to yield a homomorphism δ n with the desired properties.
3
Proof of Theorem 6.5 : We first verify that for n ≥ 1, the kernel of δ n is equal to the image of
ιn . If ζ = λ(a1 ) . . . λ(an ) is a generator of kn G, then (see Lemma 3.3)
ιn (ζ) = λ(a1 , 0)λ(a2 , 0) . . . λ(an , 0),
and it is clear that ιn (ζ) is in the kernel of δ n .
Fact 1. For m ≥ 2 and x1 , . . . , xm ∈ 2 = {0, 1},
λ(1, x1 ) . . . λ(1, xm ) = x1 . . . xm λ(−1, 0)m−1 λ(1, 1).
Proof : It is enough to verify the statement for m = 2, i.e., that
λ(1, x)λ(1, y) = xyλ(− 1 , 0)λ(1, 1).
If x = 0, then λ(1, x) = 0 and the desired equality is true. The case y = 0 is similar. If x = y = 1,
then 2.1(d) (or (a)) yields λ(1, x)λ(1, y) = λ(1, 1)2 = λ(− 1 , 0)λ(1, 1).
Fact 2. For all η ∈ kn G[2], η = ιn (πn (η)) + ιn−1 (δ n (η)) λ(1, 1).
Proof : a) It suffices to verify the equation for a generator of kn G[2], η = λ(a1 , x1 ) . . . λ(an , xn ).
Because λ(ai , xi ) = λ(ai , 0) + λ(1, xi ), Fact 1 yields
Qn
η =
i=1 [λ(ai , 0) + λ(1, xi )]
Q
Q
P
λ(a
,
0)
= λ(a1 , 0) . . . λ(an , 0) + p∈Ln
λ(1,
x
)
i
i
i6∈p
i∈p
Q
P
= λ(a1 , 0) . . . λ(an , 0) + p∈Ln xp λ(− 1 , 0)n−1−c(p) λ(1, 1)
i∈p λ(ai , 0)
= ιn (πn (η)) + ιn−1 (δ n (η))λ(1, 1),
as desired.
It follows directly from Fact 2 that the kernel of δ n is equal to the image of ιn . Thus, the
sequence in the first diagram of Theorem 6.5 is indeed exact. That it is split comes from the
observation that for all n ≥ 1, πn ◦ ιn = Idkn G .
18
To end the proof, it remains to check that the right square of the two-row diagram in the
statement is commutative. Since ι is a SG-morphism, we can use Fact 2 together with Lemma
3.3.(c) and Proposition 6.6.(b) to get, for η ∈ kn G[2] :
δ n+1 (λ(− 1 , 0)η) = δ n+1 (λ(− 1 , 0) ιn (πn (η)) + λ(− 1 , 0)λ(1, 1)ιn−1 (δ n (η)))
= δ n+1 (ιn+1 (λ(− 1 ) πn (η)) + λ(1, 1)λ(− 1 , 0)ιn−1 (δ n (η)))
= δ n+1 (λ(1, 1)λ(− 1 , 0)(ιn−1 (δ n (η)))
= δ n+1 (λ(1, 1)ιn (λ(− 1 )δ n (η))) = λ(− 1 )δ n (η).
3
The preceding results yield the following concrete description of the K-theory ring of G[2].
Proposition 6.7 With notation as in Theorem 6.5, let n, m ≥ 1 be integers. Then,
a) For η ∈ kn G[2] and ξ ∈ km G[2]
δ n+m (ηξ) = πn (η)δ m (ξ) + πm (ξ)δ n (η) + λ(−1)δ n (η)δ m (ξ).
b) Let E = (F2 , . . . , E n , . . .) be the sequence whose nth -component, n ≥ 1, is the Abelian group
E n = kn G ⊕ kn−1 G.
Define an operation · : E n × E m −→ E n+m by the following rule :
h η 1 , η 2 i · h ξ1 , ξ2 i = h η 1 ξ1 , η 1 ξ2 + η 2 ξ1 + λ(−1)η 2 ξ2 i.
Then, E is a graded ring and there is a graded ring isomorphism, f = (fn ) : k∗ G[2] −→ E, given
by f0 = IdF2 and fn (β) = h πn (β), δ n (β) i for n ≥ 1.
c) If ι : G −→ G[2] is the canonical SG-embedding, then the composition f∗ ◦ i∗ : k∗ G −→ E is
precisely the natural embedding of k∗ G into the first summand of E.
Proof: a) It is enough to verify the stated equation for generators η = λ(a1 ) · · · λ(an ) and
ξ = λ(b1 ) · · · λ(bm ) of degree n and m, respectively. Write p for a subset of cardinal at most
n − 1 of {1, . . . , n} and q for a subset of cardinal at most m − 1 of {n + 1, . . . , n + m}. The subsets
of cardinality at most m + n − 1 of {1, . . . , n + m} can be partitioned into three classes:
(I) Those of the form {1, . . . , n} ∪ q;
(II) Those of the form p ∪ {n + 1, . . . , n + m};
(III) Those of the form p ∪ q.
