CORRESPONDENCES BETWEEN VALUED DIVISION ALGEBRAS AND GRADED DIVISION ALGEBRAS

Y.-S. Hwang^{1} and A. R. Wadsworth^{2}

Abstract. If D is a tame central division algebra over a Henselian valued field F, then the valuation on D yields an associated graded ring GD which is a graded division ring and is also central and graded simple over GF. After proving some properties of graded central simple algebras over a graded field (including a cohomological characterization of its graded Brauer group), it is proved that the map [D] ↦ [GD]_{g} yields an index-preserving isomorphism from the tame part of the Brauer group of F to the graded Brauer group of GF. This isomorphism is shown to be functorial with respect to field extensions and corestrictions, and using this it is shown that there is a correspondence between F -subalgebras of D (with center tame over F) and graded GF -subalgebras of GD.

Introduction

If D is a division ring finite-dimensional over its center F, and the field F has a Henselian valuation v, then v is known to extend uniquely to a valuation on D. The features associated with the valuation on D, especially the residue division algebra \overline{D} and the value group Γ_{D} carry much information about the structure of D, and often can be used to settle questions such as decomposability, and which fields can be subfields of D. However, \overline{D} and Γ_{D} do not determine D, and there are many subtleties in the way they interact.

Associated to the valuation on D there is a filtration of D by the principal fractional ideals of the valuation ring, which allows one to build an associated graded ring GD = ⨁_{γ∈Γ_{D}} GD_{γ}, where

GD_{0} = \overline{D} and the grade group of GD is precisely the value group Γ_{D} of D. Furthermore, GD is a graded division ring, i.e., its homogeneous elements are all units. In addition, as shown in [B_{2}], the total ordering on Γ_{D} allows one to define a valuation on GD which extends to the ring of quotients QGD of GD, which is a division algebra. The valued division algebra QGD is usually not isomorphic to D, not even after Henselization, but we will see that their structures are closely related. The very presence of a valuation on QGD suggests that not so much is lost in the passage from D to its graded ring GD, even though GD appears to have a much simpler structure than D. We will show, in fact, that if D is tame then it is completely determined by GD, and its subalgebra structure is faithfully mirrored in that of GD.

Specifically, let TBr(F) denote the tame part of the Brauer group of the Henselian field F, and let GBr(GF) denote the graded Brauer group of the graded field GF determined by the valuation

Typeset by AMS-TEX

1Supported in part by the Non-directed Research Fund, Korea.

2Supported in part by the NSF.

on F. We will show in Th. 5.3 that the map [D] ↦ [GD]_{g} gives a Schur-index-preserving group isomorphism TBr(F) → GBr(GF), which (see Cor. 5.7 and Th. 6.1) is functorial with respect to scalar extensions and corestrictions. The index-preserving and functorial properties allow us to deduce (see Th. 5.9) that if K is a tame valued field extension of F, and D and A are tame division algebras with center F, then K (resp. A) embeds in D iff GK (resp. GA) embeds in GD.

These results show that much of what is known about tame valued division algebras can be carried over readily to graded division algebras finite-dimensional over their centers, when the grade group is torsion-free. Beyond that, it lays the foundation for proving theorems about valued division algebras by first proving corresponding results in the relatively easier setting of graded division algebras. This approach has previously been applied successfully for wildly ramified valued division algebras by Tignol in [T].

This paper is organized as follows: Before considering connections between valued and graded division algebras, we develop the graded theory in the first three sections. In §1 we recall basic properties of graded division algebras and graded central simple algebras (GCSA’s) over a graded field with torsion-free grade group, and point out the analogues of Wedderburn’s theorem and the double centralizer theorem. We also prove a version of the Skolem-Noether theorem for GCSA’s, which is somewhat delicate. In §2 we prove properties for graded division algebras which are analogous to known properties of tame valued division algebras. This is used in §3 to prove a cohomological characterization of the graded Brauer group GBr(R) of a graded field R. In §4, we show how to get back and forth between tame valued division algebras and graded division algebras. If we start with a graded field R with totally ordered grade group Γ_{R}, then GHR ≅_{g} R (graded isomorphism) canonically, where GHR is the graded field obtained from the valuation on the Henselian field HR obtained from the valuation on the quotient field of R determined by the grading on R. But, if we start with a Henselian valued field F, and take the Henselization HGF of the quotient field of the graded field GF (with respect to the valuation determined by the grading on GF), where GF is built from the valuation on F, then usually HGF ≇ F. (These fields need not even have the same characteristic.) Nonetheless, we prove in Th. 4.4 that TBr(HGF) ≅ TBr(F). In §5 we prove the isomorphism TBr(F) ≅ GBr(GF) mentioned above, and the correspondences between tame subalgebras and graded subalgebras. Finally, the compatibility with the corestriction is given in §6.

§1 Graded division algebras and graded central simple algebras

We begin by setting up notation and recalling some results about graded division algebras and graded central simple algebras. Except for the graded Skolem-Noether theorem, Prop. 1.6, most of what we say in this section can be found in the literature somewhere (see especially [B_{2}], [CvO], [NvO]), though not always in the generality we need.

Let A = ⨁_{γ∈Γ} A_{γ} be a graded ring. This means for us that A is an associative ring with 1, Γ is an abelian group, each A_{γ} is a subgroup of the additive group of A, and A_{γ} · A_{δ} ⊆ A_{γ+δ} for all γ, δ ∈ Γ. Because we are interested in the graded rings associated to valuation rings, we will assume throughout that Γ is torsion-free. We set

Γ_{A} = {γ ∈ Γ | A_{γ} ≠ (0)}, the grade set of A,

and

A^{h} = ⋃_{γ∈Γ_{A}}A_{γ}, the set of homogeneous elements of A.

For a ∈ A^{h}, a ≠ 0, we write deg(a) = γ if a ∈ A_{γ}. Each c ∈ A is uniquely expressible as c = ∑_{γ∈Γ_{A}} c_{γ} with each c_{γ} ∈ A_{γ}. The c_{γ} are called the homogeneous components of c. Let A^{∗} denote the group of units of A. A subring S of A is a graded subring if S = ⨁_{γ∈Γ_{A}}(S ∩ A_{γ}) (iff for each s ∈ S,

all the homogeneous components of s lie in S). Note that if S is a graded subring of A, then its centralizer C_{A}(S) is also a graded subring of A. In particular, the center of A, Z(A) = C_{A}(A) is a graded subring of A. A (left, right, or two-sided) ideal I of A is said to be homogeneous if I = ⨁_{γ∈Γ_{A}}(I ∩ A_{γ}) (iff I is generated as a left or … ideal of A by homogeneous elements). Suppose

B = ⨁_{γ∈Γ′} B_{γ} is another graded ring, and suppose there is a torsion-free group Δ containing Γ and

Γ′ as subgroups. A graded ring homomorphism f: A → B is a ring homomorphism such that f(A_{δ}) ⊆ B_{δ} for all δ ∈ Δ. (It is understood that A = ⨁_{δ∈Δ} A_{δ}, where A_{δ} = (0) for δ ∈ Δ − Γ_{A}; likewise for B.) If, further, f is bijective, then f is a graded isomorphism, and we write A ≅_{g} B. We frequently abbreviate “graded” by gr. The graded ring A is said to be graded simple if |A| > 1 (i.e., 1_{A} ≠ 0_{A}) and the only homogeneous two-sided ideals of A are A and (0).

A graded left A-module M is a left A-module with a direct sum decomposition as abelian groups M = ⨁_{γ∈Γ′} M_{γ}, where Γ′ is some torsion-free abelian group containing Γ, such that A_{γ} ·M_{δ} ⊆

M_{γ+δ}, for all γ ∈ Γ_{A}, δ ∈ Γ′. Then Γ_{M}, M^{h}, and graded submodules are defined just as above for rings. We can make M into a graded A-module in other ways by shifting the grading: For any γ ∈ Γ′, the γ-shift of M, denoted s_{γ}(M) is defined by

s_{γ}(M) = M as an A-module, and s_{γ}(M)_{δ} = M_{γ+δ}, for all δ ∈ Γ′.

So, Γ_{s_{γ}(M)} = −γ +Γ_{M}. Now, let N = ⨁_{γ∈Γ′′} N_{γ} be another graded left A-module, such that there is

a torsion-free abelian group Δ containing Γ′ and Γ′′ as subgroups. A graded A-module homomorphism f: M → N is an A-module homomorphism such that f(M_{δ}) ⊆ N_{δ} for all δ ∈ Δ. There is the corresponding notion of graded isomorphism, and when there is one between M and N we write M ≅_{g} N. Let GHom_{A}(M, N) denote the group of graded A-module homomorphisms from M to N, so GHom_{A}(M, N) is a subgroup of the group Hom_{A}(M, N) of all A-module homomorphisms from M to N. For each δ ∈ Δ, we have a subgroup of Hom_{A}(M, N) of δ-shifted homomorphisms

Hom_{A}(M, N)_{δ} = {f ∈ Hom_{A}(M, N) | f(M_{γ}) ⊆ N_{γ+δ} for all γ, δ ∈ Δ}.

Of course, Hom_{A}(M, N)_{δ} = GHom_{A}(M, s_{δ}(N)) = GHom_{A}(s_{−δ}(M), N). Clearly, ⨁_{δ∈Δ} Hom_{A}(M, N)_{δ} is subgroup of Hom_{A}(M, N); if M is a finitely-generated A-module, then

Hom_{A}(M, N) = ⨁_{δ∈Δ} Hom_{A}(M, N)_{δ}

(cf. [NvO, Lemma I.6.1, p. 26]). Indeed, for f ∈ Hom_{A}(M, N), δ ∈ Δ, define f_{δ} ∈ Hom_{A}(M, N)_{δ} by, for m = ∑_{γ∈Δ} m_{γ} with m_{γ} ∈ M_{γ}, setting f_{δ}(m) = ∑_{ε∈Δ}(f(m_{ε−δ}))_{ε}. When M is finitely-generated,

all but finitely many f_{δ} = 0, and f = ∑_{δ} f_{δ}. In particular, for any finitely-generated graded left A-module M, End_{A}(M) = Hom_{A}(M, M) is a graded ring. When A acts on M on the left, we view End_{A}(M) as acting on M on the right; so M is a graded A-End_{A}(M)-bimodule.

Now, let M = ⨁_{γ∈Γ′} M_{γ} be a graded right A-module and N = ⨁_{γ∈Γ′′} N_{γ} a graded left A-module,

with Γ′, Γ′′ ⊆ Δ for some torsion-free-abelian group Δ. Then, M ⊗_{A} N has a natural grading as Z(A)-module given by

(M ⊗_{A} N)_{δ} = { ∑_{i} m_{i} ⊗ n_{i} | m_{i} ∈ M^{h}, n_{i} ∈ N^{h}, deg(m_{i}) + deg(n_{i}) = δ}, δ ∈ Δ.

One can see that this gives a grading on M ⊗_{A} N by observing that the corresponding grading on M ⊗_{A_{0}} N is clearly well-defined, and the grading on M ⊗_{A_{0}} N is inherited by M ⊗_{A} N ≅ (M ⊗_{A_{0}} N)/J, since the subgroup J of M ⊗_{A_{0}} N is generated by the homogeneous elements {ma ⊗ n − m ⊗ an | m ∈ M^{h}, n ∈ N^{h}, a ∈ A^{h}}.

