Multidimensional local skew-fields (792481), страница 7
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Indeed, by proposition 0.7 we haveNote that δn+1(ab)δn+1=n+1kδn+1−k(a)σ(Sn+1α)(b),a, b ∈ K̄k=0But δj = 0 if j < n. Therefore,(ab)δn+1=nδn+1(a)αn+2 (b)+δn (a)(αk δ1 αn−k )(b)+α(a)δn+1(b) = δn+1(a)α2 (b)+α(a)δn+1(b)k=0and by lemma 0.12, δn+1is an inner derivation. Using lemma 0.11 with z → z =n+2for an appropriate b, we havez + bzz uz −1 = uα + uδn z n + uδn+2 z n+2 + . . .with δn+1= 0.
By induction we can assume that there exists a parameter z such thatz uz −1 = uα + uδn z n + uδ2n z 2n + . . . + uδk+1 z k+1 + . . .Then if n |k + 1, δk+1is a (αk+2 , α)-derivation. Indeed,δk+1(ab) =k+1lσ(δk+1−lα)(a)σ(Sk+1α)(b) =l=0δk+1(a)αk+2 (b)+xk+1−mnδmn(a)σ(Sk+1α)(b) + α(a)δk+1(b)m=1where x ∈ N : xn ≤ k + 1, (x + 1)n > k + 1, because δj = 0 if j < k + 1 and n |j.k+1−mnEvery monomial σ(Sk+1α) contain an element δj with j < k + 1 and n | j.ltogether with n |k + 1 − mn. Therefore,It follows from the definition of Sk+1k+1−mnk+2α)(b) = 0 ∀m and δk+1 is a (α , α)-derivation.σ(Sk+1If n|k + 1, we can apply the same arguments and conclude that δk+2is a (αk+2 , α)k+2(z +bz k+3derivation.
Therefore, by lemma 0.11 there exists a parameter z = z +bzif n|k + 1) such thatz uz −1 = uα + uδn z n + uδ2n z 2n + . . . + uδk+2 z k+2 + . . .(oruδk+1 z k+1 + uδk+3 z k+3 + . . .Since zl+1 = (1 +zll )zl , the sequenceof the lemma.2{zl }∞l=1if n|k + 1)converges in K. Therefore, we get the proof30Lemma 0.30 There existsu ∈ K such that α(u) = ξu, where ξ n = 1, a parameternkand for all j δjn (u) = u( k yjk u ) ∈ uk((un )), where yjk ∈ k, and for k not divisibleby n δk = 0.Proof We can assume that the relation from lemma 0.29 holds. We will do changesof the form u → u = u + bjn z jn . We have seen in the proof of lemma 0.24 that themaps δk , n | k don’t change under such substitutions. By lemma 0.24, (i) we can seethat uδjn = uδjn + bα − ∂/∂u(uα )b.
By corollary 6 we can assume α(u) = ξu, whereξ n = 1. Therefore, uδjn = uδjn + bα − ξb and we can find an element b to satisfy theconditions of the lemma.2As in the case α = Id we can define in , rn and an .Definition 0.31 Putin = ν((φzn − 1)(u))∈N∞rn = ν̄[((φzn − 1)(u))z −in mod ℘] mod inin +1 2uδ2in − 2 δindu∈kan = resu(uδin )2∈ Z/in Zwhere u, z are arbitrary local parameters in K, φz : K → K, φz (a) = ad(z)(a).From the previous two lemmas we can derive that if z is a local parameter fromlemma 0.29 then in ∈ nN and rn = 1 mod n. It is easy to see that the number inis the number of the first non-zero map δin in lemma 0.30.
In the same way as in theproof of proposition 0.26 we can get the following result:Proposition 0.32 We have inan (uδin +1 , . . . , uδ2in −1 ).=in (uδj , j∈/nN), rn=rn (in ), an=Proposition 0.33 Let K be a two-dimensional local skew field which satisfy the conditions in the beginning of this paragraph.Let chark = 0 and αn = Id for some natural n.Then K is isomorphic to a skew field k((u))((z)) with the relationzuz −1 = ξu + uδin z in + uδ2in z 2in ,where ξ n = 1, in = in (0, . . . , 0),δin (u) = curn , c ∈ k ∗ /(k ∗ )e , e = (rn − 1, i),(u) = (an (0, . . .
, 0) + rn (in + 1)/2)u−1 (δin (u))2 .δ2in31Proof. We can assume that the conditions of lemma 0.30 hold. Then, because ofspecial choice of the element uδjn it suffice to repeat the proof of proposition 0.28.2Combining all these results we get:Theorem 0.34 Let K and K be local skew fields of characteristic zero. Suppose theysatisfy the conditions in the beginning of this chapter.
Then K is isomorphic K iffk∼= k and the sets (n, ξ, in , rn , c, an ), (n , ξ , in , rn , c , an ) coinside.Remark. If n = 1 and in = ∞, then K is a commutative two-dimensional localfield k((u))((z)).Let us now summarise all the classification results we have got above.Theorem 0.35 (I) Let K be a two-dimensional local skew field with a commutativeresidue skew field.It splits if the canonical automorphism α satisfy the condition αn = Id for all n. Ifthis condition does not hold, there are examples of non-splittable skew fields.(II) Let K, K be skew fields as in (I). Assume αn = Id, αn = Id for all n. Then(a) K is isomorphic to a two-dimensional local skew field K̄((z)) where za = aα z,a ∈ K̄ and K̄ is a one-dimensional local field with the residue field k.(b) K and K are isomorphic iff k ∼= k and there exists an isomorphism−1 f : K̄ → K̄ such that α = f α f .(c) If charK = chark, charK = chark and k, k are algebraicallyclosed fields of characteristic 0, then K is isomorphic to K iff k ∼= k and(a1 , iα , y(α)) = (a1 , iα , y(α )).(III) Let K, K be two-dimensional splittable local skew fields of characteristic 0,k ⊂ Z(K), k ⊂ Z(K ), and αn = Id, αn = Id for some natural n, n ≥ 1.
