OVER HENSELIAN FIELDS

OVER HENSELIAN FIELDS

KARIM MOUNIRH Received 5 July 2004

This paper deals with the structure of nicely semiramified valued division algebras. We prove that any defectless finite-dimensional central division algebra over a Henselian field Ewith an inertial maximal subfield and a totally ramified maximal subfield (not neces- sarily of radical type) (resp., split by inertial and totally ramified field extensions ofE) is nicely semiramified.

1. Introduction

We recall that a nicely semiramified division algebra is defined to be a defectless finite- dimensional valued central division algebraDover a fieldEwith inertial and totally ram- ified radical-type (TRRT) maximal subfields [7, Definition, page 149]. Equivalent state- ments to this definition were given in [7, Theorem 4.4] when the fieldEis Henselian.

These division algebras, as claimed in [7, page 128], appeared in [10] as examples of di- vision algebras with nonzeroSK1. The main purpose of this paper is to prove that over a Henselian fieldE, any central division algebra with inertial and totally ramified max- imal subfields (resp., split by inertial and totally ramified field extensions ofE) is nicely semiramified.

We precise that all rings considered in this work are assumed to be associative with a unit and all free modules are assumed to be finite-dimensional. A valued division algebra Dover a fieldE—we adopt the same valuative definitions as in [7]—is called defectless (overE) if [D:E]=[ ¯D: ¯E](ΓD:ΓE), where [ ¯D: ¯E] (resp., (ΓD:ΓE)) is the residue degree (resp., ramification index) ofDoverE.

We recall that for any valued central division algebraDover a fieldE, the centerZ( ¯D) of ¯Dis a normal field extension of ¯Eand the mapping

θD:ΓD/ΓE−→GalZ( ¯D)/E¯, γ+ΓE−→θ_Dγ+ΓE

: ¯a−→dad⁻¹, (1.1)

wheredis an arbitrary element ofDsuch thatv(d)=γ(vbeing the valuation ofD), is a surjective group homomorphism [7, Proposition 1.7]. We say thatDis tame (overE) if

Copyright©2005 Hindawi Publishing Corporation

International Journal of Mathematics and Mathematical Sciences 2005:4 (2005) 571–577 DOI:10.1155/IJMMS.2005.571

it is defectless overE,Z( ¯D) is separable (hence Galois) over ¯E, and the characteristic of ¯E does not divide the cardinal of the kernel ofθ_D. A subfieldK(⊇E) ofDis called inertial overEif [K:E]=[ ¯K: ¯E] and ¯Kis separable over ¯E.

Now, letF be an associative ring with a unit, andΓa totally ordered abelian group.

We say thatFis a graded ring of typeΓif there are subgroupsF_γ(γ∈Γ) ofFsuch that F=

γ∈ΓFγandFγFδ⊆Fγ+δ, for allγ,δ∈Γ. In this case, the setΓF= {γ∈Γ|Fγ=0} is called the support ofF. If x∈Fγ for someγ∈ΓF, we say thatx is a homogeneous element ofF. In particular, ifxis a nonzero element ofF_γ, we say thatxhas gradeγand we write gr(x)=γ. We denote byF^h (resp.,F^∗) the set of homogeneous (resp., nonzero homogeneous) elements ofF. A graded ringFwhich is commutative and for which all nonzero homogeneous elements are invertible is called a graded field.

LetF be a commutative graded ring of typeΓ. A (left)F-module (resp.,F-algebra) Ais called a gradedF-module (resp., a gradedF-algebra) (of typeΓ) ifA=

δ∈ΓAδ

andFγAδ⊆Aγ+δ, for allγ,δ∈Γ(resp., ifAis a graded ring of typeΓandFγ⊆Aγ, for allγ∈Γ). In particular, ifF is a graded field, then gradedF-algebras (resp., commuta- tive gradedF-algebras) for which homogeneous elements are invertible are called graded divisionF-algebras (resp., graded field extensions ofF).