For example, suppose F is a graded free right A-module of finite rank, i.e., F is graded right A-module which is free as an A-module with a finite base {b_{1}, …, b_{n}} ⊆ F^{h}. Let δ_{i} = deg(b_{i}) ∈ Γ_{F}. Of course, End_{A}(F) ≅ M_{n}(A) (n × n matrices over A) if we ignore the grading, and by convention End_{A}(F) acts on F on the left. In this isomorphism the ij-matrix unit E_{ij} ∈ M_{n}(A) corresponds to the map e_{ij} ∈ End_{A}(F), defined by e_{ij}(b_{j}) = b_{i} and e_{ij}(b_{k}) = 0, for k ≠ j. Clearly, e_{ij} ∈ End_{A}(F)_{δ_{i}−δ_{j}}. So, when we take the grading into account, we find that

End_{A}(F) ≅_{g} M_{n}(A)(d), for d = (δ_{1}, …, δ_{n}),  (1.1)

where M_{n}(A)(d) means n×n matrices over A but with the degree of the ij-entry shifted by δ_{i} −δ_{j}, i.e.,

M_{n}(A)(d) = ( \begin{matrix}s_{δ_{1}−δ_{1}}(A) & … & s_{δ_{n}−δ_{1}}(A) \\ · & · & · \\ s_{δ_{1}−δ_{n}}(A) & … & s_{δ_{n}−δ_{n}}(A) \end{matrix} ) .  (1.2)

So, the ij-entry of M_{n}(A)(d) is s_{δ_{j}−δ_{i}}(A) (as s_{δ_{j}−δ_{i}}(A)_{δ_{i}−δ_{j}} = A_{0}). Thus, the ε-component of M_{n}(A)(d) consists of matrices with ij-entry in A_{ε+δ_{j}−δ_{i}}.

For future reference, we point out a few elementary properties of these shifted graded matrix rings. Let A be any graded ring. Then,

(i) If π ∈ S_{n} is any permutation, then

M_{n}(A)(δ_{1}, …, δ_{n}) ≅_{g} M_{n}(A)(δ_{π(1)}, …, δ_{π(n)}).  (1.3)

(ii) If γ_{1}, …, γ_{n} ∈ Γ_{A}, with γ_{i} = deg(a_{i}) for some unit a_{i} ∈ A^{h}, then

M_{n}(A)(δ_{1}, …, δ_{n}) ≅_{g} M_{n}(A)(δ_{1} + γ_{1}, …, δ_{n} + γ_{n}).  (1.4)

(iii) If A is commutative, and d = (δ_{1}, …, δ_{n}), e = (ε_{1}, …, ε_{m}), then

M_{n}(A)(d) ⊗_{A} M_{m}(A)(e) ≅_{g} M_{mn}(A)(f),  (1.5)

where f = {δ_{i} + ε_{j} | 1 ≤ i ≤ n, 1 ≤ j ≤ m}. (The order of the terms is immaterial, in view of (1.3).)

For, M_{n}(A)(δ_{1}, …, δ_{n}) ≅_{g} End_{A}(F), where F is a graded free graded right A-module with homogeneous base b_{1}, …, b_{n} with deg(b_{i}) = δ_{i}. Since b_{π(1)}, …, b_{π(n)} is also a homogeneous base of F, we also have M_{n}(A)(δ_{π(1)}, …, δ_{π(n)}) ≅_{g} End_{A}(F), yielding (1.3). Likewise, if a_{i} is a homogeneous unit of A with deg(a_{i}) = γ_{i}, then b_{1}a_{1}, …, b_{n}a_{n} is another homogeneous base of F, with deg(b_{i}a_{i}) = δ_{i} + γ_{i}. So, M_{n}(A)(δ_{1} + γ_{1}, …, δ_{n} + γ_{n}) ≅_{g} End_{A}(F), proving (1.4). Now, assuming A is commutative, let F′ be another graded free A-module, with base c_{1}, …, c_{m} with deg(c_{j}) = ε_{j}. Then, F ⊗_{A} F′ is a free graded A-module with base {b_{i} ⊗ c_{j}}, where deg(b_{i} ⊗ c_{j}) = δ_{i} + ε_{j}. So

M_{n}(A)(d) ⊗_{A} M_{m}(A)(e) ≅_{g} End_{A}(F) ⊗_{A} End_{A}(F′) ≅_{g} End_{A}(F ⊗_{A} F′) ≅_{g} M_{mn}(A)(f),

showing (1.5).

A graded ring E = ⨁_{γ∈Γ_{E}} E_{γ} is called a graded division ring if every nonzero homogeneous element of E is a unit, and 1_{E} ≠ 0_{E}. Note that the grade set Γ_{E} is actually a group. Further, since Γ_{E} is torsion-free, it follows that E has no zero divisors and E^{∗} = E^{h} − {0}. (This is easy to see by recalling that the torsion-free abelian group Γ_{E} can be given a total ordering compatible with the group operation. Thus, if a ≠ 0, a = a_{γ} + terms of higher degree and b ≠ 0, b = b_{δ} + terms of higher degree, then ab = a_{γ}b_{δ} + terms of higher degree, so ab ≠ 0.) Also, E_{0} must be a division ring, and for each γ ∈ Γ_{E}, the group E_{γ} is a one-dimensional left and right vector space over E_{0}. Note further that every graded left (resp. right) E-module M is a graded free E-module (cf. [B_{1}, Th. 3, p. 29]). For, it is easy to check that a maximal homogeneous Elinearly independent subset of M is actually a base. We call M a graded vector space over E, and write dim_{E}(M) for the rank of M as a graded free E-module. (This is well-defined, since one can apply the usual exchange argument to see that any two homogeneous bases of M have the same cardinality.) Note that if N is a graded submodule of M, then

dim_{E}(N) + dim_{E}(M/N) = dim_{E}(M).  (1.6)

Consequently, if dim_{E}(M) < ∞ and N is a proper submodule of M, then dim_{E}(N) < dim_{E}(M). Let S be a graded subring of E such that S is also a graded division ring, and let [E : S] = dim_{S}(E) (left dimension) and likewise [E_{0} : S_{0}] = dim_{S_{0}}E_{0} (left dimension). Note the easy but fundamental formula (cf. [B_{2}, p. 4278])

[E : S] = [E_{0} : S_{0}] · |Γ_{E} : Γ_{S}|.  (1.7)

This holds since if {a_{i}} is a base of E_{0} as left S_{0}-vector space and if {b_{j}} ⊆ E^{h} − {0} is chosen so that {deg(b_{j})} is a set of coset representatives for Γ_{S} in Γ_{E}, then {a_{i}b_{j}} is a homogeneous base of E as a left S-vector space.

A commutative graded division ring is called a graded field. For example, if L is any field and Γ is any torsion-free abelian group, then the group ring R = L[Γ] is a graded field with R_{0} = L and Γ_{R} = Γ. In fact, Γ is a free abelian group, then every graded field S with Γ_{S} = Γ is a group ring (cf. [HW, Prop. 1.1]). However, there do exist graded fields which are not group rings (cf. [HW, Ex. 1.2]).

Let R be a graded field. A graded R-algebra A is graded ring which is an R-algebra such that the associated ring homomorphism φ: R → Z(A) is a gr-homomorphism. This φ is necessarily injective (assuming 1 ≠ 0 in A), as R is a graded field. We have A_{0} is an R_{0}-algebra. Also, while Γ_{A} need not be a group, it is a union of cosets of the group Γ_{R} in some ambient torsion-free abelian group Γ′. We write

|Γ_{A} : Γ_{R}| = the number of cosets of Γ_{R} in Γ_{A}

and

[A : R] = dim_{R}(A).

It is easy to check that

[A : R] ≥ [A_{0} : R_{0}] · |Γ_{A} : Γ_{R}|,  (1.8)

but equality often does not hold (see Prop. 1.4 below).

For our graded field R, let Δ_{R} be the divisible hull of the torsion-free abelian group Γ_{R}, so

Δ_{R} ≅ ℚ ⊗_{ℤ} Γ_{R},

and fix some ℚ-vector space Δ′ containing Δ_{R} with dim_{ℚ}(Δ′/Δ_{R}) = ∞. Then (1.8) shows that if A is any finite dimensional graded R-algebra, then Γ_{A} is Γ_{R}-isomorphic to a subset of Δ′. Indeed, if Γ_{A} is a group (which occurs, e.g., whenever A is a graded division algebra) then, as Γ_{A} is torsion-free and Γ_{A}/Γ_{R} is torsion by (1.8), there is a unique group homomorphism Γ_{A} → Δ_{R} which restricts to the identity on Γ_{R}. So we will assume henceforth that all graded R-algebras A satisfy Γ_{A} ⊆ Δ′.

Note that if A and B are graded R-algebras, then A ⊗_{R} B is also a graded R-algebra. If A′ is a graded R-subalgebra of A and B′ a graded R-subalgebra of B, then it is easy to check that

C_{A⊗_{R}B}(A′ ⊗_{R} B′) = C_{A}(A′) ⊗_{R} C_{B}(B′).  (1.9)

A graded algebra A over a graded field R is said to be a graded central simple algebra (GCSA) over R if A is a simple graded ring, [A : R] < ∞, and Z(A) = R. There is a theory of GCSA’s over a graded field analogous to the theory of central simple algebras (CSA’s) over a field, and we recall some basic properties here.

Proposition 1.1. Let A be a GCSA over a graded field R, and let B be any graded R-algebra. If I is a homogeneous ideal of A ⊗_{R} B, then I = A ⊗_{R} J, where J = I ∩ B, and J is a homogeneous ideal of B. Hence, if B is graded simple, then A ⊗_{R} B is a GCSA over Z(B).

Of course, in Prop 1.1 we are identifying B with its gr-isomorphic copy R⊗_{R}B in A⊗_{R}B. This proposition can be proved analogously to the ungraded result. One can first show the special case: if I ∩ B = (0), then I = (0). The general result follows by applying the special case to B′ = B/J (after noting that J is homogeneous, so B′/J is graded); since A ⊗_{R} B′ ≅_{g} (A ⊗_{R} B)/(A ⊗_{R} J) and (I/(A ⊗_{R} J)) ∩ B′ ≅ (I ∩ B)/J = (0), we obtain I/(A ⊗_{R} J) = (0), as desired. Formula (1.9) shows Z(A ⊗_{R} B) = Z(A) ⊗_{R} Z(B) ≅_{g} Z(B).

Corollary 1.2 (cf. [B_{2}, Prop. 5.1]). Let A be an algebra over a graded field R. Then, A is a GCSA over R iff A is both an Azumaya algebra over R and also a graded R-algebra.