Then (a)K is isomorphic to a two-dimensional local skew field k((u))((z)) wherezuz −1 = ξu + uδin z in + uδ2in z 2in ,where ξ n = 1, in = in (0, . . . , 0),δin (u) = curn , c ∈ k ∗ /(k ∗ )e , e = (rn − 1, i),(u) = (an (0, . .
. , 0) + rn (in + 1)/2)u−1 (δin (u))2δ2in(in , rn , an were defined in 0.31).If n = 1, in = ∞, then K is commutative.(b) K is isomorphic to K iff k ∼= k and the sets(n, ξ, in , rn , c, an ), (n , ξ , in , rn , c , an ) coinside.32Corollary 7 Every two-dimensional local skew field K with the ordered set(n, ξ, in , rn , c, an )is a finite-dimensional extension of a skew field with the ordered set (1, 1, 1, 0, 1, a) ifin < ∞.Remark.
It’s easy to see from the corollary that skew fields in the theorem aboveare almost always infinite dimensional over the centre. Namely, the only finite dimensional skew fields are the skew fields with in = ∞. In the case of skew fields of positivecharacteristic the situation is much more complicated.0.4Splittable skew fields of characteristic p > 0.It is difficult to classify all splittable two-dimensional skew fields with the canonicalautomorphism of finite order in positive characteristic even if we consider only skewfields with α = id, at least because there are infinitely many maps δj which can notbe removed by any change of parameters.
Nevertheless, our methods give some usefultools for studying splittable skew fields finite dimensional over their centre.For splittable skew fields in positive characteristic one can define an invariant whichis in some sense a replacement of the invariant an for skew fields of characteristic 0.Certainly, there are infinitely many of other invariants.Namely, if K is a splittable two-dimensional local skew field of positive characteristic¯ ⊂ K, k ⊂ K̄, k ⊂ Z(K), α of finite order we definewith K̄ commutative, k = K̄(z)dK = max ν(zuz −1 − α(u) − δin (u)z in ),u,z(z)where δin is a map defined by a parameter z, and in is a number defined in 0.31.In the case of a skew field of characteristic 0 we have dK = 2in (0, .
. . , 0) or dK = ∞,that is why it is in some sense a replacement of an : in characteristic 0 it reflects theproperty of an to be zero or not.In this section we will prove the following theorem:Theorem 0.36 Suppose that a two-dimensional local skew field K splits, K̄ is a field,¯ ⊂ Z(K), char(K) = char(K̄¯ ) = p > 2, α = id, and d ≤ 2i = 2i or d = ∞.k = K̄K1KThen K is a finite dimensional vector space over its centre if and only if K isisomorphic to a two-dimensional local skew field k((u))((z)), wherez −1 uz = u + xz iwith x ∈ K̄ p , (i, p) = 1.33To prove this theorem we prove more general result about finite dimensionalalgebras, which generalises some known results of Jacob and Wadsworth in [9] andSaltman [28].
As a corollary we get the positive answer on the conjecture aboutexponent and index of a finite dimensional division algebra over a C2 -field for sometypes of C2 -fields. These results will be proved in the subsection below. Now we proveonly the ”if” part. Indeed, since x ∈ K̄ p , we have δi2 (u) = 0. Hence, by corollary 1 wej/ihave δj = cδi , c ∈ k if i|j, and δj = 0 if i |j. But then zap z −1 = ap for any a ∈ K̄, soK is a finite dimensional skew field over its centre and the index indK = p.To prove the ”only if” part we need results from the following subsection:0.4.1Wild division algebras over Laurent series fieldsIn this subsection we prove a decomposition theorem for some class of wild divisionalgebras over a Laurent series field with arbitrary residue field of characteristic greaterthan two.
Namely, we prove this theorem for wild division algebras which satisfy the following condition: there exists a section D̄ → D of the residue homomorphism D → D̄,where D is a central division algebra. This theorem is a generalisation of the decomposition theorems for tame division algebras given by Jacob and Wadsworth in [9]. Anextensive analysis of the wild division algebras of degree p over a field F with completediscrete rank 1 valuation with char(F̄ ) = p was given by Saltman in [28] ( Tignol in[32] analysed more general case of the defectless division algebras of degree p over afield F with Henselian valuation).The main result of this subsection is Theorem 0.55; it is a corollary of Theorem 0.43and propositions 0.51-0.54.
Theorem 0.43 is a key tool in the proof of Theorem 0.55.As a corollary we get the positive answer on the following conjecture: the exponent ofA is equal to its index for any division algebra A over a C2 -field F = F1 ((t2 )) (corollary8) (see also [37], corollary 4, §8.3.2.), where F1 is a C1 -field. We note that the proof ofthe conjecture does not depend on the statement of theorem 0.55, but uses only severallemmas from it’s proof.
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