Multidimensional local skew-fields (792481), страница 8
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In fact, these lemmas are most important technical statementsabout the maps δm and their generalisations.We change the notation in this subsection and use the notation of [9], because it ismore convenient for applications in the valuation theory. We always denote here by D adivision algebra finite dimensional over its centre F = Z(D). Recall that any Henselianvaluation on F has a unique extension to a valuation on D.Given a valuation v on D, we denote by ΓD its value group, by VD its valuationring, by MD its maximal ideal and by D̄ = VD /MD its residue division ring.By [31], p.21 one has the fundamental inequality[D : F ] ≥ |ΓD : ΓE | · [D̄ : F̄ ].34D is called defectless over F if equality holds and defective otherwise.
It is known thatD is defectless if it has a discrete valuation of rank 1.Jacob and Wadsworth in [9] introduced the basic homomorphismθD : ΓD /ΓF → Gal(Z(D̄)/F̄ )induced by conjugation by elements of D. They showed that θD is surjective and Z(D̄)is the compositum of an Abelian Galois and a purely inseparable extension of F̄ .We say D is tame division algebra if char(F̄ ) = 0 or char(F̄ ) = q = 0, D is defectlessover F , Z(D̄) is separable over F̄ , and q ||ker(θD )|. We say D is wild division algebraif it is non tame.We call a division algebra D inertially split if Z(D̄) is separable over F̄ , the mapθD is an isomorphism, and D is defectless over F .0.4.2Cohen’s theoremThere is a natural question if there exists a generalisation of Cohen’s theorem, i.e.
isany central division algebra splittable or not. It is not true if a division algebra is notfinite dimensional over its centre, as Dubrovin’s example shows. It is not true also forfinite dimensional division algebras, as we will see in Wadsworth’s example below. Butit is true for tame division algebras over complete discrete valued fields.
This easilyfollows from results of Jacob and Wadsworth [9].Theorem 0.37 Let (F, v) be a valued field which is complete with respect to a discreterank 1 valuation v. Suppose charF = charF̄ . Let D be a tame division algebra withZ(D) = F and [D : F ] < ∞.Then there exists a section D̄ → D of the residue homomorphism D → D̄.Proof.
Since F is a complete field, F is a Henselian field and v extends uniquely toa valuation w on D. Since D is tame, Z(D̄)/Z(D) is a cyclic Galois extension. Thereexists an inertial lift Z of Z(D̄) over F , Z is Galois over F , and by classical Cohen’stheorem there exists a section Z̃(D̄) → Z.Consider the centraliser C = CD (Z) of Z in D.
Then we have C̄ = D̄.Indeed, by Double Centraliser Theorem we have [D : F ] = [C : F ][Z : F ]and [Z : F ] = |Gal(Z(D̄)/F̄ )|. By [9], prop.1.7 a homomorphism θD : ΓD /ΓF →Gal(Z(D̄)/F̄ ) is surjective, so for any parameter z we have θD (w(z)) = σ, where< σ >= Gal(Z(D̄)/F̄ ). It is clear that z ∈/ C. Now let u1 , . . . , u[C:F ] be a F -basis ofC. It is easy to see that the elements uj , zuj , . .
. , z n−1 uj , j = 1, . . . , [C : F ], wheren = ord(σ), the order of σ, are linearly independent, so form a basis for D over F .Sincew(F < zuj , . . . , z n−1 uj , j = 1, . . . , [C : F ] >) ∩ ΓC = 0,35where F < zuj , . . . , z n−1 uj , j = 1, . . . , [C : F ] > denote a vector space in D over Fgenerated by elements uj z i , this implies that for any element x ∈ D with w(x) = 0 wecan find elements r1 , . .
. r[C:F ] ∈ F such that x = r1 u1 + . . . + r[C:F ] u[C:F ] mod MD .Hence C̄ = D̄.Fix an embedding F̄ → F and consider the algebra A = C̄ ⊗F Z(C). It is easy tosee that A is an unramified division algebra with Ā = C̄ = D̄. Therefore by [2], Th.31,A∼= C; so there exists a section D̄ → C.The theorem is proved.2Example (Wadsworth).
Let p be any prime number, let k = Z/pZ, the field withp elements, and let r, s be independent indeterminates over k. Let F = k(r, s)((t)), theformal Laurent series field in t over the rational function field k(r, s). F has its completediscrete t-adic valuation with residue field F̄ = k(r, s). Let f = xp + tx − r in F [x].The derivative test shows that f has no repeated roots.
Let θ be a root of f , and letK = F (θ), which is separable over F .Let M be the separable closure of K over F . So, the Galois group Gal(M/F ) isisomorphic to a subgroup of the symmetric group Sp . Let L be the fixed field of ap-Sylow subgroup of Gal(M/F ), and let σ be a generator of Gal(M/L), a cyclic groupof order p. The valuation on F extends uniquely to complete discrete valuations on Land on M . Note that L̄ doesn’t contain r1/p , since [L̄ : F̄ ] divides [L : F ], which isprime to p.