Proof. Suppose A is a GCSA over R. Then, A is a free R-module of finite rank, and the graded R-algebra homomorphism A ⊗_{R} A^{op} → End_{R}(A) is injective, since the domain is graded simple by Prop. 1.1, and surjective by dimension count (using (1.6)). Hence, by [DI, Th. 3.4, p. 52], A is an Azumaya algebra over R. Conversely, suppose A is an Azumaya algebra over R such that A is also a graded R-algebra. We identify R with its gr-isomorphic copy in A. Since A is Azumaya over R, by [DI, Prop. 21, p. 47; Cor. 3.7, p. 54], A is a finitely-generated R-module, so [A : R] < ∞, and Z(A) = R, and every ideal I of A has the form I = A(I ∩ R). If I is a homogeneous ideal of A, then I ∩ R is a homogeneous ideal of R. Hence, A is graded simple since R is graded simple. □

A graded central division algebra (GCDA) over a graded field R is a GCSA E over R such that E is also a graded division ring. Observe that the usual matrix calculations show that for any GCDA E over R, any n, and any d = (δ_{1}, …, δ_{n}), δ_{i} ∈ Δ′, we have M_{n}(E)(δ) is a GCSA over R. Our next proposition is the graded Wedderburn theorem, which says that all GCSA’s over R have this form.

Proposition 1.3. Let A be a GCSA over a graded field R. Then,

(a) There is a GCDA E over R such that A ≅_{g} M_{n}(E)(d) for some d = (δ_{1}, …, δ_{n}). Moreover, if A ≅_{g} M_{n′}(E′)(d′) for some GCDA E′ over R, then n′ = n and E′ ≅_{g} E.

(b) Every graded left (or right) A-module is a direct sum of graded simple A-modules.

(c) If L is a minimal nonzero homogeneous left ideal of A and N is a graded simple A-module, then N ≅_{g} s_{δ}(L) i.e., N is the δ-shift of L for some δ. Hence,

dim_{R}(N) = n[E : R] = [A : R]/n.  (1.10)

This can be proved analogously to the usual Wedderburn theorem. Here is a sketch. Take a minimal nonzero homogeneous left ideal L of A (which exists as [A : R] < ∞), and let E = End_{A}(L). Since L is a graded simple A-module (i.e., it has no nonzero proper graded A-submodule), the graded Schur’s Lemma shows that E is a graded division ring, and [E : R] ≤ [End_{R}(L) : R] < ∞. Let b_{1}, …, b_{n} be a homogeneous base of L as a graded free right E-module, so L = b_{1}E ⊕…⊕b_{n}E. Then, End_{E}(L) ≅_{g} M_{n}(E)(d), where d = (deg(b_{1}), …, deg(b_{n})), as noted in (1.1) above. Rieffel’s proof of Wedderburn’s Theorem ([Ri], or see [L, Th. 5, p. 449]) can be applied here to see that the graded R-algebra homomorphism A → End_{E}(L) (a ↦ left multiplication by a) is an isomorphism, so A ≅_{g} End_{E}(L) ≅_{g} M_{n}(E)(d). If N is any graded simple left A-module, there is a b ∈ N^{h} − {0} with L · b ≠ (0). Then the A-module homomorphism λ: L → N, ℓ ↦ ℓ · b is an isomorphism since im(λ) and ker(λ) are graded submodules. Since λ shifts degrees by deg(b), we have N ≅_{g} s_{−deg(b)}(L). This yields (c) and also the uniqueness part of (a). For, if A ≅_{g} M_{n′}(E′)(d′) and L′ is the set of first columns of elements of M_{n′}(E′)(d′), then L′ is a graded simple left M_{n′}(E′)(d′)- module with endomorphism ring E′. Since L′ is graded simple when viewed as an A-module, L′ ≅_{g} s_{δ}(L), for some δ, so E = End_{A}(s_{δ}(L)) ≅_{g} End_{A}(L′) ≅_{g} E′; then n′ = n by dimension count. Finally, for (b), since A is a sum of simple graded left A-modules (corresponding to the columns of M_{n}(E)(d) ), every graded left A-module M is a sum M = ∑ N_{i} where the N_{i} are graded simple submodules of M. The usual argument shows that M is a direct sum of some subset of {N_{i}}.

For any GCSA A over R, we define the Schur index of A, ind(A), analogously to the ungraded case: We have A ≅ M_{n}(E)(d) for a GCDA E over R; set

ind(A) = \sqrt{[E : R]}, a positive integer.

The graded Wedderburn theorem yields a description of Γ_{A} and of A_{0} for a GCSA. Let A = M_{n}(E)(d) where E is a GCDA over R and d = (δ_{1}, …, δ_{n}), δ_{i} ∈ Δ′. Let ε_{1} + Γ_{E}, …, ε_{k} + Γ_{E} be the distinct cosets of Γ_{E} of the form δ_{i} + Γ_{E}, 1 ≤ i ≤ n, and for each ε_{ℓ} let r_{ℓ} be the number of i with δ_{i} ≡ ε_{ℓ} (mod Γ_{E}).

Proposition 1.4. Let E be a graded division algebra, and let A = M_{n}(E)(d), for d = (δ_{1}, …, δ_{n}). Then,

(a) Γ_{A} = ⋃_{i=1}^{n} ⋃_{j=1}^{n}(δ_{i} − δ_{j}) + Γ_{E}.

(b) A_{0} ≅ M_{r_{1}}(E_{0}) × … × M_{r_{k}}(E_{0}), with the r_{i} as defined above. In particular, A_{0} is simple iff k = 1 iff Γ_{A} = Γ_{E}.

Proof. (a) is immediate from the description of the grading on M_{n}(E)(d) (see (1.2)). For (b), observe that by (1.3) and (1.4) above, A ≅_{g} M_{n}(E)(e), where e = (ε_{1}, …, ε_{1}, ε_{2}, …, ε_{2}, …, ε_{k}, …, ε_{k}) with each ε_{ℓ} occurring r_{ℓ} times. In M_{n}(E)(e)_{0} there is a contribution of E_{0} in the ij-entry when the same ε_{ℓ} appears in the i-th and the j-th position in e. This accounts for all of M_{n}(E)(e)_{0} since 0 ∉ ε_{ℓ} − ε_{m} + Γ_{E} when ℓ ≠ m. Thus, A_{0} ≅ M_{n}(E)(e)_{0} ≅ M_{r_{1}}(E_{0}) × … × M_{r_{k}}(E_{0}). □

The double centralizer theorem is also available in the graded context:

Proposition 1.5. Let A be a GCSA over a graded field R and let B be a graded simple graded R-subalgebra of A. Set C = C_{A}(B). Then,

(a) C is a graded simple R-subalgebra of A with Z(C) = Z(B), and C_{A}(C) = B.

(b) [C : R] · [B : R] = [A : R].

(c) B ⊗_{Z(B)} C ≅_{g} C_{A}(Z(B)). In particular, if Z(B) = R, then B ⊗_{R} C ≅_{g} A.

This is proved analogously to the ungraded version. We give a sketch. Let L be a minimal homogeneous left ideal of A, and let E = End_{A}(L), so E is a GCDA over R and A ≅_{g} End_{E}(L) as we saw in the discussion of Prop. 1.3. Let T = B^{op} ⊗_{R} E, which is a GCSA over Z(B) by Prop. 1.1, and view L as a graded right T -module. Now, End_{T} (L) ⊆ End_{E}(L) ≅_{g} A, and A acts faithfully on L. Hence End_{T} (L) is gr-isomorphic to the set of elements of A acting on L compatibly with the B-action, i.e., End_{T} (L) ≅_{g} C_{A}(B) = C. Let N be a minimal homogeneous right ideal of T, and let U = End_{T} (N), which is a GCDA over Z(T). Then L ≅_{g} s_{δ_{1}}(N)⊕…⊕s_{δ_{k}}(N) as graded T -modules by Prop. 1.3(b) and (c). Hence, C ≅_{g} End_{T} (L) ≅_{g} End_{T} (s_{δ_{1}}(N) ⊕ … ⊕ s_{δ_{k}}(N)) ≅_{g} M_{k}(U)(d), where d = (δ_{1}, …, δ_{k}), so C is graded simple with Z(C) ≅_{g} Z(U) ≅_{g} Z(T) ≅_{g} Z(B). The formula in (b) follows from [C : R] = k^{2}[U : R], dim_{R}(L) = k dim_{R}(N), [T : R] = [B : R] · [E : R], together with (by (1.10)) [A : R] · [E : R] = dim_{R}(L)^{2} and [T : Z(B)] · [U : Z(B)] = dim_{Z(B)}(N)^{2}. Then C_{A}(C) = B, since C_{A}(C) ⊇ B and (b) shows [C_{A}(C) : R] = [B : R]. The graded homomorphism B ⊗_{Z(B)} C → C_{A}(Z(B)) given by b⊗c ↦ bc is injective as its domain is graded simple by Prop. 1.1, and surjective by dimension count.

There is a partial graded analogue to the Skolem-Noether theorem theorem. One would prefer to be able to conjugate a GCSA by a homogeneous unit, since then the grading is preserved. We will see that this is possible in some significant cases, but not always.

Proposition 1.6. Let A be a GCSA over a graded field R, let B and B′ be graded simple Rsubalgebras of A, and let C = C_{A}(B), Z = Z(B), and C′ = C_{A}(B′), Z′ = Z(B′). Suppose there is a graded R-algebra isomorphism α: B → B′. Then,

(a) There is a ∈ A^{∗} such that α(b) = aba^{−1} for all b ∈ B.

(b) The a of part (a) can be chosen to be homogeneous iff there is a gr-isomorphism γ: C → C′ such that γ|_{Z} = α|_{Z}.

(c) If C_{0} is a division ring, then the a of part (a) can be chosen to be homogeneous. In particular, every graded R-algebra automorphism of A is given by conjugation by a homogeneous unit of A.

Proof. The proof of part (a) is analogous to the ungraded theorem (cf. [R, pp. 103–104]): Let L, E, T, N be as in the proof above of the double centralizer theorem, so T = B ⊗_{R} E^{op}. We make L into a graded left T -module in two ways, first by (b ⊗ e^{op}) · ℓ = b ℓ e, and second by (b ⊗ e^{op}) · ℓ = α(b) ℓ e. Write L′ for L with the second T -module action, while L unadorned denotes L with the first T -action. By Prop. 1.3, L ≅_{g} s_{δ_{1}}(N)⊕…⊕s_{δ_{n}}(N) and L′ ≅_{g} s_{ε_{1}}(N)⊕…⊕s_{ε_{m}}(N) as graded T -modules, and m = n by dimension count. Since each s_{δ}(N) = N as T -modules when we ignore the grading, we have L′ ≅ L as ungraded T -modules. The ungraded argument as in [R] then shows there is a ∈ A^{∗} with

ab = α(b)a  (1.11)

for all b ∈ B. Thus, α(b) = aba^{−1}, for all b ∈ B, proving part (a). We proceed to the proof of (c). Let c = a^{−1}, and let a = ∑ a_{γ} and c = ∑ c_{δ} be the homogeneous decompositions of a and c. Because α is a graded homomorphism, for each b ∈ B^{h}, formula (1.11) yields by comparing homogeneous components

a_{γ}b = α(b)a_{γ},  (1.12)

for each a_{γ}. Hence, (1.12) holds for all b ∈ B. Likewise, since bc = cα(b) for all b ∈ B, we find that bc_{δ} = c_{δ}α(b) for all c_{δ}. These equations show c_{δ}a_{γ} ∈ C for all a_{γ} and c_{δ}. So, the equation 1 = ca = ∑_{δ} ∑_{γ} c_{δ}a_{γ} has all its summands in C^{h}. Therefore, there must be a nonzero summand

c_{δ}a_{γ} ∈ C_{0}. When C_{0} is a division ring, c_{δ}a_{γ} ∈ C_{0}^{∗}, so a_{γ} ∈ A^{∗}. Then (1.12) shows α(b) = a_{γ}ba_{γ}^{−1} for all b ∈ B, proving (c).