The sets of type (I) give rise to the term πn (η)δ m (ξ) in the expression of δ n+m (ηξ) (cf. Proposition
6.6); those of type (II) combine to yield the term πm (ξ)δ n (η), while those of type (III) account for
λ(−1)δ n (η)δ m (ξ).
b) The verification that E is a graded ring with the operation defined in the statement is routine;
that f is a graded ring isomorphism follows from Theorem 6.5 and item (a). Item (c) follows readily
by a straightforward computation.
3
Corollary 6.8 Any extension of a [SMC]-group is [SMC].
Proof : It follows from the two-row diagram in Theorem 6.5 and the Short Five Lemma (Lemma
3.1, p. 13, [15]), that if G is [SMC], then G[2] is [SMC]. Proposition 6.1.(b) guarantees that an
extension of a [SMC]-group by a finite group of exponent two is [SMC]. Finally, Proposition 6.1.(d)
and Corollary 4.6 imply that any extension of a [SMC]-group is [SMC].
3
Every reduced special group of finite chain length (in particular, all fans and all finite reduced
special groups) can be obtained from the reduced special group Z2 by iterating the operations of
finite product and extension ([18], Thm. 4.2.2.(1)). Hence, Corollaries 6.8 and 5.5 yield a direct
proof of the following result, without resorting to the well-known fact that RSGs of finite chain
length are representable by Pythagorean fields.
19
Corollary 6.9 All reduced special groups of finite chain length are [SMC]. In particular, fans and
finite reduced special groups are [SMC].
3
We shall now give an explicit description of the K-theoretic groups of extensions of a SG by
finite or countable groups of exponent two. Preliminary to this, we register
Definition and Remarks 6.10 (a) To simplify statements, if G is a SG, set kn G = 0 for all
n < 0.
L
(b)
L If I is aI set and T is a group write I T for the direct sum of I copies of T . It is clear that
moreover, ifL
λ is the cardinal of I, then any bijection between I
I T = T , whenever I is finite; L
∼
and λ will yield an isomorphism
T
=
I
λ T.
(c) If A, B are groups, are write αA : A −→ A ⊕ B for the group homomorphism that embeds A
in the first summand, i.e., αA (a) = h a, 0 i. Similarly for αB : B −→ A ⊕ B.
f
g
(d) If A −→ B and C −→ D are group homomorphisms, then there is a group homomorphism
h f, g i : A ⊕ C −→ B ⊕ D given by h f, g i(h a, c i) = h f (a), g(c) i. With notation as in (c), it is
clear that h f, g i ◦ αA = αB ◦ f and h f, g i ◦ αC = αD ◦ g.
3
Theorem 6.11 Let G be a SG and ∆ be a group of exponent two of finite or countable dimension
d ≥ 1 as a vector space over F2 . Then, for each n ≥ 1

d

L

j
d

(k
G)
if d is finite;
n−j
j=0
kn G[∆] =


 k G ⊕ Ln L k G
if d is countably infinite.
n
j=1
d n−j
Proof : We first treat the finite dimensional case by induction on d ≥ 1. For d = 1, the result
follows, for all n ≥ 1, from Proposition 6.7. Assume the result true for d ≥ 1 and all n ≥ 1,
and
dim
∆= d + 1. Then, ∆ = S ⊕ 2, where dim S = d. Then, recalling that
suppose
d
d+1
d +
p
p+1 = p+1 , Theorem 6.5 and the induction hypothesis yield,
Ld
kn G[∆] = kn G[S][2] = kn G[S] ⊕ kn−1 G[S] =
= kn G ⊕
Ld−1
p=0
(kn−1−p G)
d
p+1
= kn G ⊕
Ld−1
= kn G ⊕
Ld−1
p=0
(kn−1−p G)
p=0
(kn−1−p G)
= kn G ⊕
Ld+1
j=1
(kn−j G)
j=0
Ld−1
⊕
d
p+1
p=0
⊕
(kn−j G)
d
p
d+1
p+1
d+1
j
d
j
(kn−1−p G)
!
⊕ (kn−1−p G)
d
p
Ld
p=0
d
p
(kn−1−p G)
⊕ kn−1−d G
⊕ kn−1−d G
⊕ kn−1−d G
,
as desired. Observe that if d ≥ n, the convention in 6.10(a) yields
kn G[∆] = kn G ⊕
Ln
j=1
d
j
(kn−j G)
.