(For the same reason, L̄ doesn’t contain a p-th root of s.) But M̄ contains θ̄, which is a p-th root of r. So, [M̄ : L̄] = [M : L] = p, and M̄ = L̄(r1/p ), which is purelyinseparable over L̄. Since σ acts trivially on M̄ , the norm map from M to L inducesthe p-th power map from M̄ to L̄. So, s is not in the image of the norm from M to L.Therefore, the cyclic algebra D = (M/L, σ, s) is nonsplit of degree p, so it is a divisionalgebra.
With respect to the unique extension of the valuation on L to D, we have D̄contains a pth root of r and also of s, so p2 = [D : L] ≥ [D̄ : L̄] ≥ [L̄(r1/p , s1/p )] = p2 .This shows that D̄ is the field L̄(r1/p , s1/p ), which is purely inseparable over L̄.Hence also, the ramification index of D over L must be 1.0.4.3Decomposition theoremIn this part we prove a generalisation of Jacob-Wadsworth’s decomposition theorem([9], Th.6.3., lemma 6.2) for finite dimensional splittable division algebras over a Laurent series field k((t)), chark > 2.So, in this section we consider only splittable division algebras.
Moreover, we will needmore strong condition:Definition 0.38 A division algebra D is called good splittable if there exists a sectionD̄ → D compatible with an embedding Z(D) → Z(D), i.e. Z(D) ⊃ Z(D) ⊂ D̄.36It’s easy to see that all tame division algebras are good splittable, because byCohen’s theorem any embedding Z(D) → Z(D) can be uniquely extended to anyseparable extension of Z(D).We note that the skew field K from theorem 0.36 is good splittable if K is a finitedimensional division algebra over its center. Indeed, because of the condition of thetheorem, we can assume k is an algebraically closed field.
Then the center of K is aC2 -field by Tsen’s theorem (see the definition and the properties of C2 -fields below,at the end of this subsection). Then it will be shown in corollary 8 that all divisionalgebras over C2 -fields are good splittable.For division algebras of index p over a Laurent series field the condition to be splittable is equivalent to the condition to be good splittable, see the end of this subsection.Let D be a finite dimensional division algebra over a complete valued field F =k((t)). Let w be a unique extension of the valuation v to D. We will denote by z anyparameter of D, i.e. any element with < w(z) >= ΓD .Proposition 0.39 D is isomorphic to a local skew-field D̄((z)), wherezaz −1 = α(a) + δ1 (a)z + δ2 (a)z 2 + .
. . ,a ∈ D̄;here α : D̄ → D̄ is an automorphism and δi : D̄ → D̄ are linear maps such that themap δi satisfy the identityδi (ab) =iσ(δ i−k α)(a)σ(Sik α)(b),a, b ∈ D̄k=0The proof is an easy combination of the proofs of propositions in section 1 andCohen’s theorem 0.37.Definition 0.40 Let us define mapsm δi: D̄ → D̄, m ∈ Z, i ∈ N as follows.z m az −m = αm (a) + m δ1 (a)z + m δ2 (a)z 2 + . . .
,If m = 0, putm δia ∈ D̄.= 0.Note that if α = id, then m δi = 0 for m = pk , where k is sufficiently large, kdepends on i. Moreover, m δi = m+pk δi for k sufficiently large. We will use also thefollowing notation:˜m δi = −m δi .Proposition 0.41 (i) The mapsm δi (ab)= m δi (a)αi+mm δisatisfy the following identities:m(b) + α (a)m δi (b) +i−1k=137m δi−k (a)i−k+m δk (b)(ii) Suppose α = id. Then the mapsm δi (ab)= m δi (a)b + am δi (b) +m δii−1satisfy the following identities:m δi−k (a)lCi−k+mδj1 . . . δjl (b)(j1 ,...,jl )k=1where the second sumall the vectors (j1 , . .
. , jl ) such that 0 < l ≤ min{i−is taken overkjm = k; Cjk = 0 if j = 0, and Cjk = Cj+pk + m, k}, jm ≥ 1,q for q >> 0 if j ≤ 0.Proof. For any a, b ∈ D̄ we haveαm (ab)z m + m δ1 (ab)z m+1 + m δ2 (ab)z m+2 + . . . = z m (ab) =(∗)(αm (a)z m + m δ1 (a)z m+1 + m δ2 (a)z m+2 + . . .)bIf we represent the right-hand side of (∗) as a series with coefficients shifted to the leftand then compare the corresponding coefficients on the left-hand side and right-handside, we get some formulas for m δi (ab). We have to prove that these formulas are thesame as in our proposition.Letz i+m−k b = αi+m−k (b)z i+m−k + . .
. + xk z i+m + . . .and(αm (a)z m + m δ1 (a)z m+1 + m δ2 (a)z m+2 + . . .)b = αm (ab)z m + ym+1 z m+1 + ym+2 z m+2 + . . .Then we haveyi+m = αm(a)xi+i−1m δi−k (a)xkk=0In the proof of prop. 0.7 we have shown thatz i+1−k b = αi+1−k (b)z i+1−k + . . . + σ(Sik α)(b)z i+1 + . . .kα)(b) for k < i. It is easy to see that xi = m δi (b), x0 = αi+m (b)Hence xk = σ(Si+m−1kand σ(Si+m−1 α) = i+m−k δk , which proves (i).For α = id, by corollary 1,klα)(b) =Ci−k+mδj1 .
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