aa r X i v : . [ m a t h . N T ] S e p Thoughts on the reduced Whitehead groupof the Iwasawa algebra
Irene Lau
Abstract
Let l be an odd prime and K/k a Galois extension of totally real number fields withGalois group G such that K/k ∞ and k/ Q are finite. We reduce the conjectured triviality ofthe reduced Whitehead group SK ( Q G ) of the algebra Q G = Quot(Λ G ) with the Iwasawaalgebra Λ G = Z l [[ G ]] to the case of pro- l Galois groups G and finite unramified coefficientextensions. We fix an an odd prime number l and a Galois extension K/k of totally real fields with Galoisgroup G such that k/ Q and K/k ∞ are finite. As usual, k ∞ denotes the cyclotomic Z l -extension of k . Next, the Iwasawa algebra Λ G = Z l [[ G ]] = lim ←− N ⊳ G Z l [ G/N ], where N runs through the opennormal subgroups of G , denotes the completed group ring of G over Z l and Q G = Quot( Z l [[ G ]])is its total ring of fractions with respect to all central non-zero divisors. Let K T (Λ G ) be theGrothendieck group of the category of finitely generated torsion Λ G -modules of finite projectivedimension. Then, the localization sequence of K -theory → K (Λ G ) → K ( Q G ) ∂ → K T (Λ G ) → is exact. Q G finds its way into non-commutative Iwasawa theory via this localization sequenceand a determinant map Det : K ( Q G ) → Hom( R l G, ( Q cl ⊗ Q l Q Γ k ) × ) , where Q cl is a fixed algebraic closure of Q l , the Z l -span of the irreducible Q cl -characters of G withopen kernel is named R l G and Γ k = G ( k ∞ /k ). This determinant is the translation of the reducednorm nr : K ( Q G ) → Z ( Q G ) × to Hom groups, where Z ( Q G ) is the centre of Q G . We refer to[6] for a precise definition of Det.As in the classical case of Iwasawa, Ritter and Weiss link a K -theoretic substitute ℧ ∈ K T (Λ G )of the Iwasawa module X to the Iwasawa L -function which is derived from the S -truncated Artin L -function for a finite set S of places of k containing all archimedian ones and those which ramifyin K (see e.g. [6]). This Iwasawa L -function lies in the upper Hom-group.1ith this, the main conjecture of equivariant Iwasawa theory says T here exists a unique element Θ ∈ K ( Q G ) s.t. Det(Θ) =
L. M oreover ∂ (Θ) = ℧ . The uniqueness of Θ would follow from the conjecture by Suslin that the reduced Whiteheadgroup SK ( A ) is trivial for central simple algebras A over fields with cohomological dimension ≤ Q G .Recently (compare [9]), Ritter and Weiss gave a complete proof of this main conjecture up to itsuniqueness statement whenever Iwasawa’s µ -invariant vanishes. In [3], Kakde also gave a proof.In fact, he does not restrict to 1-dimensional l -adic Lie groups as Ritter and Weiss do but gives aproof for higher dimensional admissible l -adic Lie groups. Yet, he does not consider the full ringof fractions Q G but the localization (Λ G ) S by the canonical Ore set S and proves uniquenessup to the quotient of K ((Λ G ) S ) by the image of SK (Λ G )). Thus, the question whether Θ isunique in K ( Q G ) is still open.In this paper, we reduce the Suslin conjecture for our Iwasawa algebra Q G for profinite Galoisgroups G to the conjecture for N ⊗ Q l Q U for pro- l groups U and finite unramified extensions N of Q l . Therefore, the proof of the uniqueness statement of the main conjecture is completelyreduced to pro- l groups provided that the studied objects are unaffected by passing to finiteunramified extensions of Q l .This paper contains some of the results of my PhD thesis. I would like to thank my supervisorJ¨urgen Ritter for his aid, encouragement and patience during my work on this paper. First, we recall some facts on the structure of Q G and formulate the Suslin conjecture.We keep the notation of the introduction, in particular we fix an odd prime l and a Galoisextension K/k of totally real fields with Galois group G such that k/ Q and K/k ∞ are finite.First, G splits (see [6, p. 551]): G = H ⋊ Γ with H = G ( K/k ∞ ) and Γ = h γ i ∼ = G ( k ∞ /k ) ∼ = Z l .Thus, for a central subgroup Γ l m =: Γ we get Q G = l m − M i =0 ( Q Γ )[ H ] γ i . This algebra is a finite dimensional Q Γ -algebra; in fact, it is a semisimple algebra, since theJacobson radical is trivial by [6, p. 553]. Now, let χ ∈ R l G be an irreducible Q cl -character of G with open kernel. Note that it is sufficient to regard the finite set of irreducible characters of G/ Γ because, by inflation and twist with irreducible characters ρ which fulfil res HG ρ = 1, everyirreducible χ ∈ R l G can be obtained from this set. These characters ρ will be called of type W.Because G is an l -group with l = 2, this implies that χ has a representation over Q l ( χ ) by [10].2urthermore, with η an absolutely irreducible constituent of res HG ( χ ), we define St ( η ) := { g ∈ G : η g = η } , w χ := [ G : St ( η )]and e ( η ) := η (1) | H | X h ∈ H η ( h − ) h. Ritter and Weiss showed in [6] that e χ := X η | res HG χ e ( η )is a central primitive idempotent in Q c G , that every central primitive idempotent is of the form e χ and that two central primitive idempotents e χ and e χ coincide if and only if χ = χ ⊗ ρ fora character ρ of type W .In the special case of a pro- l group G , the structure of Q G is completely known by [4]: Lemma 1