(I)
Now suppose the dimension of ∆ over F2 is countably infinite. Using the result just proved we
build an inductive system of finite extensions of G having kn G[∆] as canonical inductive limit for
each n ≥ 1 (Theorem 4.5), in such a way that this limit is isomorphic to the group in the statement.
The constructions that follow are tailored for that purpose.
Let {ej : j ≥ 1} be a basis for ∆. For m ≥ 1, set
∆m = subgroup generated by {ej : 1 ≤ j ≤ m}.
20
As a group of exponent two,
L
L
k1 G[∆] = G ⊕ ∆ ∼
= G ⊕ d F2 = k1 G ⊕ d k0 G,
and so the desired formula holds for n = 1. From here on, assume that n ≥ 2 and fix
m ≥ 1. The maps and the groups defined in (1)–(5) below depend on this m; the superscript
m indicating this dependence is frequently omitted to ease notation. For each p ≥ 0,
(1) Let ιm = ιp : G[∆m+p ] −→ G[∆m+p+1 ] ∼
= G[∆m+p ][2] be the canonical injection as in 6.1(a).
p
It follows from Theorem 4.5 that G[∆] = lim
−→
p≥1
G[∆m+p ]. Write
ιm
pn = ιpn : kn G[∆m+p ] −→ kn G[∆m+p+1 ]
for the homomorphism of degree n induced by ιm
p in K-theory.
(2) By induction on p ≥ 0, define a sequence of groups of exponent two and connecting homomorαp
phisms, Tp −→ Tp+1 , as follows
(
T0m
= T0
= kn G[∆m ]
m
Tp+1
= Tp+1 = Tpm ⊕ kn−1 G[∆m+p ] ,
m
m
and αp = αm
p embeds Tp into the first summand of Tp+1 , as in 6.10(c).
m = f
(3) fpn
pn : kn G[∆m+p+1 ] −→ kn G[∆m+p ] ⊕ kn−1 G[∆m+p ] is the degree n isomorphism given
0
0
by Proposition 6.7(b). Let (αm
p ) = αp be the embedding of kn G[∆m+p ] into the first summand of
the image of fpn ; by 6.7(c), we have fpn ◦ ιpn = α0p .
m
(4) Write γ 0 = γ m
0 for the identity of kn G[∆m ] and, for p ≥ 1, γ p = γ p for the identity of
kn−1 G[∆m+p ].
m
(5) By induction p ≥ 0, we define group isomorphisms, β m
p = β p : kn G[∆m+p ] −→ Tp , by the
following rules (notation as in 6.10(d)):
β 0 = γ 0 , β 1 = f0n and β p+1 = h β p , γ p i ◦ fpn , i.e.,
fpn
−→ kn G[∆m+p ] ⊕ kn−1 G[∆m+p ]
kn G[∆m+p+1 ]
hβp ,γp i
−→ Tp ⊕ kn−1 G[∆m+p ].
Clearly, β 0 and β 1 are isomorphisms with values in T0m and T1m , respectively. By induction, if β p
is an isomorphism, so is h β p , γ p i, and hence β p+1 is an isomorphism as well. The crucial step is
Fact 6.12 For each p ≥ 0, the following diagram is commutative
kn G[∆m+p ]
ιpn kn G[∆m+p+1 ]
βp
β p+1
?
?
- Tp+1
Tp
αp
Proof : Recalling that fpn ◦ ιpn = α0p (see item (3) above), as well as 6.10(d), we get
β p+1 ◦ ιpn = h β p , γ p i ◦ fpn ◦ ιpn = h β p , γ p i ◦ α0p = αp ◦ β p ,
as needed.
Since G[∆] = lim
−→
p≥0
h G[∆m+p ] ; {ιp : p ≥ 0} i, from Theorem 4.5 we have, for all n ≥ 1,
kn G[∆] = lim
−→
p≥0
h kn G[∆m+p ] ; {ιpn : p ≥ 0} i,
By the Fact, the inductive system h kn G[∆m+p ] ; {ιpn : p ≥ 0} i is isomorphic, via {β p : p ≥ 0}, to
the inductive system h Tp ; {αp : p ≥ 0} i. Hence, kn G[∆] ∼
h Tp ; {αp : p ≥ 0} i. Straight= lim
−→
forwardly, for any m ≥1 we have:
21
lim
−→
p≥0
Tpm = kn G[∆m ] ⊕
L
kn−1 G[∆m+p ].
p≥1
With m = 1, using Proposition 6.7.(b) we get:
kn G[∆] = lim
−→
p≥0
Tp1 = kn G[∆1 ] ⊕
L
kn−1 G[∆p+1 ]
L
∼
= kn G ⊕ kn−1 G ⊕
p≥2 kn−1 G[∆p ].
p≥1
The result for finite extensions proves, for p ≥ 2,
kn−1 G[∆p ] ∼
= kn−1 G ⊕
Lp
p
j
j=1
(kn−j−1 G)
,
whence,
kn G[∆] = kn G ⊕
L
p≥1
L
kn−1 G ⊕
p≥2
Lp
j=1
p
j
(kn−j−1 G)
.