A gradedF-module (resp., a gradedF-algebra) is also called a graded module (resp., a graded algebra) overF.

LetFbe a graded field of typeΓ. SinceΓis totally ordered, then any graded division algebraAoverFis a domain. We may then consider the algebra of central quotients ofA that we denote by Cq(A). Remark that ifAis a graded field extension ofF, then Cq(A) coincides with the fraction field Frac(A) ofA.

One can easily see that if F is a graded field, then any gradedF-module M is free overF. Indeed, by [1, Theorem 3, page 29] any maximalE-linearly independent subset of homogeneous elements ofMis a basis ofMoverE.

A graded field extensionK of a graded fieldF is called totally ramified overFif [K: F]=(ΓK:ΓF). It is called unramified overFif [K:F]=[K0:F0] andK0is separable over F0. Finally,Kis called tame overFifK0is separable overF0andΓK/ΓFhas nop-torsion, wherep=char(F).

IfFis the center of a graded division algebraD, thenDis called a graded central divi- sion algebra (GCDA) overF. We recall that in the same way as for (ungraded) fields, we can define the graded Brauer group GBr(F) ofF, where GCDAs play the same role as cen- tral division algebras (CDAs) over (ungraded) fields (see [2, Section 5] or [6, Section 3]).

LetF be a graded field of typeΓand Da GCDA over F. Then a natural valuation vcan be defined on Cq(D) by settingv(d)=gr(d) for anyd∈D^∗ (see [3, Section 4]

or [6, Section 4]). Throughout the rest of the paper, suchvwill be called the canonical valuation of Cq(D). We recall that Cq(D), with respect tov, is a tame CDA over Frac(F) [3, Proposition 4.1(3)].

Conversely, for any defectless valued division algebraDwith valuationvover a fieldE, we defineE^γ= {x∈E|v(x)≥γ},E^>γ= {x∈E|v(x)> γ},D^γ= {x∈D|v(x)≥γ}, and D^>γ= {x∈D|v(x)> γ}. Obviously,E^>γ(resp.,D^>γ) is a subgroup of the additive group E^γ(resp.,D^γ). So, we can define the quotient groupsGE_γ=E^γ/E^>γ andGD_γ=D^γ/D^>γ. For x∈E\ {0} (resp., x∈D\ {0}), we denote by x the element x+E^>v(x) of GE_v(x)

(resp.,x+D^>v(x)ofGD_v(x)). One can easily check that the additive groupGE=

γ∈ΓGE_γ endowed with the multiplication law defined byxy=xy is a graded field. Analogously, GD=

γ∈ΓGDγis a graded division algebra overGE[2, page 4281]. We recall thatDis tame overEif and only ifGDis a GCDA overGE[2, Corollary 4.4(3)] (remark that since Dis defectless overE, then [D:E]=[GD:GE]).

Now, letF be a graded field and letDbe a GCDA overF, and denote byHFrac(F) the Henselization of Frac(F) with respect to the canonical valuation of Frac(F) [4, Sec- tion 16], and letHCq(D)=Cq(D)⊗Frac(F)HFrac(F). Then,Dis graded isomorphic to GHCq(D) by means of the mappingx→x, wherexis an arbitrary homogeneous element ofD. Indeed, we haveD0=(GHCq(D))0andΓD=ΓGHCq(D). We writeD^∼=gGHCq(D).

We also haveF^∼_=gGHFrac(F).

We recall that ifEis Henselian, then the tame part TBr(E) of the Brauer group Br(E) ofE(i.e., TBr(E)= {[D]∈Br(E)|Dis a tame CDA overE}) is isomorphic to GBr(GE) [6, Theorem 5.3]. Also, for any graded fieldF, GBr(F) is isomorphic to TBr(HFrac(F)) [6, Theorem 5.1]. These isomorphisms are index-preserving. We call them the canonical isomorphisms.