For part (b), observe first that if the a of part (a) is homogeneous, then a^{−1} is also homogeneous, so conjugation by a is a graded automorphism of A. Since this map sends B to B′, it also sends C to C′. Hence, we can take γ to be the restriction to C of conjugation by a. Conversely, suppose there is γ: C → C′ as in (b). Let Y = C_{A}(Z) and Y′ = C_{A}(Z′). Then, as Y = BC ≅_{g} B ⊗_{Z} C by Prop. 1.5(c), we obtain a graded R-algebra isomorphism β: Y → Y′ as the composition

Y ≅_{g} B ⊗_{Z} C ⟶^{α⊗γ} B′ ⊗_{Z} C′ ≅_{g} Y′.

Now, Y is graded simple and C_{A}(Y) = Z by Prop. 1.5, and Z is a graded field. Therefore, we may apply parts (a) and (c) with Y, γ replacing B, α, to see that there is a homogeneous unit a′ ∈ A such that β(y) = a′ya^{′−1} for all y ∈ Y. Since β|_{B} = α, we can use a′ for the a of part (a) for B, α, as desired.

The final assertion of the proposition follows by taking B = A (so C = R) and invoking (a) and (c) (or (b)). □

Remark 1.7. Note that for the C of Prop. 1.6, C_{0} is a division ring iff C is a division ring, by Prop. 1.4(b).

Example 1.8. Let R be a graded field with Γ_{R} = ℤ. Let A = M_{4}(R)(0, 0, \frac{1}{2}, \frac{1}{2}), B_{1} = C_{1} = M_{2}(R)(0, \frac{1}{2}), and C_{2} = M_{2}(R)(0, 0). Then, by (1.5), (1.3), (1.4) above,

B_{1} ⊗_{R} C_{1} ≅_{g} A ≅_{g} B_{1} ⊗_{R} C_{2}.

Let B be the copy of B_{1} in A given by the first graded isomorphism, and B′ the copy of B_{1} in A given by the second. Then C_{A}(B) ≅_{g} C_{1} by (1.9), and C_{A}(B′) ≅_{g} C_{2}. However, C_{1} ≇_{g} C_{2}, e.g. since Γ_{C_{1}} = \frac{1}{2}ℤ while Γ_{C_{2}} = ℤ. Thus, Prop. 1.6(a) and (b) show that although B′ is obtainable from B by conjugating by some a ∈ A^{∗}, there is no homogeneous such a. Furthermore, a graded R-isomorphism B → B′ cannot be extended to a graded R-automorphism of A.

§2 Valuation-like properties of graded division algebras

Let R = ⨁_{γ∈Γ_{R}} R_{γ} be a graded field (with Γ_{R} torsion-free, as we are always assuming), and let E be a GCDA over R. In this section we will describe some properties of E which are analogous to known properties for tame division algebras over a Henselian valued field. We will use them in §3 in proving a cohomological characterization of GBr(R), see Prop. 3.3 below.

Before considering an arbitrary GCDA over R, we note a couple of extreme cases. First, a GCDA I over R is said to be unramified (or inertial) if Γ_{I} = Γ_{R} (iff by (1.7), [I : R] = [I_{0} : R_{0}]). In this situation the graded homomorphism I_{0} ⊗_{R_{0}} R → I is actually an isomorphism, since it is clearly surjective, and a dimension comparison then shows it is also injective. Since Z(I_{0} ⊗_{R_{0}} R) = Z(I_{0}) ⊗_{R_{0}} R, it follows that Z(I_{0}) = R_{0}. Thus, there is a one-to-one correspondence (I ↔ I_{0}) between isomorphism classes of unramified GCDA’s over R and isomorphism classes of central division algebras (CDA’s) over R_{0}. Also, if S is a graded R-subalgebra of I, then Γ_{S} = Γ_{R}, so S = S_{0} ⊗_{R} R. Thus, graded R-subalgebras of I are in canonical one-to-one correspondence (not just up to isomorphism) with R_{0}-subalgebras of I_{0}.

At the other extreme, a GCDA T over R is said to be totally ramified if T_{0} = R_{0} (iff |Γ_{T} : Γ_{R}| = [T : R], by (1.7)). In this case, there is a pairing γ_{T} : T^{∗} × T^{∗} → R_{0}^{∗} given by (s, t) ↦ [s, t] (= sts^{−1}t^{−1}). (Recall that T^{∗} = T^{h} − {0}.) The pairing is clearly skew-symmetric, and since the image of γ_{T} is central, the commutator identity [s, tu] = [s, t]t[s, u]t^{−1} shows that γ_{T} is bimultiplicative as well. Because γ_{T} (s, t) = 1 if s or t is central, the pairing is actually well-defined on T^{∗}/R^{∗} × T^{∗}/R^{∗}. But, as T is totally ramified T^{∗}/R^{∗} ≅ Γ_{T} /Γ_{R} which is finite; so every element of im(γ_{T}) has finite order in R_{0}^{∗}. Thus, γ_{T} induces a well-defined biadditive skew-symmetric pairing called the canonical pairing

β_{T} : Γ_{T} /Γ_{R} × Γ_{T} /Γ_{R} → μ(R_{0}),  (2.1)

given by (deg(s) + Γ_{R}, deg(t) + Γ_{R}) ↦ sts^{−1}t^{−1}, where μ(R_{0}) denotes the group of all roots of unity in the field R_{0}. (We will use the further notation: If F is a field and ℓ a positive integer, then μ_{ℓ}(F) is the group of all ℓ-th roots of unity in F.)

Proposition 2.1 (cf. [B_{2}, Prop. 2.6]). Let T be a totally ramified GCDA over R. Then the canonical pairing β_{T} of (2.1) is nondegenerate. The image of β_{T} is a cyclic subgroup of μ(R_{0}) of order equal to the exponent exp(Γ_{T} /Γ_{R}). Hence, char(R_{0}) ∤ |Γ_{T} /Γ_{R}|.

Proof. For s ∈ T^{∗}, if γ_{T} (s, t) = 1 for all t ∈ T^{∗}, then s ∈ Z(T) = R. This shows β_{T} is nondegenerate. Let ℓ = exp(Γ_{T} /Γ_{R}), i.e., the exponent of the finite abelian group Γ_{T} /Γ_{R}; then im(β_{T}) ⊆ μ_{ℓ}(R_{0}) as β_{T} is biadditive. But, if we take any \overline{α} ∈ Γ_{T} /Γ_{R} of order ℓ, then the nondegeneracy of β_{T} forces the homomorphism β_{T} (\overline{α}, −): Γ_{T} /Γ_{R} → μ_{ℓ}(R_{0}) to be surjective, and forces |μ_{ℓ}(R_{0})| = ℓ. Hence, im(β_{T}) = μ_{ℓ}(R_{0}), and |im(β_{T})| = ℓ = exp(Γ_{T} /Γ_{R}). If p = char(R_{0}), then as μ(R_{0}) has no p-torsion, we must have p ∤ ℓ. □

Remarks 2.2. (i) If Λ is any group with Γ_{R} ⊆ Λ ⊆ Γ_{T}, then T_{Λ} = ⨁_{λ∈Λ} T_{λ} is a graded R-subalgebra of T with Γ_{T_{Λ}} = Λ; furthermore, since T_{0} = R_{0}, every R-subalgebra of T has the form T_{Λ} for some Λ. Thus, the subgroups of Γ_{T} /Γ_{R} classify the R-subalgebras of T. Note also that Z(T_{Λ}) = T_{Λ′}, where Λ′/Γ_{R} = Λ/Γ_{R} ∩ (Λ/Γ_{R})^{⊥}, where (Λ/Γ_{R})^{⊥} denotes the orthogonal subgroup to Λ/Γ_{R} in Γ_{T} /Γ_{R} with respect to β_{T}.

(ii) Relative to the skew-symmetric nondegenerate biadditive pairing β_{T} there always exists a symplectic base for Γ_{T} /Γ_{R} (cf. [TW, Prop. 3.1]). This implies that the distinct invariant factors of the finite abelian group Γ_{T} /Γ_{R} each occur with even multiplicity. Also, the symplectic base allows one to decompose T into a tensor product of graded symbol algebras over R, analogous to the decomposition for tame totally ramified valued division algebras described in [TW, Prop. 4.2]. Furthermore, one can use the symplectic base to see that if R_{0}^{ℓ} = R_{0} where ℓ = exp(Γ_{T} /Γ_{R}), then T is determined up to isomorphism by Γ_{T} and β_{T}. Also, one can easily see that for any group Γ with Γ_{R} ⊆ Γ ⊆ Δ_{R}, such that Γ/Γ_{R} is finite and all the distinct invariant factors of Γ/Γ_{R} occur with even multiplicity and |μ_{ℓ}(R_{0})| = ℓ, where ℓ = exp(Γ/Γ_{R}), there exists a nondegenerate skew-symmetric biadditive pairing β: Γ/Γ_{R} × Γ/Γ_{R} → μ_{ℓ}(R_{0}); for any such Γ and β one can use a symplectic base for β on Γ/Γ_{R} to construct a totally ramified graded division algebra T over R such that Γ_{T} = Γ and β_{T} = β.

Now, let E be any GCDA over the graded field R. Observe that there is a well-defined group homomorphism

θ_{E} : Γ_{E}/Γ_{R} → G(Z(E_{0})/R_{0}) given by deg(e) + Γ_{R} ↦ (z ↦ eze^{−1}),  (2.2)

for all e ∈ E^{∗} and z ∈ Z(E_{0}), where G(Z(E_{0})/R_{0}) denotes the Galois group of Z(E_{0}) over R_{0}.

There are some graded R-subalgebras of E canonically determined by E_{0}: Let

Z = Z(E_{0}) · R ≅_{g} Z(E_{0}) ⊗_{R_{0}} R,
C = C_{E}(Z),
I = E_{0} · R ≅_{g} E_{0} ⊗_{R_{0}} R,
T = C_{C}(I) = C_{E}(I) = C_{E}(E_{0}).

Diagram (2.3) shows the inclusion relations among these algebras. Note that Z is a graded field and C, I, and T are graded division algebras. Clearly, Γ_{Z} = Γ_{R} and Z_{0} = Z(E_{0}). The double centralizer theorem, Prop. 1.5, shows that Z = Z(C), and [E : C] = [Z : R]. Since C = C_{E}(Z) = C_{E}(Z_{0}), the definition of θ_{E} shows that Γ_{C}/Γ_{R} = ker(θ_{E}); also clearly C_{0} = E_{0}. As for I, we have I_{0} = E_{0} and Γ_{I} = Γ_{R}. Also, Z(I) = Z(E_{0} ⊗_{R_{0}} R) = Z(E_{0})⊗_{R_{0}} R = Z, so I is unramified over its center Z. Turning to T, the double centralizer theorem shows Z(T) = Z and C ≅_{g} I ⊗_{Z} T. Also, T_{0} = Z_{0} as T_{0} centralizes I_{0}. Hence, T is totally ramified over its center Z. A dimension count using (1.7) shows Γ_{T} /Γ_{R} = Γ_{C}/Γ_{R} = ker(θ_{E}).