(II)
For each 0 ≤ j ≤ n − 1, the term kn−j−1 G occurs ℵ0 times in the right-hand side
Lnof (II).
L Hence,
up to an isomorphism permuting coordinates, kn G[∆] is isomorphic to kn G ⊕
j=1
d kn−j G,
ending the proof.
3
Remark 6.13 a) From the description of the K-theoretic groups of extensions given by Theorem
6.11 the reader should be able, using Proposition 6.7, to obtain an explicit expression for the graded
multiplication. For reasons of space we omit this point.
b) It is conjectured that the result proven above for extensions by groups of dimension ≤ ℵ0 , holds
for extensions by groups of arbitrary infinite dimension over F2 .
c) The map δ of Theorem 6.5 is connected to map δ v constructed in Lemma 2.1 of [20].
d) Let F be a formally real field, T a preorder of F and v a valuation on F , fully compatible with
T . Let Fv be the residue field, so that the push-down Tv of T is a preorder on Fv . Let Γv be the
value group of v and set Γ0v = Γv /v(Ṫ ), where Ṫ = T \ {0}. Note that Γ0v is a group of exponent
two. By Theorem 1.33 of [5], a special group version of the Baer-Krull Theorem, we have
∼ GT (Fv )[Γ0 ]
GT (F ) =
v
v
i.e., GT (F ) is the the extension of GTv (Fv ) by the group of exponent two Γ0v . Hence, in case Γ0v is
finite or countably infinite, Theorem 6.11 yields a description of the K-theory of GT (F ) in terms
of the K-theory of the SG associated to the pair h Fv , Tv i.
3
7
The K-theory of groups of continuous functions
In this section we determine the K-theory of filtered powers of a special group. The filtered
power construction originated with Arens and Kaplansky ([2]) as a means to represent algebraic
structures by continuous maps. For other examples of its usefulness the reader may consult [14]
and [21] (Def. 11.3, p. 46 ff).
In Chapter 6 of [5] there is a detailed account of filtered powers and closely related structures of
special groups. Many of the constructions therein are more general than the ones considered here
and so we shall, to ease reading, indicate the background needed for the business at hand.
Let X be a topological space. A partition of X is a collection of non-empty, pairwise disjoint
clopens, whose union is X. Write P for the set of partitions of X. Note that if X is compact, every
element of P is finite. If P and Q are partitions of X, Q is finer than P , written P ≺ Q, if there
is a map α : Q −→ P such that for all q ∈ Q, q ⊆ α(q); α is called the refinement map from Q
to P . The map α is uniquely determined, because
∀ p ∈ P , ∀ q ∈ Q, α(q) = p iff q ∩ p 6= ∅.
Thus, we speak of the refinement map from Q to P . It is readily verified that for all p ∈ P ,
22
p=
S
{q ∈ Q : α(q) = p} (disjoint union),
as well as that α must be onto P . Clearly, ≺ is a partial order on P. If P , Q are partitions of X,
there is a least upper bound for P and Q in P namely
P ∨ Q = {p ∩ q : p ∈ P , q ∈ Q and p ∩ q 6= ∅},
with refinement maps given by p ∩ q 7→ p and p ∩ q 7→ q. Thus, P is a join-semilattice and a
right-directed poset under ≺.
Let X be a topological space and G a special group, considered as a topological space with the
discrete topology (all points are open). Let C(X, G) be the set of all continuous functions from
X to G. There is a natural bijection between C(X, G) and the set of all maps P −→ G, P ∈ P;
indeed, a function f : X −→ G is in C(X, G) iff it is locally constant, that is, there is a partition
P of X such that, for all p ∈ P , f is constant on p.
Definition and Remarks 7.1 If ϕ = h f1 , . . . , fm i is a form over C(X, G) and x ∈ X, define
ϕ(x) = h f1 (x), . . . , fm (x) i.
If p is a clopen set in X, such that all fk ’s are constant on p, say with value apk , define
ϕp = h ap1 , . . . , apm i,
that is, ϕp is the m-form whose k th component is the constant value of fk on p.
If f ∈ C(X, G) and S is a subset of X, write f|S for the restriction of f to S.
If a ∈ G and u is a clopen in X, write [u, a] for the element of C(X, G) that has constant value
a on u and is constantly equal to 1 on the complement of u.
If a ∈ G, write b
a for the constant a-valued function on X. If ϕ = h a1 , . . . , am i is a form over
ϕ
b
G, write for the form h b
a1 , . . . , b
am i over C(X, G).