Let now G be a pro- l group and let W ′ be the simple component of Q G correspondingto the irreducible character χ ∈ R l G . We moreover choose an absolutely irreducible constituent η of res HG ( χ ) . Then,(i) W ′ ∼ = l m − M i =0 v χ − M j =0 ( Q l ( η ) ⊗ Q l Q Γ ) η (1) × η (1) γ i for v χ := min { ≤ j ≤ w χ − η γ j = η σ for some σ ∈ G ( Q l ( η ) / Q l ) } ,(ii) W ′ has centre Z ( W ′ ) ∼ = L ⊗ Q l Q Γ w χ with L = Q l ( η ) G and G = { σ ∈ G ( Q l ( η ) / Q l ) : η σ = η γ j for a ≤ j ≤ w χ − } .Moreover, G =: h σ v χ i is a cyclic group of order w χ v χ .(iii) Z ( W ′ ) has cohomological dimension cd( Z ( A )) = 3 ,(iv) dim Z ( W ′ ) W ′ = χ (1) ,(v) W ′ is split by Q l ( η ) ⊗ Q l Q Γ w χ ,(vi) W ′ has Schur index s D = w χ /v χ and(vii) W ′ ∼ = D n × n with n = χ (1) /s D and the skew field D is cyclic: D ∼ = w χ /v χ − M i =0 ( Q l ( η ) ⊗ Q l Q Γ w χ ) γ v χ i =: ( Q l ( η ) ⊗ Q l Q Γ w χ /L ⊗ Q l Q Γ w χ , σ v χ , γ w χ ) . roof: Statements (i) and (ii) can be found in [4, Prop 1], (iii) is [4, Thm 2] and [4, Thm 1]contains (iv) to (vii). (cid:3)
Because H is a finite l -group, Q l ( η ) is generated by a primitive l -power root of unity. Therefore, L = Q l ( η ) G ⊆ Q l ( η ) also is, i.e. we can fix a primitive l -power root of unity ξ s.t. L = Q l ( ξ ) . We now focus on the Suslin conjecture.
Definition 1 (i) For a field F and a central simple F -algebra A of finite degree [ A : F ] , let nr A/K denote the reduced norm from A to K . The group SK ( A ) := ker(nr A/F ) / [ A × , A × ] is called the reduced Whitehead group of A .(ii) For a semisimple algebra A = L i A i of finite degree with simple components A i , we set SK ( A ) := M i SK ( A i ) for the reduced Whitehead group of A . The reduced norm nr
A/F on A induces a homomorphism on K ( A ), which we will call reducednorm, too. We state the following well-known results without proof. Lemma 2 (i) Let A be a central simple F -algebra of finite degree. Then SK ( A ) = ker(nr A/F : K ( A ) → K ( F )) . (ii) Let A ∼ = D n × n be the full matrix ring of finite degree over a skew field D . Then SK ( A ) = SK ( D ) . (iii) For a field F , we have SK ( F ) = 1 . For further details, see e.g. [2, Part III].
Remark 1
The determonant map
Det in the main conjecture of equivariant Iwasawa theory isthe translation of the reduced norm to the language of Hom-groups. For a detailed definition ofthis
Det , we refer to [6, p. 558].
4e are now ready to state the
Conjecture
Let F be a field with cohomological dimension cd( F ) ≤ and A a central simple F -algebra of finite degree [ A : F ] . Then SK ( A ) = 1 . In the following, we will call this Suslin’s conjecture, although this is not literally Suslin’s for-mulation. But in the case of a field of cohomological dimension less than or equal to 3, this isexactly the statement of his conjecture. For details, we refer to [12].The centres of the Wedderburn components of Q G , i.e. the simple components W ′ , are ofcohomological dimension 3 for pro- l groups G by Lemma 1. As we will see in this paper, this isthe crucial case for the triviality of SK ( Q G ).Next, we list the cases for Q G which are known to have trivial reduced Whitehead group: Lemma 3
Let G be as above. Then, SK ( Q G ) = 1 in the following cases:(i) G = H × Γ is a direct product with H an l -group or of order prime to l .(ii) G is a pro- l group G with abelian subgroup of index l .(iii) G = H ⋊ Γ , where H is a finite group of order prime to l . Proof:
For l -groups H , Roquette has shown in [10] that Q l [ H ] is the direct sum of some matrixrings over fields. Therefore, Q G = ( Q Γ)[ H ] = Q Γ ⊗ Q l Q l [ H ] also is a direct sum of matrix ringsover fields and thus SK ( Q G ) = 1 by Lemma 2. This shows the first case of (i). The latterstatement of (i) is a special case of (iii).(ii) is shown in [8, p. 118].(iii) can be found in [7, Example 2, p. 169]. (cid:3) SK ( Q G ) to thepro- l case As main ingredient of our reduction, we cite the following lemma (see [7, Cor, p. 167]). Forthis, recall that, for a prime number q = l , G is a Q l - q -elementary group if G = H × Γ withΓ a central open subgroup of G isomorphic to G ( k ∞ /k ) and H a finite Q l - q -elementary group;i.e. H = h s i ⋊ H q is the semidirect product of a cyclic group h s i of order prime to q and a q -group H q whose action on h s i induces a homomorphism H q → G ( Q l ( ζ ) / Q l ). Here, ζ is a primitive root We will see in subsection 3.2 that the restrictions on H are not necessary.