2. Nicely semiramified division algebras over Henselian fields

LetF be a graded field andK a finite-dimensional graded field extension ofF. For an arbitrary abelian groupA—namely forA=ΓK/ΓF—and a familya1,a2,. . .,a_rof elements ofA, we say thata1,a2,. . .,arare independent if the subgroupa1,a2,. . .,arofA, gener- ated bya1,a2,. . .,ar, equals^r_i=1ai. We recall thatKis called totally ramified of radical type (TRRT) overFif there are homogeneous elementst1,. . .,t_r∈F^∗and nonnegative integersn1,. . .,n_rsuch that the following conditions are satisfied:

(1)K=F[t₁^1/n1,. . .,tr^1/nr],

(2) gr(t^1/n_i ⁱ) +ΓF (1≤i≤r) are independent elements ofΓK/ΓF, with ordern_i, re- spectively.

One can see that in the same way as for TRRT valued field extensions (see [7, Lemma 4.1]), a totally ramified finite-dimensional graded field extensionKofFis TRRT overF if and only if there is a subgroupGofK^∗/F^∗such that the mappingG→ΓK/ΓF, defined byxF^∗→gr(x) +ΓF, is a group isomorphism.

Lemma2.1. LetFbe a graded field andKa totally ramified finite-dimensional graded field extension ofF. Then,Kis TRRT overF.

Proof. Let (xi)^r_i=1be a family of nonzero homogeneous elements ofKsuch that (gr(xi) + ΓF)^r_i=1is a basis ofΓK/ΓF(i.e.,ΓK/ΓF=_r

i=1gr(x_i) +ΓF). Letn_ibe the order of gr(x_i) + ΓF(1≤i≤r) and setI= {m¯ =(m1,. . .,mr)∈N^r|0≤mi< nifor all 1≤i≤r}. LetGbe the subgroup ofK^∗/F^∗, generated by the elementsxiF^∗(1≤i≤r). For ¯m=(m1,. . .,mr)∈ I, letx^m¯ =x₁^m1···x^mr^r. SinceK/Fis totally ramified, then for any 1≤i≤r,xⁿ_iⁱ∈F^∗(in- deed, we haveK0=F0and gr(xⁿ_iⁱ)∈ΓF). Clearly, the elementsx^m¯ ( ¯m∈I) are pairwise distinct. Hence,G= {x^m¯F^∗|m¯ ∈I}and the mapφ:G→ΓK/ΓF,x^m¯F^∗→gr(x^m¯) +ΓF, is

a group isomorphism.

LetFbe a graded field andDa GCDA overF. We recall that in the same way as for valued central division algebras,Dis called nicely semiramified (NSR) (overF) if it has

unramified and totally ramified radical-type maximal graded subfields. We recall also that Dis NSR (overF) if and only if Cq(D) is NSR (over Frac(F)) (see [3, Proposition 6.4]).

Moreover, Cq(D) is NSR (over Frac(F)) if and only ifHCq(D) is NSR (overHFrac(F)).

Indeed, assume that Cq(D) is NSR and letK (resp., L) be an inertial (resp., a TRRT) maximal subfield of Cq(D). Then, by [8, Theorem 1],HK(=K⊗Frac(F)HFrac(F)) (resp., HL(=L⊗Frac(F)HFrac(F))) is an inertial (resp., a TRRT) maximal subfield ofHCq(D).

Conversely, ifHCq(D) is NSR, then it has an inertial (resp., a TRRT) maximal subfieldK (resp.,L). SoGK(resp.,GL) is an unramified (resp., a totally ramified) maximal graded subfield ofD(^∼₌gGHCq(D)). HenceDis NSR. Therefore, by the above, Cq(D) is NSR.