α
For a map G −→ H, let α
b be the map from C(X, G) to C(X, H) given by α
b (f ) = α ◦ f . For
x ∈ X, define evx : C(X, G) −→ G by evx (f ) = f (x) (evaluation at x).
With pointwise defined operation, C(X, G) is a group of exponent two, whose neutral element is
b
c =def −11 as the distinguished element of C(X, G); define isometry
1 =def 11. We shall consider −1
in C(X, G) by the following clause :
[≡]
h f, g i ≡C(X,G) h u, v i
iff
For all x ∈ X, h f (x), g(x) i ≡G h u(x), v(x) i.
It is straightforward to verify that C(X, G) is a pre-special group, that is, it satisfies axioms
[SG 1] – [SG 5] in Definition 1.1 of [5].
3
The following definition, a special case of Definitions 6.5 and 6.8 in [5], describes the objects
that will interest us here.
Definition 7.2 Let X be a topological space, H be a SG and ` ≥ 0 be an integer. A filtration
of length ` on h X, H i is a set Σ = {(Fk , Hk ) : 1 ≤ k ≤ `} such that for 1 ≤ k ≤ `
[fp 1] : Fk is a non-empty closed set in X, strictly contained in Fk+1 , where F`+1 =def X.
[fp 2] : There is a clopen U in X disjoint from Fk , such that U ∩ Fk+1 6= ∅.
[fp 3] : Hk is a special subgroup of H, strictly contained in Hk+1 , where H`+1 =def H.
If Σ is a filtration on (X, H), the filtered power of H by X modulo Σ, written C(X, H; Σ), is
defined as
C(X, H; Σ) = {f ∈ C(X, H) : For all k ≤ ` and x ∈ X, x ∈ Fk implies f (x) ∈ Hk }.
C(X, H) is considered a filtered power of length 0.
3
A filtered power T = C(X, H; Σ) has a natural structure of pre-special group induced by that
of C(X, H) (see clause [≡] in 7.1 above).
23
Remark 7.3 Condition [fp 2] above is no restriction at all: given any set Σ satisfying requirements
[fp 1] and [fp 3], it is easy to construct a subset Σ0 ⊆ Σ verifying in addition [fp 2] and such that
the filtered powers C(X, H; Σ) and C(X, H; Σ0 ) are isomorphic.
3
A filtration Σ of length ` on (X, H) induces a map µ : 2X −→ {1, . . . , ` + 1}, as follows :
For S ⊆ X,
µ(S) = min {k ≤ ` + 1 : S ∩ Fk 6= ∅},
called the index of S with respect to Σ. If S = {x}, write µ(x) instead of µ({x}). Note that µ is
decreasing, that is,
For all S, T ⊆ X,
S ⊆ T implies µ(T ) ≤ µ(S).
From Lemma 6.9 and Theorem 6.10 in [5] we get
Theorem 7.4 Let X be a topological space and H be a special group. For a filtration Σ of finite
length on (X, H), let T = C(X, H; Σ) be the associated filtered power. Let ϕ, ψ be forms of the
same dimension over T .
a) For f ∈ C(X, H), let P be a partition of X such that f is constant on each p ∈ P , with value
fp . The following conditions are equivalent :
1. f ∈ T ;
2. For all x ∈ X, f (x) ∈ Hµ(x) ;
3. For all p ∈ P , fp ∈ Hµ(p) .
b) (Local-Global Principle) If P is a partition of X such that every component of ϕ and ψ is
constant on all elements of P , then the following are equivalent :
1. ϕ ≡T ψ ;
2. For all x ∈ X, ϕ(x) ≡Hµ(x) ψ (x);
c) For f ∈ T , f ∈ DT (ϕ)
iff
3. For all p ∈ P , ϕp ≡Hµ(p) ψ p .
for all x ∈ X, f (x) ∈ DHµ(x) (ϕ(x)).
d) T is a special group, which is reduced (resp., formally real) iff H is reduced (resp., formally real).
This applies, in particular, to C(X, H).
e) If all Hk ’s are complete subgroups of H, then T is a complete subgroup of C(X, H).
3
For C(X, H) we also have the following facts whose proof is straightforward :
Lemma 7.5 Let X be a topological space and G be a special group. With notation as in 7.1,
a) For x ∈ X, the map evx : C(X, G) −→ G, is a SG-morphism, while b· : G −→ C(X, G) is a
complete embedding, such that evx ◦ b· = IdG .
α
b) For a map G −→ H
1. α is a qSG-morphism (SG-morphism) iff α
b is a qSG-morphism (resp., SG-morphism).
2. α is an embedding, or a complete embedding iff the same is true of α
b.