E C = I ⊗_{Z} T I  T Z R  (2.3)

grade  degree 0
algebra  group /Γ_{R}  component
E  Γ_{E}/Γ_{R}  E_{0}
C  ker(θ_{E})  E_{0}
T  ker(θ_{E})  Z(E_{0})
I  (0)  E_{0}
Z  (0)  Z(E_{0})
R  (0)  R_{0}

Proposition 2.3 (cf. [B_{2}, Prop. 2.4]). For any GCDA E over a graded field R, the field Z(E_{0}) is Galois over R_{0} and the homomorphism θ_{E} of (2.2) maps Γ_{E}/Γ_{R} onto the Galois group G(Z(E_{0})/R_{0}), so G(Z(E_{0})/R_{0}) is abelian. Also, char(R_{0}) ∤ |ker(θ_{E})|.

Proof. We give a different proof from the one in [B_{2}]. We use the information accumulated above about Z, C, I, T. Observe that

|G(Z(E_{0})/R_{0})| ≤ [Z(E_{0}) : R_{0}] = [Z : R] = [E : C] = |Γ_{E} : Γ_{C}|
= |Γ_{E}/Γ_{R} : ker(θ_{E})| = |im(θ_{E})| ≤ |G(Z(E_{0})/R_{0})|.

Hence, equality holds throughout. This shows that θ_{E} is surjective and that |G(Z(E_{0})/R_{0})| = [Z(E_{0}) : R_{0}], hence Z(E_{0}) is Galois over R_{0}. Because T is totally ramified, Prop. 2.1 shows that char(R_{0}) = char(Z_{0}) ∤ |Γ_{T} /Γ_{Z}| = |ker(θ_{E})|. □

Remarks 2.4. (i) In [B_{2}, p. 4279] Boulagouaz defines a canonical pairing C_{E}: ker(θ_{E}) × ker(θ_{E}) → μ(Z(E_{0})). This pairing is just the pairing β_{T} : Γ_{T} /Γ_{Z} × Γ_{T} /Γ_{Z} → μ(Z_{0}) of (2.1) for the totally ramified T in E shown in (2.3). This pairing is canonically determined by E, since T is built canonically from E.

(ii) The graded subalgebras Z, C, I, T of E described here are analogous to valued subalgebras of a division algebra tame over a Henselian valued field, cf. [JW, §§1–2]. But notice that the subalgebras of E defined here are unique, not just unique up to isomorphism (as in the valued situation). Also the existence and properties of the subalgebras are easier to prove in the graded case than in the corresponding valued case.

(iii) There is a slight variation of the map θ_{E} of (2.2), which will appear in §4: If E is a graded division algebra over a graded field S with [E : S] < ∞ (so S ⊆ Z(E), but possibly S ≠ Z(E) ), define

θ_{E,S}: Γ_{E}/Γ_{S} → G(Z(E_{0})/S_{0}) by deg(e) + Γ_{S} ↦ (z ↦ eze^{−1})  (2.4)

for e ∈ E^{∗} and z ∈ Z(E_{0}). This map is clearly well-defined.

§3 The graded Brauer group of a graded field

We can now consider the graded Brauer group of a graded field R. Define an equivalence relation ∼_{g} on GCSA’s over R by: A ∼_{g} B if there are finitely-generated (hence graded free) graded R-modules M and N such that A ⊗_{R} End_{R}(N) ≅_{g} B ⊗_{R} End_{R}(M) as graded R-algebras. So, ∼_{g} is clearly an equivalence relation which is compatible with tensor products. Let [A]_{g} denote the equivalence class of A with respect to ∼_{g}. Then, the graded Brauer group of R is defined to be

GBr(R) = {[A]_{g} | A is a GCSA over R}.

(See [B_{2}, §5]; see also [CvO, III.4–IV.1] for the case Γ_{R} = ℤ, but note that our GBr(R) is their UBr_{g}(R), see [CvO, p. 139], since we allow Γ_{A} ⫌ Γ_{R}.) The operation on GBr(R) is induced by the tensor product, and, as we noted earlier for Cor. 1.2, A ⊗_{R} A^{op} ≅_{g} End_{R}(A). Thus, GBr(R) is a group with identity element [R]_{g} and [A]_{g}^{−1} = [A^{op}]_{g}. Now, if E is any GCDA over R, L is any finitely-generated graded right E-vector space, and N is any finitely-generated graded Rvector space, then End_{E}(N ⊗_{R} L) ≅_{g} End_{R}(N) ⊗_{R} End_{E}(L). It follows from this and the graded Wedderburn theorem, Prop. 1.3, that for GCSA’s A ≅_{g} M_{n}(E)(d) and A′ ≅_{g} M_{n′}(E′)(d′) with E, E′ GCDA’s over R, we have A ∼_{g} A′ iff E ≅_{g} E′. Thus, GBr(R) classifies GCDA’s over R up to graded isomorphism. Note that, unlike the case of central simple algebras over a field, we can have GCSA’s A, B over R with [A]_{g} = [B]_{g} and [A : R] = [B : R], but A ≇_{g} B. This occurs when A ≅_{g} M_{n}(E)(d) and B ≅_{g} M_{n}(E)(d′) with d and d′ sufficiently different.

As we noted above for graded division algebras, the assumption that Γ_{R} is torsion-free implies that a graded field R is an integral domain. Let

QR = the quotient field of R.

Likewise, for any graded R-algebra B, let

QB = QR ⊗_{R} B,

an algebra over the field QR. Observe that as B is graded-free as a graded R-module, B is R-torsion-free, so the canonical map B → QB is injective; also

[QB : QR] = [B : R].  (3.1)

Note in particular that if B is a graded division algebra over R with [B : R] < ∞, then since B has no zero divisors the same is true for QB; since also [QB : QR] < ∞ it follows that QB is a division ring.

Now, suppose A is a GCSA over R. Then as A is an Azumaya algebra over R (see Cor. 1.2), A determines a class [A] in the (ungraded) Brauer group Br(R); also QA is Azumaya over the field QR, i.e., QA is a central simple algebra over QR. Indeed QA is the classical ring of quotients of the prime p.i. ring A. There are canonical group homomorphisms

GBr(R) → Br(R) → Br(QR),  (3.2)

given by [A]_{g} ↦ [A] and [C] ↦ [QR ⊗_{R} C], and the composition is injective since if E is a GCDA over R, then QE is a CDA over QR of the same degree as E over R. So GBr(R) injects into Br(R). In general, Br(R) and Br(QR) may be much larger than GBr(R)(but not always, see [CvO, Th. IV.1.11, p. 139]). We will see below that if we give a total order to Γ_{R}, then there is a valued field extension of QR whose tame Brauer group coincides with GBr(R).

From Prop. 1.1 it is clear that for any graded field extension S of R, there is a well-defined scalar extension group homomorphism

res_{S/R}: GBr(R) → GBr(S) given by [A]_{g} ↦ [S ⊗_{R} A]_{g}.

We have therefore a corresponding relative graded Brauer group

GBr(S/R) = ker(res_{S/R}).

We will give in Prop. 3.2 below a cohomological description of GBr(S/R) when S is Galois over R, by adapting the usual crossed product construction.

Let R ⊆ S be graded fields with [S : R] < ∞. Then QS (= QR ⊗_{R} S, as above) is the quotient field of S, and [QS : QR] = [S : R]. Recall from [HW, Th. 3.11] (or see [B_{1}, Th. 4, p. 33]) that S is tame over R (i.e., S_{0} is separable over R_{0} and char(R_{0}) ∤ |Γ_{S} : Γ_{R}| ) iff QS is separable over QR, iff S is separable over R. Furthermore, every QR-automorphism of QS restricts to a graded R-automorphism of S (since S is the integral closure of R in QS and for every s ∈ S^{h}, deg(s) is determined by its minimal polynomial over QR, cf. [HW, Cor. 2.5 (a), (d)]). Indeed, by [HW, Th. 3.11(b)], S is G-Galois over R for some group G iff QS is Galois over QR, and when this occurs, G is canonically isomorphic to the Galois group G(QS/QR). There is thus no ambiguity in writing G = G(S/R). Furthermore, the preceding comments show that every element of G(S/R) is a graded automorphism of S.

The results in [HW] quoted in the preceding paragraph were stated there with the added assumption that Γ_{R} is totally ordered. However, they are valid when one only assumes that Γ_{R} is torsion-free (our standing hypothesis here), since any torsion-free abelian group can be given a total ordering, and the quoted results are independent of the choice of total ordering on Γ_{R}. This remark applies a number of times below when we quote [HW].

We next construct graded crossed products. Assume the graded field S is Galois over R, and let G = G(S/R). Then S^{∗}(= S^{h} − {0}) is a G-submodule of QS^{∗}. We write Z^{i}(G, S^{∗}), B^{i}(G, S^{∗}), H^{i}(G, S^{∗}) for the i-th cocycle group, i-th coboundary group, i-th cohomology group of G with coefficients in S^{∗}. Take any f ∈ Z^{2}(G, S^{∗}). We construct the crossed product algebra (S/R, G, f) in the usual way: Let {x_{σ} | σ ∈ G} be new symbols, and on the free S-module ⨁_{σ∈G} Sx_{σ}, with base {x_{σ} | σ ∈ G}, define multiplication by

(ax_{σ})(bx_{τ}) = aσ(b)f(σ, τ)x_{στ} ,  (3.3)

for all a, b ∈ S, σ, τ ∈ G (and extended distributively to all of ⨁_{σ} Sx_{σ}). It is well-known that (S/R, G, f) is an associative R-algebra since f is a 2-cocycle, and that it is an Azumaya algebra over R, since all f(σ, τ) ∈ S^{∗}. We now make it into a graded R-algebra.

Lemma 3.1. There is a unique way of assigning degrees to the x_{σ} so that (S/R, G, f) is a graded R-algebra with grading extending the grading on S, such that the x_{σ} are all homogeneous. With this grading, (S/R, G, f) is a GCSA over R.

Proof. Since x_{σ}x_{τ} = f(σ, τ)x_{στ}, we need to assign degrees to the x_{σ} so that

deg(x_{σ}) + deg(x_{τ}) = deg(x_{στ}) + deg(f(σ, τ)),  (3.4)

for all σ, τ ∈ G. Once this is done, define, for any a ∈ S^{h}, deg(ax_{σ}) = deg(a) + deg(x_{σ}). Then formula (3.4) assures that the multiplication on (S/R, G, f) given in (3.3) is compatible with this assignment of degrees. We obtain a grading

(S/R, G, f) = ⨁_{δ∈Δ_{R}}(S/R, G, f)_{δ}, where (S/R, G, f)_{δ} = ⨁_{σ∈G}S_{δ−deg(x_{σ})}x_{σ},

where Δ_{R} = ℚ ⊗_{ℤ} Γ_{R}. This makes (S/R, G, f) into a graded R-algebra. By arguing as in the ungraded case (or invoking Cor. 1.2, since the crossed product algebra is an Azumaya algebra over R), we see that A is a GCSA over R.