The following lemma is analogous to [7, Theorem 4.4]. It gives equivalent statements for a GCDA over a graded field to be NSR. In condition (3)(i) of this lemma, the graded field extensionsL⁽ⁱ⁾are said to be linearly disjoint if the graded ringL⁽¹⁾⊗F··· ⊗FL^(k)is a graded field.

Lemma2.2. LetFbe a graded field andDa GCDA overF. Then the following statements are equivalent:

(1)DisNSR;

(2)Dis split by unramified and totally ramified graded field extensions ofF;

(3)D^∼_=g(L⁽¹⁾/F,σ1,t1)⊗F··· ⊗F(L^(k)/F,σ_k,t_k), whereL⁽ⁱ⁾,σ_i, andt_isatisfy the follow- ing conditions:

(i)L⁽ⁱ⁾are linearly disjoint cyclic unramified graded field extensions ofFwith dimen- sion[L⁽ⁱ⁾:F]=niand with Galois group generated byσi(1≤i≤k),

(ii)t_iare nonzero homogeneous elements ofF such thatgr(t^n/n_i ⁱ) +nΓFare indepen- dent elements ofΓF/nΓF, with orderni(1≤i≤k), respectively.

Proof. (1)⇒(2). Obvious.

(2)⇒(3). LetK(resp.,K) be an unramified (resp., a totally ramified—hence TRRT) graded field extension ofFsplittingD. It follows from the commutative diagrams

GBr(F) ^Ext

∼=

GBr(K)

∼=

TBrHFrac(F) ^Ext TBrHFrac(K)

GBr(F) ^Ext

∼=

GBr(K)

∼=

TBrHFrac(F) ^Ext TBrHFrac(K)

(2.1)

where the horizontal maps are scalar extension homomorphisms and the vertical ones are canonical isomorphisms (see [6, Remark 5.10]), thatHFrac(K) (resp.,HFrac(K) (=K⊗Frac(F)HFrac(F))) is an inertial (resp., a TRRT) field extension ofHFrac(F) split- tingHCq(D). So, by [7, Theorem 4.4]HCq(D) is NSR and we can writeHCq(D)= (M⁽¹⁾/HFrac(F),τ1,s1)⊗HFrac(F)··· ⊗HFrac(F)(M^(k)/HFrac(F),τ_k,s_k), whereM⁽ⁱ⁾,τ_i, and

s_isatisfy the following conditions:

(i)M⁽ⁱ⁾are linearly disjoint cyclic inertial field extensions ofHFrac(F) with dimen- sionniand with Galois group generated byτi, respectively (1≤i≤k),

(ii)s_i(1≤i≤k) are elements ofHFrac(F)^∗such that, if we denote byvthe canon- ical valuation ofHFrac(F), then (n/ni)v(si) +nΓFare independent elements of ΓF/nΓF, with orderni, respectively.

Therefore, D^∼=gGHCq(D)

∼=g

GM⁽¹⁾/GHFrac(F),τ1,s1)⊗GHFrac(F)··· ⊗GHFrac(F)

GM^(k)/GHFrac(F),τ_k,s_k).

(2.2) Indeed, letDi=(M⁽ⁱ⁾/HFrac(F),τi,si)=_ni

j=1M⁽ⁱ⁾x_i^j, wherex_iⁿⁱ=siandxia=τi(a)xifor alla∈M⁽ⁱ⁾(1≤i≤k). SinceM⁽ⁱ⁾/HFrac(F) is cyclic inertial, then by [5, Remark 3.1]

the unramified graded field extensionGM⁽ⁱ⁾/GHFrac(F) is cyclic and, up to a group iso- morphism, Gal(GM⁽ⁱ⁾/GHFrac(F))= τi. Moreover, it is clear that for anyd∈M⁽ⁱ⁾, we havex_idx⁻_i¹=τ_i(d)=τ_i(d). Thus, GD_i=(GM⁽ⁱ⁾/GHFrac(F),τ_i,s_i). It follows from the canonical isomorphism TBr(HFrac(F))^∼₌GBr(GHFrac(F)) that

D^∼=gGHCq(D)^∼=gGD1⊗GHFrac(F)··· ⊗GHFrac(F)GDk

=

GM⁽¹⁾/GHFrac(F),τ1,s1

⊗GHFrac(F)··· ⊗GHFrac(F)

GM^(k)/GHFrac(F),τk,sk

. (2.3) The conditions of (3) in this lemma are then satisfied.