3
b and ⊥ =
If B is a Boolean algebra, it is clear that C(X, B) is a Boolean algebra, with > = >
b
⊥. The following result gives a description of the Boolean hull of C(X, G), for G reduced. It is a
consequence of Theorem 6.34 in [5], but we shall include a proof in this simpler context to illustrate
the methods used in dealing with these structures.
Lemma 7.6 Let G be a reduced special group and X a topological space. Let εG : G −→ BG be
the Boolean hull of G. Then,
a) εc
G : C(X, G) −→ C(X, BG ) is a complete embedding.
b) The Boolean hull of C(X, G) is the Boolean algebra generated by the image of εc
G in C(X, BG ).
c) If X is compact, then εc
G : C(X, G) −→ C(X, BG ) is the Boolean hull of C(X, G).
24
Proof : Item (a) follows from Lemma 7.5.(b.2), while (b) is an immediate consequence of the
universal property of the Boolean hull (Theorem 4.17) and Proposition 4.10 in [5]. Now suppose
that X is compact. It is enough to show that if U is clopen in X and b ∈ BG , then the function
[b, U ] that has constant value b on U and is ⊥ on U c =def X \ U , is in the BA generated by
the image of εc
G . To see this, if β ∈ C(X, BG ), let P be a finite partition of X such that β is
constant
with
value
bp on p. Let β p ∈ C(X, BG ) have constant value bp on p and ⊥ on pc . Since
W
p
β = p∈P β and each β p is in the BA generated by the image of εc
G , the same must be true of β.
By Proposition 4.10 in [5], there is a collection of finite subsets of G, {Fi : 1 ≤ i ≤ m} such
that
W
V
b= m
i=1
a∈Fi εG (a).
For each i ≤ m and a ∈ Fi , let fa ∈ C(X, G) have constant value a in U and 1 on U c . Then
W
V
[b, U ] = m
c
G (fa ),
i=1
a∈Fi ε
3
as required.
Recall that a space is Boolean if it is compact, Hausdorff and has a basis of clopen sets.
From here on, fix a filtration Σ of length ` on a pair (X, H), X a Boolean space and H a special
group. Let T = C(X, H; Σ) be the associated filtered power.
For a partition P of X, define the special group TP by
Q
TP = p∈P Hµ(p) .
(P)
A typical element of TP is written aP = h ap : p ∈ P i. There is a natural map fP : TP −→ T ,
defined, for aP = h ap : p ∈ P i, as follows :
fP (aP ) : X −→ H is the function whose value on p is constantly equal to ap .
Theorem 7.4.(a) shows that fP takes values in T . It is clear that fP is injective. Since product and
binary isometry are defined coordinatewise in TP and pointwise in T , fP is a group homomorphism,
taking −1 (the constant P -sequence −1) to −11 in T and preserving binary isometry. In fact, it
follows from Theorem 7.4.(b) that for all aP , bP , cP , dP in TP ,
h aP , bP i ≡TP h cP , dP i iff fP ? h aP , bP i ≡T fP ? h cP , dP i.
(E)
Thus, fP is an embedding of reduced special groups.
Let α : Q −→ P be a refinement map. For all q ∈ Q, we have µ(α(q)) ≤ µ(q). Thus, given
aP = h ap : p ∈ P i in TP , the Q-sequence h bq : q ∈ Q i defined by bq = aα(q) is in TQ . This
shows that α induces a map fα : TP −→ TQ , sending aP to h aα(q) : q ∈ Q i ∈ TQ . Clearly,
fα (aP ) = aP ◦ α (aP ∈ TP ).
fα
α - TQ
Q
P
TP
A
aP
A
fα (aP ) A
A
AA
U
H
A
A
fP A
A
AAU
T
fQ
It is clear that fα is injective. Routine verification proves fQ ◦ fα = fP ; this and (E) above, imply
that fα is, in fact, an embedding of reduced special groups.
Proposition 7.7 Let Σ be a finite filtration on the pair (X, H), with X a Boolean space. Let T
be the associated filtered power. With notation as above,
a) For all refinements α : Q −→ P , the map fα : TP −→ TQ is a complete embedding.
b) fP : TP −→ T is a complete embedding, for all P ∈ P.
25
c)
h T, {fP : P ∈ P} i is the inductive limit of the inductive system of complete embeddings
fα
α
T = {TP −→ TQ : Q −→ P is a refinement map in P}.
Proof : a) For p ∈ P , define Ap = {q ∈ Q : α(q) = p}. The family Ap is non-empty S
(α is onto),
pairwise disjoint and its union is p. Note that for all q ∈ Ap , µ(p) ≤ µ(q). Since p = Ap , there
must be qp ∈ Ap such that µ(qp ) = µ(p).