5f unity of order |h s i| . For q = l , the group G is called Q l - l -elementary if G = h s i ⋊ U is thesemidirect product of a finite cyclic group h s i of order prime to l and an open pro- l subgroup U whose action on h s i induces a homomorphism U → G ( Q l ( ζ ) / Q l ) with again ζ a primitive root ofunity of order |h s i| . Lemma 4 (Ritter, Weiss)
Let
K/k be a Galois extension of totally real fields with Galoisgroup G such that K/k ∞ and k/ Q are finite. Then, SK ( Q G ) = 1 if SK ( Q G ′ ) = 1 for all open Q l - q -elementary subgroups G ′ of G and all prime numbers q ( q might be equal to l ). Thus, we have to compute SK ( Q G ) for Q l - q -elementary groups G with q running through theset of all prime numbers. Q l - l -elementary groups G We begin with the case q = l , i.e. G = h s i ⋊ U with a finite cyclic group h s i of order prime to l and U an open pro- l subgroup.We fix a finite set { β i } of representatives of the G ( Q cl / Q l )-orbits of the irreducible Q cl -charactersof h s i . Let also ζ i denote a fixed primitive l -prime root of unity with β i ( s ) = ζ i .Let U i := { u ∈ U : β ui = β i } denote the stabilizer group of β i . Clearly, U i ⊳ U and A i := U/U i ≤ G ( Q l ( β i ) / Q l ) = G ( Q l ( ζ i ) / Q l ). Thus, A i is cyclic because Q l ( β i ) = Q l ( ζ i ) is unramified over Q l .We fix a representative x i ∈ U with h x i i = U/U i = A i . Then, x i maps to some τ i under theinjection U/U i G ( Q l ( β i ) / Q l ) = G ( Q l ( ζ i ) / Q l ) and therefore the order of x i clearly is a powerof l , say l n := | U/U i | . Although n depends on i , we omit this in the notation. Moreover, for thesake of brevity, we set x := x i and τ := τ i , but still keep in mind the underlying β i . Finally, weset G i := h s i ⋊ U i .We next read the structure of Q G in these terms. For this, recall that the e i := 1 |h s i| X ν mod |h s i| tr Q l ( ζ i ) / Q l ( ζ i ( s − ν )) s ν ∈ Z l h s i are the primitive central idempotents of the group algebra Q l h s i , and furthermore they are centralidempotents of Q G . Because the e i are orthogonal in Q l h s i , we have e i e j = 0 for i = j in Q G ,too. Therefore, we conclude L i e i Q G ⊆ Q G . For Q G = M i e i Q G, it remains to show the other inclusion Q G ⊆ L i e i Q G . We use that P i e i = 1 is true in Q l h s i and therefore it is true in Q G , too. Thus, Q G = 1 · Q G ⊆ L i e i Q G . We are now ready to state Lemma 5
With the above notations, we have:(i) e i Q G i ∼ = Q l ( ζ i ) ⊗ Q l Q U i , ii) e i Q G ∼ = L l n − j =0 ( Q l ( ζ i ) ⊗ Q l Q U i ) x j ,where x acts on U i by conjugation and on Q l ( ζ i ) via τ . Proof: (i) is stated in [7, p. 160] and (ii) follows immediately by (i) and the definition of U i . (cid:3) To point out the importance of the operation of x , we will also use the notation( Q l ( ζ i ) ⊗ Q l Q U i ) ⋆ h x i := l n − M j =0 ( Q l ( ζ i ) ⊗ Q l Q U i ) x j . Proposition 1
With the above notations, the following are equivalent:(i) SK ( Q G ) = 1 .(ii) SK ( e i Q G ) = SK (( Q l ( ζ i ) ⊗ Q l Q U i ) ⋆ h x i ) = 1 for all characters β i of h s i . Proof:
This follows immediately by Q G = L i e i Q G and Lemma 5. (cid:3) As the structure of Q U i is well known from Section 2, we now examine ( Q l ( ζ i ) ⊗ Q l Q U i ) ⋆ h x i .Because ( Q l ( ζ i ) ⊗ Q l Q U i ) ⋆ h x i is isomorphic to e i Q G , this algebra is semisimple.Let W ′ be the Wedderburn component, i.e. the simple component, of Q U i corresponding to χ ∈ R l U i and set W = Q l ( ζ i ) ⊗ Q l W ′ = ( Q l ( ζ i ) ⊗ Q l Z ( W ′ )) ⊗ Z ( W ′ ) W ′ ⊆ Q l ( ζ i ) ⊗ Q l Q U i . As Q l ( ζ i ) and F ′ := Z ( W ′ ) = L ⊗ Q l Q Γ w χ are linearly disjoint over Q l , the tensor product Q l ( ζ i ) ⊗ Q l F ′ is a field and thus W is still a simple algebra and therefore a Wedderburn componentof Q l ( ζ i ) ⊗ Q l Q U i with centre F := Z ( W ) = Q l ( ζ i ) ⊗ F ′ .Then, x acts on W as it acts on Q l ( ζ i ) ⊗ Q l Q U i . This action fixes the algebra Q l ( ζ i ) ⊗ Q l Q U i as a whole, but might not fix W . If W x = W , then W x is another Wedderburn componentof Q l ( ζ i ) ⊗ Q l Q U i by the following: W x is a two-sided ideal of Q l ( ζ i ) ⊗ Q l Q U i because W is atwo-sided ideal of Q l ( ζ i ) ⊗ Q l Q U i . Furthermore, it has centre F x with F = Z ( W ). As seenabove, F = Q l ( ζ i ) ⊗ Q l L ⊗ Q l Q Γ w χ is a field and therefore F x is a field, too. But as a semisimplealgebra with a field as centre, W x is already a simple algebra. Thus, x permutes the Wedderburncomponents of Q l ( ζ i ) ⊗ Q l Q U i and W x · W = 0 if W x = W because of the orthogonality ofWedderburn components.Note that the minimal j , such that W x j = W , is an l -power because this is the length of theorbit of W in the set of Wedderburn components of Q l ( ζ i ) ⊗ Q l Q U i under the action of h x i . Proposition 2
Let W be a simple component of Q l ( ζ i ) ⊗ Q l Q U i with centre F . Set ≤ d ≤ n to be minimal such that W x ld = W . Then, ˜ W := l d − M j =0 ( W x j ⊕ W x j x ⊕ ... ⊕ W x j x l n − )7 s a simple component of ( Q l ( ζ i ) ⊗ Q l Q U i ) ⋆ h x i with centre Z ( ˜ W ) = F h x ld i =: E .Furthermore, ˜ W is the full matrix ring ˜ W = V l d × l d with V := W ⊕ W x l d ⊕ ... ⊕ W x l d ( l n − d − . Proof:
We set y := x l d and m := n − d , i.e. y l m = x l n ∈ U i .First, ˜ W is a two-sided ideal of ( Q l ( ζ i ) ⊗ Q l Q U i ) ⋆ h x i ; for this, we only have to check thatit is closed under multiplication with x , which is obvious. Thus it is the direct sum of someWedderburn components.Next, we show that the centre of ˜ W is a field, which automatically implies that ˜ W is a simplealgebra. We start with the computation of the centre Z ( V ) of V . Here, we do not consider thetrivial case d = n , i.e. V = W and therefore Z ( V ) = Z ( W ) = F is a field.We assume 0 ≤ d < n and take an element z = w + w y + ... + w l m − y l m − ∈ Z ( V ). For any w ∈ W ⊆ V , we see zw = w w + w w y − y + ... + w l m − w y − ( lm − y l m − ,wz = ww + ww y + ... + ww l m − y l m − . Because zw = wz , we conclude that w ∈ Z ( W ) = F and w w y − = ww , ... , w l m − w y − ( lm − = ww l m − . (1)Assume for the moment that w ∈ Z ( W ) = F . Then, (1) implies w w y − = w w , ... , w l m − w y − ( lm − = w l m − w. But as F = Q l ( ζ i ) ⊗ Q l L ⊗ Q l Q Γ w χ , we can specialize to w = ζ i . By definition, y does not acttrivially on ζ i (otherwise y ∈ U i ) and thus w = ... = w l m − = 0.Moreover, z fulfils yz = zy . As we have already seen that z = w ∈ F , this implies that z ∈ F h y i .Thus Z ( V ) ⊆ F h y i . Because the other inclusion Z ( V ) ⊇ F h y i is trivially true, we finally conclude Z ( V ) = F h y i . Now, we are ready to show that Z ( ˜ W ) = Z ( V ) = F h y i . For the rest of the proof, we will againallow the trivial case, i.e. 0 ≤ d ≤ n . We use the relation˜ W = l d − M j =0 ( W x j ⊕ W x j x ⊕ ... ⊕ W x j x l n − )= l d − M j =0 ( V x j ⊕ V x j x ⊕ ... ⊕ V x j x l d − ) . ≤ j ≤ l d −
1. Because W x j = W , we have seen W x j · W = 0 and therefore V x j · V = 0.We choose z = P l d − i,j =0 v x j ij x i ∈ Z ( ˜ W ), and v, v ′ ∈ V . Then zv = X i,j v x j ij v x − i x i = v v + X i> ( v i,l d − i v x − ld ) x ld − i x i ∈ V ⊕ l d − M i =1 V x ld − i x i ,vz = X i,j vv x j ij x i = X i vv i x i = vv + X i> vv i x i ∈ V ⊕ l d − M i =1 V x i . Thus, v ∈ Z ( V ) and the orthogonality of the V x j implies v i,l d − i = 0 = v i for all i >
0. Next, zv x = X i,j v x j ij v x − i +1 x i = ( v v ) x + v vx + X i> ( v i,l d − i +1 v x − ld ) x ld − i +1 x i ,v x z = X i,j v x v x j ij x i = X i ( vv i ) x x i = ( vv ) x + ( vv ) x x + X i> ( vv i ) x x i . Thus, v ∈ Z ( V ), v = 0 = v and v i,l d − i +1 = 0 = v i for all i >
1. Analogous computationsfor zv x ν = v x ν z finally lead to z = v + v x + ... + v x ld − l d − with v i ∈ Z ( V ). We apply this together with the orthogonality and compute z ( v + v ′ x i ) = v v + v v ′ x, ( v + v ′ x i ) z = vv + v ′ v i x = v v + v i v ′ x with 1 ≤ i ≤ l d −
1. Thus, v i = v for all 1 ≤ i ≤ l d − z ∈ { P l d − j =0 v x j : v ∈ Z ( V ) } ∼ = Z ( V ), i.e. Z ( ˜ W ) ⊆ Z ( V ). For theother inclusion, it remains to show that elements of { P l d − j =0 v x j : v ∈ Z ( V ) } are already central in˜ W . For this, we only have to check that P l d − j =0 v x j commutes with x for every v ∈ Z ( V ) = F h x ld i : l d − X j =0 v x j x = l d − X j =0 v x j +1 = l d − X j =1 v x j + v x ld = l d − X j =1 v x j + v = l d − X j =0 v x j . Hence, Z ( ˜ W ) = Z ( V ) is true. This moreover shows that ˜ W is a Wedderburn component of( Q l ( ζ i ) ⊗ Q l Q U i ) ⋆ h x i .It remains to show that ˜ W is the claimed matrix ring. First, if ˜ W is a matrix ring over V , thenthe dimension is clear becausedim Z ( ˜ W ) ˜ W = dim Z ( V ) ˜ W = dim Z ( V ) l d − M j =0 ( V x j ⊕ V x j x ⊕ ... ⊕ V x j x l d − )= l d dim Z ( V ) V and therefore dim V ˜ W = dim Z ( V ) ˜ W / dim Z ( V ) V = l d .9oth V and ˜ W are central simple algebras over F h y i . We show that V ∼ ˜ W in Br( F h y i ), i.e. that V and ˜ W are full matrix rings over the same skew field D of centre F h y i .For the computation of the skew field D in ˜ W , we recall the fact that there exists a primitiveidempotent ε of ˜ W such that D ∼ = ε ˜ W ε and ˜ W ∼ = B n for a minimal right ideal B = ε ˜ W of ˜ W . Analogously, there exists a primitive idempotent ε V ∈ V with ε V V ε V ∼ = D V a skewfield and S = ε V V a minimal right ideal of V . Then, we get ε V ˜ W ε V = ε V V ε V because for P l d − i,j =0 v x j ij x i ∈ ˜ W , we achieve ε V · l d − X i,j =0 v x j ij x i · ε V = ε V · l d − X i,j =0 v x j ij ε x − i V x i = ε V v ε V + ε V · l d − X i =1 ( v x ld i,l d − i ) x − i ε x − i V x i = ε V v ε V + l d − X i =1 ε V ( v x ld i,l d − i ) x − i x i · ε V = ε V v ε V . For = and =, we have again used V x j · V = 0 for 1 ≤ j ≤ l d − ε V ˜ W is a right ideal of ˜ W , there exists a 0 < r ∈ N with B r ∼ = ε V · ˜ W for the minimalright ideal B . Because End ˜ W ( B ) ∼ = D is the skew field lying in ˜ W , we get D V ∼ = ε V V ε V = ε V ˜ W ε V ∼ = End ˜ W ( ε V ˜ W ) ∼ = End ˜ W ( B r ) ∼ = End ˜ W ( B ) r × r ∼ = D r × r . This forces r = 1 (because, for example, D V does not have zero divisors whereas D r × r has for r > V and ˜ W are equal, i.e. V ∼ ˜ W in Br( E ). By V ⊆ ˜ W ,this implies the claim and concludes the proof. (cid:3) Remark 2
The ˜ W in Proposition 2 exhaust all simple components of ( Q l ( ζ i ) ⊗ Q l Q U i ) ⋆ h x i because Q l ( ζ i ) ⊗ Q l Q U i = M W W and therefore ( Q l ( ζ i ) ⊗ Q l Q U i ) ⋆ h x i = M W W ! ⋆ h x i = M ˜ W ˜ W .
