Multidimensional local skew-fields, страница 2

In §2 we give a sufficient condition for a skew field to besplit. Namely, a local skew field splits if a canonical automorphism has infinite order.The canonical automorphism can be defined as a restriction of an inner automorphismad(z) on the residue field, where z is any local parameter.

We show that there existcounterexamples when this condition does not hold. We note that this condition andcounterexamples are true even in more general situation when the skew field is nottwo-dimensional skew field or the residue skew field is not commutative. We classifyall the skew fields which possess this condition up to isomorphism. The results of §2don’t depend on the characteristic of a skew field.In §3 we classify all the local splittable skew fields of characteristic 0 with commutative residue skew field and with the canonical automorphism of finite order.In §4 we study splittable local skew fields of characteristic p > 2 with commutativeresidue skew field and with the canonical automorphism of finite order. We give acriterium when such a skew field is finite dimensional over its centre. Then we prove thatevery tame finite dimensional division algebra over a local complete field splits.

Usingthis fact we prove the decomposition theorem for splittable algebras. As a corollary weget the proof of the conjecture mentioned above.In §5 we study some properties of local skew fields described in §3. In particular,we give a criterium when two elements from such a skew field conjugate. This is ageneralisation of analogous theorems from [23]. As a corollary we prove that almostall such skew fields are infinite dimensional over their centre. Also we prove that theScolem-Noether theorem holds only in the case of the classical ring of pseudo-differentialoperators.In §6 we get new KP-hierarchies for every class of isomorphic two-dimensional local4skew fields, which were studied in section 3. We derive new equations of the KP-typefor some hierarchies.In the second chapter we classify conjugacy classes in the group of continuousautomorphisms of a two-dimensional local field of characteristic zero with the residuefield of the same characteristic.

Some facts about automorphisms of a local field ofcharacteristic p > 0 one can find in lemma 1.3. Also in this chapter we show how thisclassification can be generalised to the case of a n-dimensional local field, n > 2.Acknowledgements. At this point, I would like my advisor Prof. Ernst-WilhelmZink for the warm atmosphere and valuable discussions, Prof. A. N. Parshin, my scientific advisor during my undergraduate studies in Moscow State University, Prof. V.I.Yanchevskii for stimulating discussions and advises.

I would further like to thank Martin Grabitz, who allowed me to more easily orient myself in the new environment,country and language and who spent a lot of time for our interesting discussions. Ithank Prof. Helmut Koch for the invitation to HU zu Berlin in 1999 that made itpossible for me to stay here for a long time. I would also like to thank my wife Olgafor her help in preparing the final version of this thesis and her love, and I thankGraduiertenkolleg ”Geometry und Nichtlineare Analisys”, which provided the doctoralstipend that allowed me to complete this work.5Chapter 0The structure of two-dimensionallocal skew fields.0.1General.Definition 0.1 Let K and k be arbitrary skew fields. A skew field K is called a complete discrete valuation skew field if K is complete with respect to a discrete valuation.

A skew field K is called an n-dimensional local skew field if there are skew fieldsK = Kn , Kn−1 , . . . K0 = k such that each Ki for i > 0 is a complete discrete valuationskew field with residue skew field Ki−1 .The following properties are well known from the valuation theory of division algebras (see for ex. [31]).Lemma 0.2 Let K be a complete diskrete valuation skew field. Then the followingproperties hold:i) The valuation ring O is a topological group and a metric space under the naturaltopology;ii) The ring O is a local ring and a principal ideal domain.For every two-dimensional local skew field we haveK ⊃ O → K̄ ⊃ Ō → kwhere Ō is a valuation ring in K̄.

There are two filtrationsK ⊃ O ⊃ ℘ ⊃ ℘2 ⊃ . . .K̄ ⊃ Ō ⊃ ℘¯ ⊃ ℘¯2 ⊃ . . .where ℘¯ is a maximal ideal in Ō, ν̄ is a discrete valuation on K̄.6Definition 0.3 Two two-dimensional local skew fields K and K are isomorphic ifthere is an isomorphism which preserves the filtrations above, i.e. it maps OK ontoOK , ℘ onto .℘ and OK̄ onto OK̄ , ℘K̄ onto ℘K̄ .Definition 0.4 A two-dimensional skew field K is said to split if there is a section ofthe homomorphism OK → K̄.Elements z ∈ O, ν(z) = 1 and u ∈ Ō ⊂ K̄, ν̄(u) = 1 are called local parameters(or variables) in K.Proposition 0.5 Suppose K splits.

Fix some local parameters z and u. Then K isisomorphic to a two-dimensional local skew field K̄((z)) whereza = aα z + aδ1 z 2 + aδ2 z 3 + . . .where a ∈ K̄, α is an automorphism, δi : K̄ → K̄ are linear maps.Proof. Suppose a ∈ K, ν(a) = j. Then we have ν(az −j ) = 0 and az −j :=az −j mod ℘ ∈ K̄. We will assume that the last element lies in O, since there isa section.we have ν(az −j − az −j ) ≥ 1. Continuing this line of reasonings, we get∞ Thenia = i=j ai z , ai ∈ K̄.automorNow define aα = zaz −1 mod ℘, where a ∈ K̄. It’s clear that α is an −1−1iphism. Since ν(zaz ) = 0, the element zaz can be written as a series ∞i=0 ai z ,αδiwhere ai ∈ K̄. Here we have a0 = a .