For p ∈ P , consider the diagonal embedding
Q
∆p : Hµ(p) −→ q∈Ap Hµ(q) , a 7→ the constant Ap -sequence a.
∆p is a complete embedding because isometry on the right-hand side is defined componentwise and
the group in the (qp )th component is precisely Hµ(p) . Since fα = h ∆p : p ∈ P i, we conclude that
fα is a complete embedding. Item (b) is a straightforward consequence of the local-global principle
in Theorem 7.4.(b).
c) We know that if f ∈ T , then there is P ∈ P such that for all p ∈ P , f|p is constant on p. Thus,
if we identify TP with its image by fP in T , we have
S
T = {TP : P ∈ P}.
Since the fP ’s and the fα ’s are injective, it follows that T must be the inductive limit of the system
T , as asserted.
3
Proposition 7.7.(c) and Theorem 4.5, together with Corollary 6.9, yield
Corollary 7.8 Let Σ be a filtration of finite length on the pair h X, H i, where X is a Boolean
space, and let T be the associated filtered power. With notation as above, we have
L
a) k∗ T = lim k∗ TP = lim p∈P k∗ Hµ(p) .
−→
−→
b) Every filtered power [SMC]-groups over X is [SMC].
3
Note that the K-theoretic morphisms appearing in the inductive limit of 7.8.(a) may not be
injective. Hence, the K-theory of a filtered power may not be the filtered power of the corresponding
K-theories. However, this is the case for special groups of the type C(X, G), as we show in the
sequel. For this, we need some notation.
If A = (F2 , A1 , . . . , An , . . . ) is a graded F2 -algebra, and X is a topological space, let C(X, A)
be the following sequence of groups of exponent two
C(X, A) = (F2 , C(X, A1 ), . . . , C(X, An ), . . . ),
where multiplication is defined for f ∈ C(X, An ) and g ∈ C(X, Am ) by
[f · g](x) = f (x) · g(x).
To see that this defines a continuous map from X to An+m , note that there is a clopen p containing
x such that f and g are constant on p. But then, f · g is constant on p. With this operation,
C(X, A) is a graded F2 -algebra.
When X has the discrete topology, write AX for C(X, A).
A graded ring homomorphism γ : A −→ B, of degree d, induces a graded ring homomorphism
γ
b : C(X, A) −→ C(X, B), of degree d, as follows :
For f ∈ C(X, An ), γ
bn (f ) = γ n ◦ f .
Theorem 7.9 Let X be a Boolean space, let G a special group and let T = C(X, G). Then, there
is a natural isomorphism, Ev : k∗ T −→ C(X, k∗ G), that sends a generator ζ = λ(f1 ) . . . λ(fn ) to
Ev(ζ) ∈ C(X, kn G), defined by
For all x ∈ X, Ev(ζ)(x) = λ(f1 (x)) . . . λ(fn (x)).
Moreover, the following diagram is commutative
26
k∗ T
Ev C(X, k∗ G)
ω
ω
b
?
k∗ T
?
-
Ev
C(X, k∗ G)
where ω is multiplication by λ(−11) in k∗ T and ω
b is the map associated to multiplication by λ(−1)
in k∗ G (b
ω n (f )(x) = λ(− 1 )f (x), for x ∈ X and f ∈ C(X, kn G), n ≥ 1).
Proof : We must verify that for each n ≥ 1, Evn is an isomorphism from kn T onto C(X, kn G).
For n ≥ 1, the map β : T n −→ C(X, kn G), defined by
For all x ∈ X, β(f1 , . . . , fn )(x) = λ(f1 (x)) . . . λ(fn (x))
Nn
is multilinear, and so it induces a group homomorphism β 0 :
i=1 k1 T −→ C(X, kn G). Now
suppose ζ = λ(f1 ) . . . λ(a)λ(ab) . . . λ(fn ), where a ∈ DT (11, b). Since isometry in T is defined
pointwise, for all x ∈ X, a(x) ∈ DG (1, b(x)). But then
β 0 (ζ)(x) = λ(f1 (x)) . . . λ(a(x))λ(a(x)b(x)) . . . λ(fn (x)) = 0 in kn G,
and so β 0 (ζ) = 0 in C(X, kn G). Therefore, β 0 factors through kn T to yield a homomorphism
Evn : kn T −→ C(X, kn G), that sends a generator ζ = λ(f1 ) . . . λ(fn ) to the continuous function
from X to kn G whose value at x ∈ X is Evn (ζ)(x) = λ(f1 (x)) . . . λ(fn (x)).
If Γ ∈ C(X, kn G), then there are {Γ1 , . . . , Γm } ⊆ kn G and a partition of X, P = {u1 , . . . , um },
such that Γ is constant, with value Γk on each uk , 1 ≤ k ≤ m; let [uk , Γk ] be the element of
C(X, kn G) that is constantly equal to Γk on uk and 0 in the complement of uk . Then,
P
Γ= m
j=1 [uk , Γk ].
