3.2 The center-valued Atiyah conjecture over subfields of C
3.2.2 Base change in the algebraic and center-valued Atiyah conjecture
that the semisimple ring DL has no non-trivial zero divisors if and only if it is a division ring. We are thus left to show that ΦLK is surjective if and only ifDLis a division ring.
IfDLis a division ring, thenΦLK is clearly an isomorphism of infinite cyclic groups and in particular surjective. If DL is not a division ring, then by Corollary 1.6.5 there exists an elementx∈K0(DL) that cannot be expressed ask·[DL]and is thus not contained in the image ofΦLK.
Step 6: (b) forR=D division ring. We again writeL/K as a directed union over its finitely generated subextensionsLi/K fori∈I for some index setI, as in Step 2.
If ΦLK is not surjective, then there is somei∈ I and x ∈K0(DLi) such that x is not in the image of ΦLKi. By Step 5, there is a non-trivial zero divisor in D⊗KLi. Since the maps D⊗KLi →D⊗KL are all injective, this element remains a non-trivial zero divisor inD⊗KL.
At last, we consider a non-trivial zero divisor z ∈ D⊗KL, witnessed by an element z′ ∈D⊗KL, z′ 6= 0such that zz′ = 0. Then there exists i∈I such that z, z′ ∈D⊗KLi, which means thatD⊗KLi has non-trivial zero divisors. By Step 5, there isx∈K0(DLi) that is not contained in the image ofΦLKi. We additionally observe that, since the canonical map
Ore(Ore(D⊗KLi)⊗LiLj)→Ore(D⊗KLj)
is an isomorphism, the maps onK0 induced by the connecting maps in the directed union of ringsDLi are themselves of the formΦLLj
i fori, j∈I, i⩽j. IfΦLK were surjective, there would bey∈K0(D)such thatΦLK(y) = ΦLL
i(x). But thenΦLL
i(ΦLKi(y)) = ΦLK(y) = ΦLL
i(x), which implies that ΦLKi(y) =x sinceΦLL
i is injective by (a) applied to the semisimple Li -algebraDLi and the field extensionL/Li. This contradicts the fact thatxis not contained in the image ofΦLKi and we have established thatΦLK is not surjective.
Step 7: (b). This reduces to the situation of Step 5 as in Step 3 and 4 by noting that a direct sum of surjective maps between abelian groups is surjective if and only if every individual map is.
3.2.2 Base change in the algebraic and center-valued Atiyah conjecture
As K realizes G, the map ΦCK is also surjective since by Proposition 1.6.4 (c) and Corollary 1.6.5 twoC-representations are isomorphic if and only if they represent the same element inK0. The same argument applies toΦCL. As bothΦCK and ΦCLare isomorphisms and ΦCK = ΦCL◦ΦLK, the map ΦLK is also an isomorphism
For a group G, we denote the order of an element g ∈ G by ord(g). Recall that exp(G):=lcm{ord(g)|g∈G,ord(g)<∞}is theexponent ofG. The exponent of Gmay be infinite, but we always have exp(G)⩽lcm(G).
Proposition 3.2.9. Let G be a group with exp(G) <∞ and ω ∈ C a primitive root of unity ω of order exp(G). Then Q(ω) realizes G.
Proof. For any finite subgroup F of G we have that exp(F)|exp(G) by definition. Thus, Q(ω)contains a primitive exp(F)-th root of unity for any finite subgroupF, which suffices to realize every linear representation of F by the Brauer induction theorem, see [Ser77, 12.3].
We will also need the following generalization of a well-known notion from field theory that will allow us to invoke Proposition 3.2.6:
Definition 3.2.10. LetDbe a division ring of characteristic 0 andK⩽Z(D) a subfield of its center. We call K totally algebraically closed in Dif for every field extension L/K the ring D⊗KLis a domain.
A reader well-versed in field theory will notice that being relatively algebraically closed usually means something else and that what we call a totally algebraically closed extension is called a regular extension in the literature. We make this deliberate choice since the term “regular” is already attached to two other concepts of relevance to this thesis. There should be no potential for confusion as we are exclusively working in characteristic 0 in this chapter, where an extension D/K is totally algebraically closed in our sense if and only if it is totally algebraically closed in the usual sense.