Now put a := ai for i ≥ 1. It is easy to see thatδi are linear maps.2In fact, the maps δi satisfy some identities. To write them we need extra notation.Consider the ring Z < α, δ > of noncommutative polynomials in two variables.Define the mapσ : Z < α, σ >→ Z < α, δ, δi ; i ≥ 1 >,σ(αa1 δ b1 . . . αan δ bn ) = αa1 δb1 . . . δbn−1 αan −1 δ bn ,where a1 , bn ≥ 0, ai , bj ≥ 1, i > 1, j < n for every word in Z < α, δ >.For exampleσ(αk ) = αkσ(αk δ l αi ) = αk δl αi−1where k, l, i are natural numbers, i, l ≥ 1.Let Sik ∈ Z < α, δ >, i ≥ k, i ≥ 1 be polynomials given by the following formula:Sik =τ (α.

. α δ .. . δ), .τ ∈Si /Gi−kkwhere Si is a permutation group and G is an isotropy subgroup.Immediately from the definition we get the following lemma7Lemma 0.6 The polynomials Sik satisfy the following property:Sii = δ i ,Si0 = αi ,k+1Si+1= αSik+1 + δSikNow we can define the identities for the maps δi :Proposition 0.7 Every map δi , i ≥ 1 satisfy the identityδi (ab) =iσ(δ i−k α)(a)σ(Sik α)(b),a, b ∈ K̄k=0Proof. For any a, b ∈ K̄. We have(ab)α z + (ab)δ1 z 2 + .

. . = z(ab) = (aα z + aδ1 z 2 + . . .)b(∗)If we represent the right-hand side of (∗) as a series with coefficients shifted to theleft and then compare the corresponding coefficients on the left-hand side and righthand side, we get some formulas for δi (ab). We have to prove that these formulas arethe same as in our proposition.Letz i+1−k b = αi+1−k (b)z i+1−k + . . .

+ xk z i+1 + . . .and(α(a)z + δ1 (a)z 2 + δ2 (a)z 3 + . . .)b = α(ab)z + y2 z 2 + y3 z 3 + . . .Then we haveyi+1 = α(a)xi +i−1δi−k (a)xk =k=0iσ(δ i−k α)(a)xkk=0Note that xk are polynomials which consist of monomials of the typeαa1 δb1 . . . αan δbn αan+1 (b),ak , bk ∈ Z,ak , b k ≥ 0(we put δ0 to be equal to 1). It is easy to see that these polynomials have integralpositive coefficients.We claim that xk = σ(Sik α)(b).To prove this fact it suffice to show that xk contains every monomial fromσ(Sik α)(b) and the sum of coefficients in xk is equal to the sum of coefficients inσ(Sik α)(b).By definition every coefficient of σ(Sik α)(b) is equal to 1. It is easy to see that thesum of coefficients is equal to Cik = i!/(i − k)!k!.8Let us show that xk contains every monomial from σ(Sik α)(b).

By definition,. . α δ .. . δ)α)(b), where τ ∈ Si , i.e. it conσ(Sik α)(b) consists of monomials σ(τ (α .i−kksists of monomials αa1 δb1 . . . αan δbn αan+1 (b), where aj ≥ 0, bj ≥ 1,n+1j=1 aj = i − k + 1 − n. We havenj=1 bj= k,z i+1−k b = z i+1−k−an+1 αan+1 (b)z an+1 + other terms,z i+1−k−an+1 αan+1 (b)z an+1 = z i+1−k−an+1 −1 [αan+1 +1 (b)z+. . .+δbn αan+1 (b)z bn +1 +. . .]z an+1 =z i+1−k−an+1 −1 αan+1 +1 (b)z an+1 +1 + z i+1−k−an+1 −1 δbn αan+1 (b)z bn +1+an+1 + . . .Now put d1 = δbn αan+1 (b). Then we havez i+1−k−an+1 −1 d1 z bn +1+an+1 = . . .

+ z i+1−k−an+1 −1−an −1 d2 z bn +1+an+1 +an +bn−1 +1 + . . . ,where d2 = δbn−1 αan δbn αan+1 (b). By induction we getz i+1−k−aj −nαa1 δb1 . . . αan δbn αan+1 (b)zbj +n+aj= αa1 δb1 . . . αan δbn αan+1 (b)z i+1 + . . . ,that is xk contains any given monomial from σ(Sik α)(b).Let us show that the sum of coefficients of xk is equal to Cik .Denote by sln the sum of coefficients in yl , wherez n az −n =∞yk z k ,a ∈ K̄k=0Then the sum of coefficients of xk is equal to ski+1−k . We claim that the followingrelation holdsddsn =sln−1l=0The proof is by induction on n. For n = 1 we have sd1 = 1 for all d ≥ 0, sl0 = 0 forl > 0 and s00 = 1.For arbitrary n putz n−1 az −n+1 = y0 + y1 z + .