To find degrees for the x_{σ} satisfying (3.4), note that the degree map deg: S^{∗} → Δ_{R} is a Gmodule homomorphism, with G acting trivially on Δ_{R}; there is an induced map deg^{∗}: Z^{2}(G, S^{∗}) → Z^{2}(G, Δ_{R}). But Z^{2}(G, Δ_{R}) = B^{2}(G, Δ_{R}), since H^{2}(G, Δ_{R}) = 0, as the ℚ-vector space Δ_{R} is uniquely divisible. So, since deg^{∗}(f) ∈ B^{2}(G, Δ_{R}), there exists {b_{σ} | σ ∈ G} ⊆ Δ_{R}, such that

deg(f(σ, τ)) = b_{σ} + b_{τ} − b_{στ} ,  (3.5)

for all σ, τ ∈ G. Then, define deg(x_{σ}) = b_{σ}, and (3.4) holds, as desired. Note also that if we have another set {b′_{σ} | σ ∈ G} ⊆ Δ_{R} satisfying deg(f(σ, τ)) = b′_{σ} + b′_{τ} − b′_{στ}, then the map σ ↦ (b_{σ} − b′_{σ}) is a group homomorphism from the finite group G to the torsion-free group Δ_{R}; therefore this homomorphism must be trivial (i.e., H^{1}(G, Δ_{R}) = 0). So, the b_{σ} satisfying (3.5) are uniquely determined; hence, there is only one way to define deg(x_{σ}) so that (3.4) holds. This gives the uniqueness asserted in the lemma. The values of deg(x_{σ}) are given explicitly by the formula

deg(x_{σ}) = \frac{1}{|G|} ∑_{τ∈G} deg(f(σ, τ)) ∈ Δ_{R}.

□

We call (S/R, G, f) with the grading of Lemma 3.1 a graded crossed product algebra.

Proposition 3.2. Let S be a Galois graded field extension of a graded field R (with [S : R] < ∞), and let G = G(S/R). Then,

GBr(S/R) ≅ H^{2}(G, S^{∗}).

Proof. Define a map ψ: Z^{2}(G, S^{∗}) → GBr(R) by f ↦ [(S/R, G, f)]_{g}, where the crossed product is given the grading of Lemma 3.1. We will show that ψ is a group homomorphism with ker(ψ) = B^{2}(G, S^{∗}) and im(ψ) = GBr(S/R). This will yield the desired isomorphism.

The following diagram is evidently commutative:

Z^{2}(G, S^{∗}) ⟶^{ψ} GBr(R) ⟶^{res_{S/R}} GBr(S)
↓  ↓  ↓
H^{2}(G, QS^{∗}) ⟶ Br(QR) ⟶ Br(QS)

(3.6)

In this diagram, the bottom row is exact, and the middle and right vertical maps are injective, by the comments after (3.2). This shows that ψ is a group homomorphism (since the other maps in the left square are homomorphisms), and also im(ψ) ⊆ GBr(S/R) and B^{2}(G, S^{∗}) ⊆ ker(ψ).

To show that this last inclusion is an equality, take any f ∈ ker(ψ). We may assume, after modifying f by a coboundary, that f is normalized. Hence, in (S/R, G, x_{σ}) = ⨁_{σ∈G} Sx_{σ}, x_{e} is the 1 (e = id_{S} = identity element of G) and the mapping S → (S/R, G, f) given by s ↦ sx_{e} is a graded R-algebra monomorphism, so we identify S with its gr-isomorphic copy Sx_{e} in the crossed product. Let n = [S : R] = |G|. Since f ∈ ker(ψ), by Prop. 1.3 there is a graded R-algebra isomorphism α: (S/R, G, f) → End_{R}(M) for some graded R-vector space M. By dimension count, dim_{R}(M) = n. The copy of S in (S/R, G, f) acts on M by s · m = α(s)(m). This action makes M into a graded S-vector space, necessarily of dimension 1. So M = S · m for any nonzero m ∈ M^{h}. Hence, as End_{R}(M) = End_{R}(S · m) ≅_{g} End_{R}(S), we may identify (S/R, G, f) with End_{R}(S) so that s ∈ S corresponds to left multiplication by s. For each σ ∈ G ⊆ End_{R}(S)_{0}, let s_{σ} = x_{σ}σ^{−1} ∈ End_{R}(S)^{h}. Then, as x_{σ}tx_{σ}^{−1} = σ(t) = σ ∘ t ∘ σ^{−1} ∈ End_{R}(S), for all t ∈ S, we have s_{σ} ∈ C_{End_{R}(S)}(S)^{h} = S^{h}. Since s_{σ} ≠ 0, s_{σ} ∈ S^{∗}. Furthermore, from x_{σ}x_{τ} = f(σ, τ)x_{στ}, we obtain s_{σ}σ(s_{τ})s_{στ}^{−1} = f(σ, τ), for all σ, τ ∈ G, proving that f ∈ B^{2}(G, S^{∗}), as desired.

Finally, to see that im(ψ) = GBr(S/R), take any GCSA A over R with [A]_{g} ∈ GBr(S/R). Let A′ = A ⊗_{R} End_{R}(S), which is also a GCSA over R. After identifying S with its gr-isomorphic copy in End_{R}(S), we have A ⊗_{R} S is a graded R-subalgebra of A′. Since S splits A, we have A ⊗_{R} S ≅_{g} M_{m}(S)(d) for some d = (δ_{1}, …, δ_{m}), where m = \sqrt{[A : R]}. Let B be a graded Rsubalgebra of A ⊗_{R} S such that B ≅_{g} M_{m}(R)(d); let C = C_{A′}(B). From B ⊆ A ⊗_{R} S ⊆ A′ we have, using (1.9), C ⊇ C_{A′}(A⊗_{R} S) = R ⊗_{R} C_{End_{R}(S)}(S) = R ⊗_{R} S ≅_{g} S. Thus, we may view S as a graded R-subalgebra of C. The double centralizer theorem, Prop. 1.5, shows that C is a GCSA over R with [C : R] = [A′ : R]/[B : R] = [End_{R}(S) : R] = [S : R]^{2}; hence, again by the double centralizer theorem, C_{C}(S) = S. Note also that in GBr(R), [A]_{g} = [A′]_{g} = [B]_{g} + [C]_{g} = [C]_{g}. Now, for each σ ∈ G, the graded Skolem-Noether theorem, Prop. 1.6, shows that the graded Ralgebra isomorphism σ: S → S is induced by conjugation by some x_{σ} ∈ C^{∗}. Moreover, we may assume x_{σ} ∈ C^{h} by Prop. 1.6(c), as C_{C}(S)_{0} = S_{0}, which is a field. Set f(σ, τ) = x_{σ}x_{τ} x_{στ}^{−1} ∈ C_{C}(S)^{h} ∩ C^{∗} ⊆ S^{h} − {0} = S^{∗}. Then f ∈ Z^{2}(G, S^{∗}). Also, the usual calculation shows that the sum ∑_{σ∈G} Sx_{σ} in C is a direct sum; hence it is all of C, by dimension count. The multiplication in ⨁_{σ∈G} Sx_{σ} is the same as that of (S/R, G, f), given in (3.3); so Lemma 3.1 shows (S/R, G, f) ≅_{g} C. Thus, ψ(f) = [C]_{g} = [A]_{g} in GBr(S/R), completing the proof. □

The description of GBr(S/R) given in Prop. 3.2 leads to a corresponding cohomological description of all of GBr(R). Recall from [HW, Prop. 3.7] that there is a maximal tame graded field extension Y of a graded field R. This Y is graded algebraic over R (though typically [Y : R] = ∞) and it contains a graded isomorphic copy of every tame graded field extension. We have that Y_{0} is the separable closure of R_{0} and Γ_{Y} /Γ_{R} is the prime-to-p primary component of the torsion group Δ_{R}/Γ_{R}, where p = char(R_{0}). Moreover, QY (= QR⊗_{R} Y = quotient field of Y) is Galois over QR, and G(QY/QR) maps bijectively (by restriction to Y) to the group G(Y/R) of all R-algebra automorphisms of Y, and every such automorphism preserves the grading on Y. Therefore, G(Y/R) inherits from G(QY/QR) the structure of a profinite group, in which the closed normal subgroups of finite index correspond one-to-one to the finite-degree Galois graded field extensions of R in Y (cf. [HW, Th. 3.9, Th. 3.11]).

Proposition 3.3. Let R be a graded field, and let Y be the maximal tame graded field extension of R. Then,

GBr(R) ≅ H^{2}(G(Y/R), Y^{∗}).

Proof. Let G = G(Y/R). Here H^{2}(G, Y^{∗}) denotes the continuous cohomology group with respect to the discrete G-module Y^{∗}. Because G = \underleftarrow{lim} G(S/R) as S ranges over the finite degree Galois graded field extensions S of R in Y, we have, in light of Prop. 3.2,

GBr(Y/R) = ⋃_{S} GBr(S/R) ≅ \underrightarrow{lim}_{S} H^{2}(G(S/R), S^{∗}) ≅ H^{2}(G, Y^{∗}).

Thus, it remains only to prove that GBr(R) = GBr(Y/R), which we do by showing that GBr(Y) = 0. For this, let E be any GCDA over Y, and form its graded Y -subalgebras Z, C, I, and T as in (2.3). Since Y_{0} is separably closed and Z_{0} is Galois over Y_{0} by Prop. 2.3, we have Z_{0} = Y_{0}, so Z = Y and C = E. Moreover, as Br(Y_{0}) = 0, we have E_{0} = Z_{0} = Y_{0}, so I = Y, hence E = T, which is totally ramified over Y. Since [E : Y] = [T : Y] = |Γ_{T} : Γ_{Y} | which is prime to char(Y_{0}) by Prop. 2.1, and Δ_{R}/Γ_{Y} is char(Y_{0})-primary, we must have E = Y. Hence, GBr(Y) = 0, as asserted. □

§4 Valuations from gradings, and vice versa

We now consider the valuation which arises when the grade group of a graded field R is given a total ordering. Since Γ_{R} is assumed torsion-free there always exists a total ordering on Γ_{R} compatible with its group operation. There are typically many such total orderings. Fix one such on Γ_{R}, and denote it ≤. Then, for any torsion free abelian group Λ containing Γ_{R} as a subgroup such that Λ/Γ_{R} is torsion, there is a unique extension of ≤ to a total ordering on Λ.

Let E be any GCDA over R. So, the fixed total ordering on Γ_{R} extends uniquely to a total ordering on Γ_{E}, again denoted ≤. A key observation in [B_{2}] (see also [HvO, Prop. 3.1] when Γ_{E} ≅ ℤ) is that the ordering on Γ_{E} induces a valuation on QE: One first defines v: E − {0} → Γ_{E}

by, for a = ∑_{γ∈Γ_{E}} a_{γ}

v(a) = min{γ ∈ Γ_{E} | a_{γ} ≠ 0}.