Corollary 1
With the notations of Proposition 2, we have SK ( ˜ W ) = SK ( V ) . roof: This is obvious, as the reduced Whitehead group of a simple algebra only depends onthe underlying skew field but not on the matrix degree. (cid:3)
Since the Wedderburn components ˜ W are classified now, we can start our study of SK ( ˜ W ). Theorem 1
Let G = h s i ⋊ U be a Q l - l -elementary group with a finite cyclic group h s i of orderprime to l and U an open pro- l subgroup. Assume that SK ( Q l ( ζ i ) ⊗ Q l Q U i ) = 1 for all i . Then SK ( Q G ) = 1 . The proof of this theorem depends on the number l d = min { ≤ j ≤ l n : W x j = W } . d = n First, let d = n , i.e. W x j = W for all 1 ≤ j ≤ l n −
1. Thus,˜ W = l n − M j =0 ( W x j ⊕ ... ⊕ W x j x l n − ) , V = W. Then, Proposition 2 implies that ˜ W = V l n × l n = W l n × l n . Furthermore, both W and ˜ W have centre Z ( W ) = Z ( ˜ W ) = F . (Observe that F commutes with x l n because x l n ∈ U i and F = Z ( W ) for aWedderburn component W of Q l ( ζ i ) ⊗ Q l Q U i .) By Corollary 1, together with the preconditionthat SK ( Q l ( ζ i ) ⊗ Q l Q U i ) = 1 and in particular SK ( W ) = 1, this also implies Proposition 3
With the above notations, assume W x j = W for all ≤ j ≤ l n − and SK ( Q l ( ζ i ) ⊗ Q l Q U i ) = 1 for all i . Then SK ( ˜ W ) = SK ( W ) = 1 . (cid:3) d = 0Next, we consider d = 0, i.e. W x = W . Thus,˜ W = l n − M j =0 W x j , V = ˜ W .
This time, Z ( ˜ W ) = F h x i = E with [ F : E ] = l n and G ( F/E ) = h σ i , where σ is induced by theconjugation by x . Note that h σ i ∼ = h x i ∼ = h τ i ⊆ G ( Q l ( ζ i ) / Q l )with τ as above induced by the action of x on Q l ( ζ i ).First, we give a brief outline of the proof of SK ( ˜ W ) = SK ( V ) = 1.11 tep 1 F ⊗ E V ∼ = W l n × l n ⊆ V l n × l n . Step 2
There exists a w ∈ W l n × l n such that the conjugation by w − x is the automorphism C w − x = σ ⊗ on W l n × l n . It is of order l n and ( w − x ) l n ∈ Z ( W l n × l n ) = F . Step 3 ( w − x ) l n = 1 and therefore A = ( F/E, σ, ( w − x ) l n ) ⊆ V l n × l n is a central simple split E -algebra. Step 4 Z V ln × ln ( A ) = ( W l n × l n ) h w − x i . Step 5 V l n × l n ∼ = A ⊗ E ( W l n × l n ) h w − x i . Step 6 SK ( V ) = SK (( W l n × l n ) h w − x i ) = 1 . We now start with the sketched computations. We can read V as free left W -module of rank l n with basis 1 , x, ..., x l n − , i.e. V = L l n − j =0 W x j . This allows us to formulate Lemma 6
With the above notations, we have an isomorphism F ⊗ E V ∼ = −→ W l n × l n = Hom W ( V, V ) , f ⊗ v l f ◦ r v , where l f resp. r v denotes the left resp. right multiplication with f ∈ F resp. v ∈ V . Remark 3
In particular, f ⊗ ∈ F ⊗ E V maps to the diagonal matrix f · with the unitmatrix. Proof: F ⊗ E V and W l n × l n are isomorphic by [5, Cor 7.14], we only have to substitute K by E , A by V and B by F . Then, we get r = l n and the centralizer B ′ = Z V ( F ) = W implies F ⊗ E V = F op ⊗ E V ∼ = W l n × l n .We will call the stated isomorphism ϕ for the moment. We read the actions of the W -endomorphismsof V by the right to ensure that ϕ is compatible with multiplication. For this, take f, f ′ ∈ F and v, v ′ , a ∈ V . Then the commutativity of F yields( a )( ϕ ( f ⊗ v ) ◦ ϕ ( f ′ ⊗ v ′ )) = ( a )(( l f ◦ r v ) ◦ ( l f ′ ◦ r v ′ ))= f ′ f avv ′ = f f ′ avv ′ = ( a )( l ff ′ ◦ r vv ′ ) = ( a ) ϕ ( f f ′ ⊗ vv ′ )= ( a ) ϕ (( f ⊗ v )( f ′ ⊗ v ′ )) . It now easily follows that ϕ is a homomorphism of E -algebras. F ⊗ E V is simple because V is a central simple E -algebra. Thus, ϕ = 0 implies that ϕ is injective.By dimension comparison, it is surjective as well. (cid:3) C w − x on W l n × l n . On the one hand, conjugation by x is anautomorphism c x on W and can therefore be extended to W l n × l n by letting it act on the matrixentries. Furthermore, we can read x as the diagonal matrix M x = x · in V l n × l n . Then, theextension of c x on W l n × l n is the conjugation by this matrix M x . This automorphism on W l n × l n will be called C x in the sequel and we remark that C x acts on F = F · = F ⊗ E E as σ , with h σ i = G ( F/E ) as above.On the other hand, σ ⊗ F ⊗ E V → F ⊗ E V is another automorphism on W l n × l n . As therestriction of σ ⊗ F ⊗ E E is by construction the old isomorphism σ , the actions of C x and σ ⊗ F = F ⊗ E E .Therefore, C x ( σ ⊗ − is a central automorphism on W l n × l n , i.e. it acts trivially on the centre Z ( W l n × l n ) = F · = F . The theorem of Skolem-Noether now implies that C x ( σ ⊗ − is theconjugation C w by some w ∈ W l n × l n , i.e. σ ⊗ C − w C x = C w − x . As ( σ ⊗ l n = id, we furthermore conclude ( C w − x ) l n = id. This means that the conjugation by( w − x ) l n = ( w − ) x − + ... + x − ln +1 x l n is trivial on W l n × l n and, as x l n ∈ W (more precisely x l n ∈ Q ( ζ i ) ⊗ Q l Q U i has a component in W but we suppress this here for the sake of brevity), we conclude( w − x ) l n ∈ Z ( W l n × l n ) = F · = F. Finally, we choose A = ( F ⊗ E E/E ⊗ E E, σ ⊗ C w − x , ( w − x ) l n ) = ( F/E, σ, ( w − x ) l n ) . By construction, we have w ∈ W l n × l n and x ∈ V . Therefore, w − x ∈ l n − M j =0 W l n × l n x j = V l n × l n and hence A ⊆ V l n × l n . Lemma 7
Let A = ( F/E, σ, ( w − x ) l n ) be as above. Then A splits, i.e. A ∼ E in Br( E ) . Proof:
The cyclic algebra A splits if ( w − x ) l n is a norm element in E , i.e. if there exists anelement f ∈ F with N F/E ( f ) = ( w − x ) l n , where N F/E = N h σ i is the Galois norm of the fieldextension F/E . To show this, we compute ( w − x ) l n explicitely.First, let k x denote the conjugation by x on V , s.t. we can study the automorphism σ ⊗ k x : F ⊗ E V → F ⊗ E V.
13y Lemma 6, we know F ⊗ E V ∼ = W l n × l n = Hom W ( V, V ) , f ⊗ v l f ◦ r v . Here, we choose a basis 1 , x, ..., x l n − of the free left W -vector space V . Then, we write v = P l n − i =0 w i x i and achieve that l f ◦ r v is represented by the matrix f w f w . . . f w l n − f w x − l n − x l n f w x − . . . . ... ... ... ... f w x − ( ln − x l n . . . . f w x − ( ln − . Here, we recall that we write the matrices from the right. Next,( σ ⊗ k x )( f ⊗ v ) = σ ( f ) ⊗ v x ↔ σ ( f ) w x σ ( f ) w x . . . σ ( f ) w xl n − σ ( f ) w l n − x l n σ ( f ) w . . . . ... ... ... ... σ ( f ) w x − ( ln − x l n . . . . . . A comparison of the two matrices shows that σ ⊗ k x is the conjugation by x on W l n × l n , i.e. σ ⊗ k x = C x . Next, we obtain C x ◦ ( σ ⊗ − = ( σ ⊗ k x ) ◦ ( σ ⊗ − = 1 ⊗ k x = C ⊗ x : F ⊗ E V → F ⊗ E V. Now, we see that 1 ⊗ x = w ∈ W l n × l n is the element s.t. C w − x = σ ⊗ V l n × l n , we can write w − x = ( x − w ) − = x − x − . . . . . . x − · . . .
00 0 1 . . . . . . x l n . . . − = x − . . .