Therefore, to prove that Evn is onto C(X, kn G), it is enough to show that if u is a clopen set in
X and ζ = λ(a1 ) . . . λ(an ) is a generator in kn G, then [u, ζ] is in the image of Evn . But it is
immediate from the definition of Evn that Evn (λ([u, a1 ]) . . . λ([u, an ])) = [u, ζ].
To show that Evn is injective,
Q we use the hypothesis that X is compact. With notation as
above, if P ∈ P, we have TP = p∈P G; by Theorem 5.1, for all n ≥ 1 there is an isomorphism
L
θPn : kn TP −→ p∈P kn G,
taking a generator λ(a1P ) . . . λ(anP ) to h λ(a1p )λ(a2p ) . . . λ(anp ) : p ∈ P i. Furthermore, there is an
L
injective homomorphism δ Pn :
p∈P kn G −→ C(X, kn G), obtained by setting for each x ∈ X
δ Pn (h η p : p ∈ P i)(x) = η q , where q is the unique element of P such that x ∈ q.
Since the following square is commutative on generators
kn TP
θPn
-
L
p∈P
kn G
δ Pn
(fP )n
?
(D)
?
- C(X, kn G)
kn T
Evn
we conclude that it is a commutative diagram of group homomorphisms for all P ∈ P, where
fP : TP −→ T is the SG-morphism defined in (P) (page 25).
Now suppose that Evn (η) = 0, for η ∈ kn T . Since k∗ T = lim k∗ TP (Corollary 7.8), there is
−→
27
P ∈ P and ξ ∈ kn TP such that (fP )n (ξ) = η. Since δ Pn and θPn are injective, the commutativity
of diagram (D) yields η = 0, as desired. It is clear that the diagram displayed in the statement is
commutative.
3
Theorem 7.9 has a number of interesting consequences. In particular, it furnishes a proof of
Theorem 2.9 and an answer to Problem 5.4 in the case of arbitrary powers of a finite group.
Corollary 7.10 If G is a finite special group and I is a set, then:
a) k∗ GI ≈ (k∗ G)I .
b) If G is reduced then GI is [SMC].
Proof : Since G is finite, it is compact endowed with the discrete topology. Let βI denote the
Stone-Čech compactification of the discrete space I; βI is a Boolean space (= the Stone space of
the power-set algebra of I). Further, every map h : I −→ G extends uniquely to a continuous map
b
h : βI −→ G (cf. [9], Ch. 6). Since the group operation and isometry are pointwise defined in
both GI and C(βI, G), the correspondence h 7→ b
h is an isomorphism of SGs. The result follows,
then, from Theorem 7.9 and Corollary 7.8(b).
3
Corollary 7.11 If B is a Boolean algebra, then k∗ B is the graded ring in 1.1(f ), that is,
k∗ B = (F2 , B, B, . . . , B, . . .).
Proof : Let Z2 = {1, − 1 } be the two element SG. It is clear that the isomorphism λ that converts
multiplication in addition is just the map λ : Z2 −→ F2 , where λ(1) = 0 and λ(− 1 ) = 1. Hence,
k∗ Z2 = (F2 , F2 , . . . , F2 , . . . ),
where graded multiplication is that induced by field multiplication in F2 . If B is a Boolean algebra,
let X = S(B) be the Stone space of B. By Stone’s Theorem we can identify, as special groups, B
with C(X, Z2 ) ∼
= C(X, F2 ); from Theorem 7.9 and the observations above we conclude that
k∗ B = (F2 , C(X, F2 ), . . . , C(X, F2 ), . . . )
= (F2 , B, . . . , B, . . . ),
3
as needed.
Remark. Some of the results above, for instance Corollary 7.10, hold for arbitrary topological
spaces X; indeed, generalizing 7.10.(a) one can prove that k∗ (C(X, G)) = C(X, k∗ G) if G is finite.
Others, for example Corollary 7.8, hold under the assumption that X is compact and H is an
arbitrary SG or that X is any topological space and H is finite. Since the proof of these results is
rather technical, we have preferred to omit them.
3
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M. Dickmann
Équipe de Logique Mathématique
Université de Paris VII, France
F. Miraglia
Departamento de Matemática
Instituto de Matemática e Estatı́stica
Universidade de São Paulo
C. P. 66.281 (Ag. Cidade São Paulo)
05311-970 S.Paulo, S.P. - Brazil
e-mail : [email protected]
Équipe de Topologie e Géométrie
Algébriques
Institut de Mathématiques de Jussieu
Paris, France
e-mail : [email protected]
29