Checking whether a given central subfieldK of a division ringDis totally algebraically closed reduces to understanding the base change to the algebraic closure of K:
Theorem 3.2.11 ([CD80, Corollary 6]). Let Dbe a division ring of characteristic 0 and K⩽Z(D) a subfield of its center. Then K is totally algebraically closed in Dif and only if D⊗KK is a division ring.
In [Jai19c], Jaikin-Zapirain has studied the base change from Q to C in the Lück approximation and the strong Atiyah conjecture for sofic groups, showing that the latter holds overC if it holds overQ. He achieves this by showing that for a sofic groupG the
∗-regular closure RCG coincides with Ore(RQ⊗QC) and is thus obtained via the type of central base change of a semisimple algebra studied in the previous section. In the final step, he uses the assumption that the strong Atiyah conjecture holds overQto apply the special case of Theorem 3.2.11 where K = K = Q. The first part of this proof applies more generally:
Theorem 3.2.12. Let G be a sofic group with lcm(G) < ∞. Let K ⩽ L ⩽ C be subfields closed under complex conjugation. Then the inclusionKG ,→LG extends to an isomorphism
Ore(RKG⊗KL)−→ R∼= LG.
If the field extension L/K is algebraic, the same statement holds also without applying Ore(?).
Proof. The inductive strategy used to show that RCG ∼=Ore(RQG⊗QC) in the proof of [Jai19c, Theorem 10.7 (2)] can be applied verbatim to the more general situation in which Q is replaced by K and C is replaced by L. Specific properties of Q are only used to conclude that extending the coefficients of a division ring from Qto some extension field does not introduce zero divisors, which is not needed for our purposes (and also far from true in our more general setting).
The second statement appears in the proof of [Jai19c, Theorem 10.7 (2)] in the induc-tive step for the algebraic closure and again does not use any specific properties of the base field.
We can now put the pieces together and obtain a base change result for the algebraic Atiyah conjecture in the presence of sufficiently many roots of unity:
Theorem 3.2.13. Let G be a sofic group with lcm(G)<∞ and K ⩽Q a subfield closed under complex conjugation. Assume that K realizes G and that G satisfies the algebraic Atiyah conjecture over Q. Then G satisfies the algebraic Atiyah conjecture over every L⩽C that contains K and is closed under complex conjugation.
Proof. We consider the following commutative diagram, where all maps are induced from the corresponding embedding of rings:
L
F⩽G
|F|<∞
K0(KF) K0(RKG)
L
F⩽G
|F|<∞
K0(QF) K0(RQG) K0(RKG⊗KQ)
f1
f2
f5
f3
f4
By Lemma 3.2.8 combined with the assumption that K realizes G, the map f2 is an isomorphism and in particular surjective. The map f3 is surjective since G satisfies the algebraic Atiyah conjecture over Q. The map f4 is an isomorphism as it is induced by the map RKG ⊗K Q → RQG, which is an isomorphism by Theorem 3.2.12 applied to the algebraic extension Q/K. Taken together, these facts imply that the concatenation f4−1◦f3◦f2 of the maps along the left and lower edge of the diagram is surjective. By commutativity, we conclude that also the concatenationf5◦f1of the maps along the upper and right edge is surjective. In particular, the map f5 is surjective.
Since G satisfies the algebraic Atiyah conjecture over Q, it also satisfies the strong Atiyah conjecture over Q by Theorem 3.1.4 and Proposition 2.5.2. As opposed to the algebraic Atiyah conjecture, the strong Atiyah conjecture clearly descends to subfields, so that it holds over every subfield of Q. We conclude from Proposition 2.4.6 that RKG
and RKG ⊗K Q∼=RQ are both semisimple. Note that a semisimple ring is in particular von Neumann regular and hence, by Proposition 1.4.5, coincides with its localization at the non-trivial zero divisors. We have thus identified f5 as the map ΦQK appearing in Proposition 3.2.6 for the semisimple ring R =RKG and L =Q. In particular, f5 is also injective, which means that the surjectivity off5◦f1 implies thatf1 is surjective. We have thus verified that the algebraic Atiyah conjecture forGholds over K and are left to show that this extends to all fields LbetweenK and C.