Clearly, for all a, b ∈ E − {0} we have (i) v(ab) = v(a) + v(b) as Γ_{E} is totally ordered, and (ii) v(a + b) ≥ min(v(a), v(b)) (if b ≠ −a). The function v has an extension to QE^{∗}, also denoted v: QE^{∗} → Γ_{E} given by v(ab^{−1}) = v(a) − v(b). (This is well-defined by property (i).) Then, properties (i) and (ii) hold for all a, b ∈ QE^{∗}. (This is very easy to verify, since every element a of QE is expressible as a = er^{−1} with e ∈ E and r ∈ R − {0}.) So, v is a valuation on QE. Clearly, for the value group of v on QE, denoted Γ_{QE}, we have

Γ_{QE} = Γ_{E}.  (4.1)

Also, for the residue division ring, denoted \overline{QE}, of the valuation ring of v on QE, we have

\overline{QE} ≅ E_{0}.  (4.2)

(For, if c ∈ QE^{∗} with v(c) = 0, then c = ab^{−1} with a = ∑_{γ∈Γ_{E}} a_{γ}, b = ∑_{γ∈Γ_{E}} b_{γ} ∈ E − {0} with

v(a) = v(b) = δ, say. Then c has the same image in \overline{QE} as a_{δ}b_{δ}^{−1} ∈ E_{0}.) The valuation on QE restricts to a valuation on its center QR, which clearly coincides with the valuation determined by the grading on R.

The properties of graded division algebras correspond most closely to those of valued vision over a Henselian field, as we will see. We can obtain such a division algebra from the GCDA E over R by Henselizing: Let HR denote “the” Henselization of the field QR with respect to the restriction of v to QR (cf. [E, p. 131, (17.11)]). So, HR is a separable algebraic field extension of QR which is uniquely determined up to isomorphism, and there is a Henselian valuation on HR extending v on QR, with residue field and value group satisfying

\overline{HR} ≅ \overline{QR} and Γ_{HR} = Γ_{QR}.  (4.3)

Define

HE = HR ⊗_{QR} QE = HR ⊗_{R} E.  (4.4)

Because QE has a valuation extending v on QR, Morandi’s Henselization theorem [M_{1}, Th. 2] shows that HE is a division ring (with center HR), and its unique valuation extending the Henselian valuation on HR restricts to v on QE. Furthermore, for the residue division algebra \overline{HE} and value group Γ_{HE} of the valuation on HE we have (using (4.2) and (4.1)),

\overline{HE} ≅ \overline{QE} ≅ E_{0} and Γ_{HE} = Γ_{QE} = Γ_{E}.  (4.5)

Moreover, by (1.7) together with (4.5) for E and for R, we have,

[HE : HR] = [QE : QR] = [E : R] = [\overline{HE} : \overline{HR}] |Γ_{HE} : Γ_{HR}|  (4.6)

so the valuation on HE is defectless over HR (likewise QE is defectless over QR). It follows easily from Prop. 2.3 that HE is also tame over HR (as described in Prop. 4.3 below). We will see in Th. 5.1 below that the map [E]_{g} ↦ [HE] is an isomorphism from GBr(R) to the tame part of the Brauer group of HR.

Remark 4.1. If A is a GCSA over the graded field R, then we have corresponding CSA’s QA = QR ⊗_{R} A over QR (so QA is the Artinian ring of quotients of the prime p.i. ring A) and HA = HR ⊗_{R} A over HR. We can define a valuation-like function on QA − {0}, but when A is not a graded division algebra, we generally obtain a ring in QA that could not be reasonably called a valuation ring. Specifically, define w: QA − {0} → Γ_{A} by first defining w(a) = min{γ | a_{γ} ≠ 0} for a = ∑ a_{γ} ∈ A−{0}, and then for c = ar^{−1} ∈ QA−{0} with a ∈ A−{0} and r ∈ R −{0}, defining w(c) = w(a) − w(r). This is a well-defined function satisfying w(cd) ≥ w(c) + w(d) (if cd ≠ 0) and w(c + d) ≥ min(w(c), w(d)) (if d ≠ −c) for all c, d ∈ QA − {0}. This function yields a subring V_{QA} of QA given by V_{QA} = {a ∈ QA − {0} | w(a) ≥ 0} ∪ {0}, but V_{QA} need not be a valuation ring, not even in the sense of Dubrovin (see after Th. 5.3 below). For example, suppose F is a field, t an indeterminate over F, and R = F [t, t^{−1}] = ⨁_{i∈ℤ} R_{i} where R_{i} = t^{i}F. So, R is a graded field with R_{0} = F and Γ_{R} = ℤ with its usual ordering. We have QR = F (t), and the valuation on QR induced by the grading on R is the t-adic valuation ring V_{QR} = F [t]_{(t)} = {fg^{−1} | f, g ∈ F [t], g(0) ≠ 0}. Let M_{QR} = tV_{QR}, the maximal ideal of V_{QR}, so the residue field is \overline{QR} = V_{QR}/M_{QR} ≅ F. Now, let A = M_{2}(R)(d) where d = (0, \frac{1}{2}). Then the ring V_{QA} obtained from w on QA − \{0\} is

V_{QA} = ( \begin{matrix}V_{QR} & M_{QR} \\ V_{QR} & V_{QR} \end{matrix} ), with Jacobson radical J(V_{QA}) = ( \begin{matrix}M_{QR} & M_{QR} \\ V_{QR} & M_{QR} \end{matrix} ). (J(V_{QA}) is also the

ideal of elements of positive w “value” in V_{QA}.) So, V_{QA}/J(V_{QA}) ≅ \overline{QR}⊕\overline{QR}, which is semisimple, but not simple. So, V_{QA} is not a Dubrovin valuation ring, and is also not a maximal order in QA. In fact, every Dubrovin valuation ring of QA contracting to V_{QR} in QR is isomorphic to M_{2}(V_{QR}).

We now turn to valued division algebras, and the graded division algebras derived from them. This will lead to consideration of the tame part of the Brauer group of a Henselian valued field F, whose central division algebras all have associated graded division algebras with center the graded field GF. For the rest of this section we will take the first steps toward proving an isomorphism between the tame part of the Brauer group of the Henselian field F and the graded Brauer group of GF; the proof of this will finally be completed in §5 (see Th. 5.3).

Let D be a division ring, and suppose there is a valuation v : D^{∗} → Γ on D. That is, Γ is a totally ordered abelian group, and v satisfies, for all a, b ∈ D^{∗},

(i) v(ab) = v(a) + v(b); (ii) v(a + b) ≥ min(v(a), v(b)) if b ≠ −a.

Let V_{D} = {a ∈ D^{∗} | v(a) ≥ 0} ∪ {0}, the valuation ring of v; M_{D} = {a ∈ D^{∗} | v(a) > 0} ∪ {0} the unique maximal left ideal (and maximal right ideal) of V_{D}; U_{D} = V_{D}^{∗} = V_{D} − M_{D}; \overline{D} = V_{D}/M_{D}, the residue division ring of v on D; and Γ_{D} = im(v), the value group. There will be no ambiguity in indexing these objects by D, since we will never consider more than one valuation on a given division ring. Let \overline{p} = char(\overline{D}).

The filtration of fractional ideals of V_{D} determined by v on D yields an associated graded ring GD. Specifically, for γ ∈ Γ_{D}, let

W^{γ} = {d ∈ D^{∗} | v(d) ≥ γ} ∪ {0} and W^{>γ} = {d ∈ D^{∗} | v(d) > γ} ∪ {0};

so W^{>γ} is a subgroup of the additive group W^{γ}. Then set

GD = ⨁_{γ∈Γ_{D}}GD_{γ}, where GD_{γ} = W^{γ}/W^{>γ}.

Because W^{>γ}W^{δ} + W^{γ}W^{>δ} ⊆ W^{>γ+δ}, for all γ, δ ∈ Γ_{D}, the multiplication on D induces a welldefined multiplication on GD, making it into a graded ring. Moreover, property (i) of the valuation assures that GD is a graded division ring. Clearly GD_{0} = \overline{D} and the grade group is Γ_{GD} = Γ_{D}. It is the basic theme of this paper that much of the structure of D is well reflected in GD.

Now, let F = Z(D), and suppose [D : F] < ∞, so D is a central division algebra (CDA) over F. The restriction v|_{F} of the valuation on D is a valuation on F (with associated structures V_{F} , M_{F} , U_{F} , \overline{F} , Γ_{F}), which induces a corresponding graded field GF. Clearly, GD is a graded GF -algebra, with

[GD : GF] = [\overline{D} : \overline{F}]|Γ_{D} : Γ_{F} | = [D : F]/δ_{D/F} ,  (4.7)

by (1.7) above, where δ_{D/F} is the defect of D over F with respect to the valuation. By Morandi’s Ostrowski theorem for valued division algebras, [M_{1}, Th. 3], δ_{D/F} is a nonnegative power of \overline{p} if \overline{p} > 0, while δ_{D/F} = 1 if \overline{p} = 0. Let Z = Z(GD), a graded GF -subalgebra of GD. Even though F = Z(D), this Z may be strictly larger than GF; the following result of Boulagouaz shows when this occurs. Recall (see [JW, (1.6)]) that there is a canonical homomorphism

θ_{D}: Γ_{D}/Γ_{F} → G(Z(\overline{D})/\overline{F})  (4.8)

given by, for any d ∈ D^{∗} and any z ∈ V_{D} with \overline{z} ∈ Z(\overline{D}), θ(v(d) + Γ_{F})(\overline{z}) = \overline{dzd^{−1}}. Note that the following diagram is evidently commutative with horizontal maps the obvious isomorphisms,

\begin{matrix}Γ_{D}/Γ_{F} & ⟶ & Γ_{GD}/Γ_{GF} \\ ^{θ_{D}} ↓ & & ^{θ_{GD,GF}} ↓ \\ G(Z(\overline{D})/\overline{F}) & ⟶ & G(Z(GD_{0})/GF_{0}) \end{matrix}  (4.9)

where θ_{GD,GF} is the map of (2.4).

Proposition 4.2. With D and F as above, for Z = Z(GD), (a) Z_{0} is the purely inseparable closure of \overline{F} in Z(\overline{D});

(b) Γ_{Z}/Γ_{F} is the \overline{p}-primary part of ker(θ_{D}).

Proof. See [B_{2}, Prop. 3.1, Th. 3.4]. □

Now, let F be a field with Henselian valuation. That is, v has a unique extension to each field L ⊇ F with L algebraic over F. Likewise, as is well known (see [S, Th. 9, p. 53] or [W_{1}, Th.], v has a unique extension to each CDA D over F (given by v(d) = \frac{1}{\sqrt{[D:F]}}v(Nrd(d)) ∈ ℚ ⊗_{ℤ} Γ_{F}, for all d ∈ D^{∗}). We will focus on tame division algebras D (defined below), which all have the property that GD is a GCDA over the graded field GF. But first we recall some terminology connected with Henselian valued fields.