(3)⇒(1). By [5, Theorem 3.11(b)] Frac(L⁽ⁱ⁾)/Frac(F) are cyclic with Gal(Frac(L⁽ⁱ⁾)/

Frac(F))= σi. Thus, applying [8, Theorem 1], one can easily see that HFrac(L⁽ⁱ⁾)/

HFrac(F) are cyclic with Gal(HFrac(L⁽ⁱ⁾)/HFrac(F))^∼₌Gal(Frac(L⁽ⁱ⁾)/Frac(F)). More- over, since HFrac(L⁽ⁱ⁾)=L⁽ⁱ⁾0 , ΓHFrac(L⁽ⁱ⁾)=ΓL⁽ⁱ⁾, HFrac(F)=F0, and ΓHFrac(F)=ΓF, it follows thatHCq(D) (^∼₌D_⊗FHFrac(F)^∼₌(HFrac(L⁽¹⁾)/HFrac(F),σ1,t1)⊗HFrac(F)···

⊗HFrac(F)(HFrac(L^(k))/HFrac(F),σ_k,t_k)) is NSR (by [7, Theorem 4.4]). So, by the argu-

ments preceding this lemma,Dis NSR.

Lemma2.3. Let Ebe a Henselian valued field and Da defectless CDA overE. Then the following statements are equivalent:

(1)DisNSR;

(2)GDisNSR.

Proof. (1)⇒(2). Since D is nicely semiramified—hence tame—over E, then GD is a graded central division algebra overGE (see [2, Corollary 4.4(3)]—remark that since Dis defectless overE, then [D:E]=[GD:GE]). LetL(resp.,M) be an inertial (resp., a TRRT) maximal subfield ofD. Then,GL(resp.,GM) is an unramified (resp., a totally ramified) maximal graded subfield ofGD. Hence,GDis nicely semiramified.

(2)⇒(1). SinceGDis nicely semiramified overGE, then byLemma 2.2,GD^∼₌g(L⁽¹⁾/GE, σ1,t1)⊗GE··· ⊗GE(L^(k)/GE,σ_k,t_k), where L⁽ⁱ⁾/GE, σ_i, and t_i satisfy the conditions of

Lemma 2.2(3). SinceL⁽ⁱ⁾/GEare unramified, then by [5, Theorem 5.2 (see the proof)], there are inertial cyclic field extensions K⁽ⁱ⁾ of E such that—up to a graded iso- morphism—L⁽ⁱ⁾=GK⁽ⁱ⁾ with Gal(L⁽ⁱ⁾/GE)^∼₌Gal(K⁽ⁱ⁾/E). Let D⁽ⁱ⁾=(L⁽ⁱ⁾/GE,σi,ti). It follows from the canonical isomorphism TBr(E)^∼₌^αGBr(GE) that there is a unique—up to an algebra isomorphism—central division algebraD(i)overEsuch thatD^(i)∼₌gGD(i). By [6, Corollary 5.8],K⁽ⁱ⁾splitsD(i). Hence, up to an isomorphism,K⁽ⁱ⁾is a maximal subfield ofD(i)(see [9, Corollary, page 241]). WriteD(i)=(K⁽ⁱ⁾/E,σi,si). Then, (L⁽ⁱ⁾/GE,σi,ti)= D^{(i) ∼}_=g GD_(i) = (GK⁽ⁱ⁾/GE,σ_i,s_i) ^∼_=g (L⁽ⁱ⁾/GE,σ_i,s_i). Accordingly, (L⁽ⁱ⁾/GE,σ_i,t_i)⊗GE