If Mn1(D1)⊕ · · · ⊕Mns(Ds) is the Artin–Wedderburn decomposition of RKG, then, using Proposition 3.2.6, we obtain as a consequence of the surjectivity ofΦQK thatDi⊗KQ is a domain for every i = 1, . . . , s. By Theorem 3.2.11, the field K is totally alge-braically closed in each of the Di. We are now in a position to apply the other direc-tion of Proposidirec-tion 3.2.6 to any extension L/K as in the statement of the theorem. In
that case, since K is totally algebraically closed in each of the Di, we conclude that the map ΦLK: K0(RKG) → K0(Ore(RKG ⊗K L)) is surjective. Another application of Theorem 3.2.12, this time to the extension L/K, yields that Ore(RKG ⊗K L) → RLG
is an isomorphism of rings. Combined with the surjectivity of ΦLK, we conclude that K0(RKG)→K0(RLG)is surjective.
The situation we have arrived at can be summarized in the following commutative diagram, where the upper horizontal map is surjective since we have already verified above that the algebraic Atiyah conjecture for Gholds over K:
L
F⩽G
|F|<∞
K0(KF) K0(RKG)
L
F⩽G
|F|<∞
K0(LF) K0(RLG)
Since the diagram is commutative, we read off that the lower horizontal map is sur-jective, which is precisely the statement of the algebraic Atiyah conjecture for G over L.
Corollary 3.2.14. Let Gbe a sofic group such thatlcm(G)<∞. IfGsatisfies the center-valued Atiyah conjecture over Q, thenG satisfies the center-valued Atiyah conjecture over any K⩾Q(ω) that is closed under complex conjugation, whereω is a primitiveexp(G)-th root of unity. In particular, it satisfies the center-valued Atiyah conjecture overC. Proof. The fieldQ(ω) realizesGby Proposition 3.2.9, hence the corresponding statement for the algebraic Atiyah conjecture follows from Theorem 3.2.13. Now use the equivalence to the center-valued Atiyah conjecture proved in Theorem 3.1.4.
While Corollary 3.2.14 allows us to deduce the center-valued Atiyah conjecture over most subfields ofConce we know it forQ, in certain situations the existence of sufficiently many roots of unity may not be guaranteed. For this reason, we also want to mention the following result, which allows passing to purely transcendental extensions without any assumption on the base field.
Proposition 3.2.15. Let G be a sofic group such that lcm(G) < ∞. Let K ⩽ C be a subfield and assume that G satisfies the algebraic Atiyah conjecture over K. Then G satisfies the algebraic Atiyah conjecture over every purely transcendental extension L of K. The corresponding statements hold for the center-valued Atiyah conjecture.
Proof. As in the proof of Theorem 3.2.13, we conclude from the assumption thatRKG is semisimple. Take its Artin–Wedderburn decomposition to be Mn1(D1)⊕ · · · ⊕Mns(Ds) and choose a transcendence basis X of LoverK. Then Theorem 3.2.12 implies that
RLG∼=Ore(RKG⊗KL)∼=⊕si=1Mni(Ore(Di⊗KK(X)).
Since Di ⊗K K(X) embeds into the domain Di(X), i.e., the ring of rational functions in central indeterminants X with coefficients in Di, it is itself a domain for i= 1, . . . , s.
We now deduce from Proposition 3.2.6 that the canonical map K0(RKG)→ K0(RLG) is surjective, which finishes the proof exactly as in the proof of Theorem 3.2.13.
Corollary 3.2.16. Let K be a subfield of C that
(1) is a purely transcendental extension of a subfield ofQ, or
(2) contains all roots of unity.
Then the center-valued Atiyah conjecture over K is true for all elementary amenable extensions of pure braid groups, of right-angled Artin groups, of primitive link groups, of cocompact special groups, or of products of such, assuming that the lcm of the orders of their finite subgroups is finite. In particular, the center-valued Atiyah conjecture over C holds for these groups.
Proof. First note that all groups in the classes under consideration are residually elemen-tary amenable, as is explained in the proof of [KLS17, Corollary 4.6], and in particular sofic.
IfK ⩽Q, then the center-valued Atiyah conjecture overKholds by [KLS17, Corollary 4.6].
It extends to purely transcendental extensions by Proposition 3.2.15 and Theorem 3.1.4 and holds for subfields of Cthat contain all roots of unity by Corollary 3.2.14.