00 0 x − . . . . . . x − x l n − . . . −
14e finally conclude that ( w − x ) l n = 1which is certainly a norm element. (cid:3) Lemma 8
Set A = ( F/E, σ, ( w − x ) l n ) and V as above. Then Z V ln × ln ( A ) = ( W l n × l n ) h w − x i . Proof:
First, we have to read A in V l n × l n . For this, we observe that E and F are to be representedby the diagonal matrices E = E · and F = F · . Then, choose a matrix ( v ij ) i,j ∈ Z V ln × ln ( A )and f · ∈ F · . We get f − ( v ij ) f = ( f − v ij f ) ! = ( v ij ) , i.e. f − v ij f ! = v ij for all i, j = 0 , ..., l n −
1. As this equation has to be fulfilled for all f ∈ F , but v ij = w + ... + w l n − x l n − ∈ V , we conclude that v ij = w ∈ W . Therefore, Z V ln × ln ( A ) ⊆ W l n × l n .Next, we conjugate by w − x and see Z V ln × ln ( A ) ⊆ ( W l n × l n ) h w − x i .It remains to show the other inclusion ( W l n × l n ) h w − x i ⊆ Z V ln × ln ( A ). But this is obvious, because A = L l n − j =0 F · ( w − x ) j and v ∈ ( W l n × l n ) h w − x i commutes with w − x as well as with a ∈ F · = Z ( W l n × l n ). Thus, v commutes with a + ... + a l n − ( w − x ) l n − ∈ L l n − j =0 F ( w − x ) j = A , too. (cid:3) Corollary 2 ( W l n × l n ) h w − x i is a central simple Z ( A ) = E -algebra. Proof:
This is true by the centralizer theorem. (cid:3)
Lemma 9
With the above notations, we have V ∼ = ( W l n × l n ) h w − x i . Moreover, F ⊗ E ( W l n × l n ) h w − x i ∼ = −→ W l n × l n , f ⊗ w f w, and A ⊗ E Z V ln × ln ( A ) ∼ = −→ V l n × l n , a ⊗ v av, are isomorphisms. Proof:
First, V ∼ ( W l n × l n ) h w − x i in Br( E ) by the centralizer theorem which states that Z V ln × ln ( A ) ∼ A op ⊗ E V l n × l n in Br( E ). By Lemma 7, we know that A ∼ E in Br( E ). Because A op is the inverse of A in Br( E ),we conclude A op ∼ E in Br( E ), too. Thus,( W l n × l n ) h w − x i = Z V ln × ln ( A ) ∼ A op ⊗ E V l n × l n ∼ E ⊗ E V l n × l n ∼ V. E :[( W l n × l n ) h w − x i : E ] = [ W l n × l n : E ] /l n = l n [ W : E ] = l n [ W : F ][ F : E ] = l n [ W : F ]and [ V : E ] = [ V : W ][ W : F ][ F : E ] = l n [ W : F ] . Thus, V and ( W l n × l n ) h w − x i are as Brauer equivalent algebras of the same degree isomorphic.We turn to the second isomorphism. As Z V ln × ln ( A ) = ( W l n × l n ) h w − x i is a central simple E -algebra, F ⊗ E ( W l n × l n ) h w − x i is a central simple F -algebra. We compute the respective degreesover F : [ V l n × l n : E ] = [ V l n × l n : W l n × l n ][ W l n × l n : F ][ F : E ] = l n [ W l n × l n : F ]and, by the centralizer theorem,[ V l n × l n : E ] = [ A : E ][ Z V ln × ln ( A ) : E ]= [ F : E ] [( W l n × l n ) h w − x i : E ] = l n [( W l n × l n ) h w − x i : E ]imply [ W l n × l n : F ] = [( W l n × l n ) h w − x i : E ] = [( W l n × l n ) h w − x i ⊗ E F : E ⊗ E F ]= [( W l n × l n ) h w − x i ⊗ E F : F ] . Next, F ⊗ E ( W l n × l n ) h w − x i → W l n × l n , f ⊗ w f w , is injective because otherwise the kernel wouldform a non-trivial two-sided ideal. But F ⊗ E ( W l n × l n ) h w − x i is a central simple F -algebra. Thus,the only non-trivial two-sided ideal is F ⊗ E ( W l n × l n ) h w − x i itself, which is impossible because E ⊗ E ( W l n × l n ) h w − x i ⊆ F ⊗ E ( W l n × l n ) h w − x i maps to ( W l n × l n ) h w − x i ⊆ W l n × l n and thus thekernel can not be F ⊗ E ( W l n × l n ) h w − x i . This implies F ⊗ E ( W l n × l n ) h w − x i ⊆ W l n × l n . As bothsides are of the same degree over F , we conclude F ⊗ E ( W l n × l n ) h w − x i = W l n × l n .Finally, we show A ⊗ E Z V ln × ln ( A ) ∼ = V l n × l n . As A and Z V ln × ln ( A ) are central simple E -algebras, A ⊗ E Z V ln × ln ( A ) is a central simple E -algebra, too. We again show that the respective degreesover E coincide: [ V l n × l n : E ] = [ A : E ][ Z V ln × ln ( A ) : E ] = [ A ⊗ E Z V ln × ln ( A ) : E ] . Next, the homomorphism A ⊗ E Z V ln × ln ( A ) → V l n × l n , a ⊗ v av , again is injective. Dimensioncomparison implies A ⊗ E Z V ln × ln ( A ) = V l n × l n . (cid:3) Proposition 4
With the above notations, assume that W x = W and moreover SK ( Q l ( ζ i ) ⊗ Q l Q U i ) = 1 for all i . Then SK ( ˜ W ) = SK (( W l n × l n ) h w − x i ) = 1 . roof: We are still in the case V = ˜ W . With V l n × l n = A ⊗ E Z V ln × ln ( A ), it therefore suffices tocompute SK ( V ) = SK ( V l n × l n ) = SK ( A ⊗ E ( W l n × l n ) h w − x i ) = SK ( A ) × SK (( W l n × l n ) h w − x i ) = 1 × SK (( W l n × l n ) h w − x i ) . in the sequel. For =, we use that A splits and therefore SK ( A ) = 1; moreover, A and( W l n × l n ) h w − x i have coprime Schur indices which implies = by [2, Lem 5, p. 160].Now, we choose a v ∈ V l n × l n with nr V ln × ln /E ( v ) = 1. It represents an element in SK ( V l n × l n ).By the above, the class of v can be read as[ v ] = (1 , [˜ v ]) = [1 ⊗ ˜ v ] = [˜ v ]with ˜ v ∈ ( W l n × l n ) h w − x i ⊆ V l n × l n andnr ( W ln × ln ) h w − x i /E (˜ v ) = 1 . Therefore, v and ˜ v only differ by a factor in [( V l n × l n ) × , ( V l n × l n ) × ]. It hence suffices to show that˜ v ∈ [( V l n × l n ) × , ( V l n × l n ) × ] for SK ( V l n × l n ) = 1.For the computation of nr ( W ln × ln ) h w − x i /E , let M be a splitting field of ( W l n × l n ) h w − x i with M ⊇ F .Thus, as ( W l n × l n ) h w − x i ⊆ F ⊗ E ( W l n × l n ) h w − x i = W l n × l n , we get M m × m = M ⊗ E ( W l n × l n ) h w − x i = M ⊗ F F ⊗ E ( W l n × l n ) h w − x i = M ⊗ F W l n × l n for a certain m ∈ N , i.e. M is also a splitting field of W l n × l n . This implies1 = nr ( W ln × ln ) h w − x i /E (˜ v ) = nr W ln × ln /F (˜ v ) , where = holds due to the isomorphism F ⊗ E ( W l n × l n ) h w − x i = W l n × l n , 1 ⊗ ˜ v · ˜ v andthe common splitting field M ⊇ F ⊇ E of ( W l n × l n ) h w − x i and W l n × l n . But, by assumption, SK ( W l n × l n ) = SK ( W ) = 1 and hence˜ v ∈ [( W l n × l n ) × , ( W l n × l n ) × ] ⊆ [( V l n × l n ) × , ( V l n × l n ) × ] . This concludes the proof. (cid:3) .1.3 The intermediate case < d < n Finally, the triviality of SK ( ˜ W ) in the intermediate cases for 0 < j < n is a consequence of theextreme cases: We fix a 0 < d < n and set y := x l d and m := n − d . Thus, V = W ⊕ W x l d ⊕ ... ⊕ W x l d ( l n − d − = W ⊕ W y ⊕ ... ⊕ W y ( l n − d − = l m − M j =0 W y j . As ˜ W = V l d × l d , it suffices to compute SK ( V ). But V is now of the same form as ˜ W in the case d = 0, with x replaced by y and n replaced by m . Thus, we only have to check that the abovearguments apply to this V in the same manner. As it can be seen easily that we can copy theabove literally, we leave this to the reader.Hence, we have seen that SK ( ˜ W ) = 1 for every Wedderburn component of Q G . This concludesthe proof of Theorem 1. (cid:3) Q l - q -elementary groups G Next, we consider the case of Q l - q -elementary groups G with q = l . Here, our result on thetriviality of the reduced Whitehead group is stronger than in the case q = l because it holdswithout assumptions: Theorem 2
Let G be a Q l - q -elementary group with q = l prime. Then SK ( Q G ) = 1 . The proof of this theorem closely follows the proof of Theorem 1. Thus, we only give a shortoutline how to adapt the ideas used for the case q = l to our new situation.To do so, we first recall that the Q l - q -elementary group G is a direct product G = H × Γ with H a finite Q l - q -elementary group. More precisely, H = h s i ⋊ H q with h s i a cyclic group of orderprime to q and a q -group H q whose action on h s i induces a homomorphism H q → G ( Q l ( ζ ) / Q l )for ζ a primitive root of unity of order |h s i| .We now take h s l i the l -Sylow subgroup of h s i , thus h s i = h s l i × h s ′ i , and obtain G = h s l i ⋊ U with U = ( h s ′ i ⋊ H q ) × Γ and h s ′ i ⋊ H q an l -prime group. Still, U acts on h s l i via Galois automorphisms.As in the Q l - l -elementary case, we fix a finite set { β i } of representatives of the G ( Q cl / Q l )-orbitsof the irreducible Q cl -characters of h s l i . Let also ζ i denote a fixed primitive root of unity with β i ( s ) = ζ i . This time, Q l ( ζ i ) is not unramified because h s l i is an l -group, but Q l ( ζ i ) / Q l remainscyclic because l is odd. Let again U i := { u ∈ U : β ui = β i } denote the stabilizer group of β i .Thus, U i ⊳ U and A i := U/U i ≤ G ( Q l ( β i ) / Q l ) = G ( Q l ( ζ i ) / Q l ) is cyclic because G ( Q l ( ζ i ) / Q l )is. We fix a representative x ∈ U with h x i = U/U i = A i . Then, x maps to some τ under the18njection U/U i G ( Q l ( ζ i ) / Q l ); and | x | = | U/U i | =: l n with again x , τ and n depending on i .Finally, we set G i := h s l i ⋊ U i .Now, we may compute the isomorphism Q G = M i e i Q G ∼ = M i ( Q l ( ζ i ) ⊗ Q l Q U i ) ⋆ h x i . In this case that q = l , the group U i is not pro- l and therefore the structure of Q U i differs fromthe structure of the analogous object in the pro- l case as stated in Section 2. Yet, we may collectall relevant information on Q U i easily. Recall that U is the direct product U = ( h s ′ i ⋊ H q ) × Γand therefore U i also is the direct product U i = H ′ × Γ with H ′ a subgroup of h s ′ i ⋊ H q . Now, [1,74.11, p. 740] implies that Q U i is the direct sum of matrix rings over the fields F ′ = Q l ( η ′ ) ⊗ Q l Q Γfor certain characters η ′ of H ′ . Because F ′ and Q l ( ζ i ) are linearly disjoint over Q l , the algebra Q l ( ζ i ) ⊗ Q l Q U i also is a direct sum of matrix rings over fields F = Q l ( ζ i ) ⊗ Q l F ′ . This proves Proposition 5
With the above notations, we have SK ( Q l ( ζ i ) ⊗ Q l Q ( U i )) = 1 . This proposition allows us to formulate Theorem 2 in the stronger form without any assumptionson SK ( Q l ( ζ i ) ⊗ Q l Q ( U i )).It now remains to examine the semisimple algebra ( Q l ( ζ i ) ⊗ Q l Q U i ) ⋆ h x i . We may transfer thecomputation for the case q = l directly to our new situation.This concludes the proof of Theorem 2. (cid:3) References [1] C. W. Curtis and I. Reiner.
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