HFrac(GE)^∼₌(L⁽ⁱ⁾/GE,σ_i,s_i)⊗GEHFrac(GE) or, equivalently, (HFrac(L⁽ⁱ⁾)/HFrac(GE), σi,ti)^∼₌(HFrac(L⁽ⁱ⁾)/HFrac(GE),σi,si). Therefore, by [9, Lemma, page 278], there isαi∈ HFrac(L⁽ⁱ⁾) such that ti =(ⁿ_j=ⁱ₁σ_i^j(αi))si. Let v be the canonical valuation of HFrac(L⁽ⁱ⁾). Sincevis Henselian, then gr(t_i) (=v(t_i))=n_iv(α_i) +v(s_i). So, (n/n_i) gr(t_i) + nΓE=(n/ni)v(si) +nΓE(sinceΓHFrac(L⁽ⁱ⁾)=ΓE). We have [D]=α⁻¹([GD]), thereforeD^∼₌ (K⁽¹⁾/E,σ1,s1)⊗E··· ⊗E(K^(k)/E,σk,sk). Hence, by [7, Example 4.3],Dis NSR.

Theorem2.4. LetEbe a Henselian valued field andDa defectless CDA overE. Then the following statements are equivalent:

(1)Dis NSR;

(2)Dhas inertial and totally ramified maximal subfields;

(3)Dis split by inertial and by totally ramified field extensions ofE.

Proof. (1)⇒(2). Obvious.

(2)⇒(3). Obvious.

(3)⇒(1). SinceDis split by an inertial field extension ofE, thenDis tame overE(see [6, page 99]). Hence,GDis a GCDA overGE. Now, letK(resp.,K) be an inertial (resp., a totally ramified) field extension ofEsplittingD. ThenGK(resp.,GK) is an unramified (resp., a totally ramified) graded field extension ofGEthat splitsGD. So, by Lemmas2.2

and2.3,Dis NSR.

References

[1] M. Boulagouaz,The graded and tame extensions, Commutative Ring Theory (F`es, 1992), Lec- ture Notes in Pure and Appl. Math., vol. 153, Marcel Dekker, New York, 1994, pp. 27–40.

[2] ,Le gradu´e d’une alg`ebre `a division valu´ee[The associated graded ring of a valued division algebra], Comm. Algebra23(1995), no. 11, 4275–4300 (French).

[3] ,Alg`ebre `a division gradu´ee centrale[Central graded division algebra], Comm. Algebra 26(1998), no. 9, 2933–2947 (French).

[4] O. Endler,Valuation Theory, Springer, New York, 1972.

[5] Y.-S. Hwang and A. R. Wadsworth,Algebraic extensions of graded and valued fields, Comm.

Algebra27(1999), no. 2, 821–840.

[6] ,Correspondences between valued division algebras and graded division algebras, J. Alge- bra220(1999), no. 1, 73–114.

[7] B. Jacob and A. R. Wadsworth,Division algebras over Henselian fields, J. Algebra128(1990), no. 1, 126–179.

[8] P. Morandi,The Henselization of a valued division algebra, J. Algebra122(1989), no. 1, 232–

243.

[9] R. S. Pierce,Associative Algebras, Graduate Texts in Mathematics, vol. 88, Springer, New York, 1982.

[10] V. P. Platonov,The Tannaka-Artin problem, and reducedK-theory, Izv. Akad. Nauk SSSR Ser.

Mat.40(1976), no. 2, 227–261, 469, translated in Math. USSR Izv.10(1976), 211–243.

Karim Mounirh: D´epartement des Math´ematiques et d’Informatique, Facult´e des Sciences Dhar El Mehraz, Universit´e Sidi Mohamed Ben Abdellah, B.P. 1796, Atlas, Morocco

E-mail address:[email protected]