For our Henselian field F, let F_{sep} denote the separable closure of F (in some fixed algebraic closure F_{alg} of F). Let F_{nr} be the inertia field of F_{sep} over F, with respect to the unique extension of the Henselian valuation v to F_{sep}. Then, (see [E, (19.12), (19.8)(b)]) Γ_{F_{nr}} = Γ_{F}, \overline{F_{nr}} ≅ (\overline{F})_{sep}, and F_{nr} is Galois over F with G(F_{nr}/F) ≅ G((\overline{F})_{sep}/\overline{F}). Moreover, for any field L with F ⊆ L ⊆ F_{sep} and [L : F] < ∞, we have (by [E, (19.14)] and an application of Hensel’s Lemma) L ⊆ F_{nr} iff L is unramified over F (i.e., [\overline{L} : \overline{F}] = [L : F] and \overline{L} is separable over \overline{F}). Note that F_{nr} is the compositum of all finite degree unramified field extensions of F in F_{sep}; F_{nr} is called the maximal unramified extension of F. Let F_{t} be the ramification field of F_{sep} over F. Then (see [E, (20.17)]) \overline{F_{t}} ≅ (\overline{F})_{sep} and Γ_{F_{t}}/Γ_{F} is the prime-to-\overline{p} primary part of Δ_{F} /Γ_{F}, where Δ_{F} = ℚ ⊗_{ℤ} Γ_{F}. That is, if \overline{p} = 0 then Γ_{F_{t}} = Δ_{F}; if \overline{p} ≠ 0, then Γ_{F_{t}}/Γ_{F} is \overline{p}-torsion-free and Δ_{F} /Γ_{F_{t}} is \overline{p}-primary torsion. Also, F_{t} is Galois over F, by [E, (21.2)]. Further, for any field L with F ⊆ L ⊆ F_{sep} and [L : F] < ∞, we have L ⊆ F_{t} iff L is tame (= tamely ramified and defectless) over F (i.e., [L : F] = [\overline{L} : \overline{F}]|Γ_{L} : Γ_{F} |, \overline{L} is separable over \overline{F}, and \overline{p} ∤ |Γ_{L} : Γ_{F} |). Here, “only if” follows by [E, (20.20), (19.10)(b), (19.14)], and “if” from [E, (20.18)], since when L is tame over F, L · F_{nr} is tame and totally ramified over F_{nr}. Note that F_{t} is the compositum of all the finite-degree tame extension fields of F in F_{sep}; F_{t} is called the maximal tame extension of F. We have F ⊆ F_{nr} ⊆ F_{t} ⊆ F_{sep}, and if \overline{p} = 0, then F_{t} = F_{sep}.

The Henselian valuation on F yields certain distinguished subgroups of its Brauer group Br(F), discussed in [JW] and denoted as follows:

IBr(F) = \{[D] | D is a CDA over F with [\overline{D} : \overline{F}] = [D : F]

and Z(\overline{D}) = \overline{F} \} ≅ Br(\overline{F}), the inertial part of Br(F);

SBr(F) = \{[D] | D is a CDA over F, [\overline{D} : \overline{F}] | Γ_{D} : Γ_{F} | = [D : F],

Z(\overline{D}) is separable over \overline{F} , and θ_{D} is injective\}

= Br(F_{nr}/F), the inertially split part of Br(F);
TBr(F) = Br(F_{t}/F), the tame part of Br(F) (further described

in Prop. 4.3 below).

So, IBr(F) ⊆ SBr(F) ⊆ TBr(F) ⊆ Br(F) and if \overline{p} = 0, then TBr(F) = Br(F). Our focus will be on the tame CDA’s D over F (i.e., those for which [D] ∈ TBr(F) ), since they are the ones for which the associated graded division rings GD carry the most complete information about D (see, e.g., Th. 5.9 below). Tame CDA’s over F were defined in a different way in [JW], but the next proposition shows that the definitions are equivalent.

Proposition 4.3. Let D be a CDA over a Henselian field F, with char(\overline{F}) = \overline{p}. Then, the following properties are equivalent:

(i) D is tame (i.e., [D] ∈ TBr(F)); (ii) [D\overline{p}] ∈ SBr(F), where D\overline{p} is the \overline{p}-primary component of D; (iii) [\overline{D} : \overline{F}]|Γ_{D} : Γ_{F} | = [D : F], Z(\overline{D}) is separable over \overline{F}, and \overline{p} ∤ ker(θ_{D}); (iv) D has a maximal subfield which is tame over F;

(v) [GD : GF] = [D : F] and Z(GD) = GF.

Proof. (i) ⇒ (ii) If [D] ∈ TBr(F), then [D\overline{p}] ∈ TBr(F), since [D\overline{p}] = [D^{⊗n}] for some n. (If \overline{p} = 0, it is understood that D\overline{p} = F.) Let B = D\overline{p} ⊗_{F} F_{nr}. Since F_{t} splits B, there is a splitting field L of B with F_{nr} ⊆ L ⊆ F_{t} and [L : F_{nr}] < ∞. Since L is tame over F_{nr} and F_{nr} is separably closed, [L : F_{nr}] = |Γ_{L} : Γ_{F_{nr}}|, which is prime to \overline{p}. Because gcd([B : F_{nr}], [L : F_{nr}]) = 1 and L splits B, B must already be split, proving (ii).

(ii) ⇔ (iii) was proved in [JW, Lemma 6.1].

(iii) ⇔ (v) Let Z = Z(GD), a graded GF -subalgebra of GD. Then, Z = GF iff Z_{0} = \overline{F} and Γ_{Z} = Γ_{F}. Therefore, (iii) ⇔ (v) follows immediately from (4.7) and Prop. 4.2, since Z(\overline{D}) is normal over \overline{F}, by [JW, Prop. 1.7].

(ii) ⇒ (iv) Assume [D\overline{p}] ∈ SBr(F). Then, D\overline{p} has a maximal subfield K with K unramified over F, by [JW, Lemma 5.1]. Let M be any maximal subfield of D′, the prime-to-\overline{p} component of D. Since \overline{p} ∤ [M : F], M is tame over F, by “Ostrowski’s theorem,” which says that [M : F]/([\overline{M} : \overline{F}]|Γ_{M} : Γ_{F} |) equals a nonnegative power of \overline{p} if \overline{p} > 0, and equals 1 if \overline{p} = 0, (by [E, (20.21)] applied to N over M and N over F, where N is the normal closure of M over F). Since K ⊆ F_{t} and M ⊆ F_{t}, the compositum K · M is also tame over F. Clearly, K · M splits D ≅ D\overline{p} ⊗_{F} D′ and [K · M : F] ≤ [K : F][M : F] = deg(D), so K · M is a maximal subfield of D.

(iv) ⇒ (i) is clear, since any maximal subfield of D is a splitting field. □

Now, let F be a Henselian field, with its associated graded field GF, and let QGF be the quotient field of GF. The total ordering on Γ_{F} gives us a total ordering on Γ_{GF} = Γ_{F}, which induces a valuation on QGF, as described at the beginning of this section. Let HGF be the Henselization of QGF with respect to this valuation. Then, HGF need not be isomorphic to F (they need not even have the same characteristic). But we have shown in [HW, Prop. 5.1, Th. 5.2] that for the maximal tame extensions HG(F_{t}) ≅ (HGF)_{t}, and the canonical map of Galois groups G(F_{t}/F) → G((HGF)_{t}/HGF) is an isomorphism. We also have homomorphisms of multiplicative groups F_{t}^{∗} → G(F_{t})^{∗} and G(F_{t})^{∗} → HG(F_{t})^{∗} → (HGF)_{t}^{∗}; these maps compose to give a group homomorphism F_{t}^{∗} → (HGF)_{t}^{∗} which is clearly compatible with the respective Galois group actions. Therefore, there is a homomorphism of continuous cohomology groups H^{i}(G(F_{t}/F), F_{t}^{∗}) → H^{i}(G((HGF)_{t}/HGF), (HGF)_{t}^{∗}). In particular, for i = 2, we obtain a group homomorphism

γ: TBr(F) → TBr(HGF).

Theorem 4.4. Let F be a Henselian valued field. Then, the map γ: TBr(F) → TBr(HGF) just defined is an isomorphism.

Proof. We do this by stages. For the maximal unramified extension F_{nr} of F, it is easy to check that HG(F_{nr}) ≅ (HGF)_{nr} canonically, and that γ maps SBr(F) into SBr(HGF). We first show that this map SBr(F) → SBr(HGF) is an isomorphism. We have SBr(F) ≅ H^{2}(\overline{G}, F_{nr}^{∗}), where \overline{G} = G(\overline{F}_{sep}/\overline{F}) ≅ G(F_{nr}/F) ≅ G((HGF)_{nr}/HGF). Let Γ = Γ_{F_{nr}} = Γ_{F} = Γ_{HGF}. We have a commutative diagram:

\begin{aligned}0 & ⟶ & H^{2}(\overline{G}, U_{F_{nr}}) & ⟶ & H^{2}(\overline{G}, F_{nr}^{∗}) & ⟶ & H^{2}(\overline{G}, Γ) & ⟶ & 0 \\ & & ↓ & & ↓ & & ↓ \\ 0 & ⟶ & H^{2}(\overline{G}, U_{(HGF)_{nr}}) & ⟶ & H^{2}(\overline{G}, (HGF)_{nr}^{∗}) & ⟶ & H^{2}(\overline{G}, Γ) & ⟶ & 0 \end{aligned}  (4.10)

The first row of (4.10) arises from the short exact sequence of \overline{G}-modules

1 → U_{F_{nr}} → F_{nr}^{∗} → Γ → 1

induced by the valuation on F_{nr}, and the second row arises likewise from (HGF)_{nr}. It was shown in [JW, (5.4)] that the rows of (4.10) are exact, since F and HGF are Henselian. It was also shown that

IBr(F) ≅ H^{2}(\overline{G}, U_{F_{nr}}) ≅ H^{2}(\overline{G}, \overline{F_{nr}}^{∗}) ≅ Br(\overline{F}),

where the middle isomorphism arises from the projection U_{F_{nr}} → \overline{F_{nr}}^{∗}. It follows from this and the isomorphism \overline{F_{nr}} ≅ \overline{(HGF)_{nr}} that the left vertical map in (4.10) is an isomorphism. Since the right vertical map is the identity, the 5-lemma shows that the middle vertical map in (4.10) is also an isomorphism. Thus,

SBr(F) ≅ H^{2}(\overline{G}, F_{nr}^{∗}) ≅ H^{2}(\overline{G}, (HGF)_{nr}^{∗}) ≅ SBr(HGF),

and the composition of these isomorphisms coincides with γ on SBr(F).

Now, we have a commutative diagram with exact rows

0 ⟶ SBr(F) ⟶ TBr(F) ⟶ TBr(F_{nr})
↓  ^{γ}↓  ↓
0 ⟶ SBr(HGF) ⟶ TBr(HGF) ⟶ TBr((HGF)_{nr})

(4.11)

We just showed that the left vertical map in (4.11) is an isomorphism. To analyze the right vertical map, first note that TBr(F_{nr}) has no \overline{p}-torsion, since [F_{t} : F_{nr}] is prime to \overline{p} as a supernatural number. Take any positive integer n such that n is prime to \overline{p} (if \overline{p} = 0, this means any positive integer at all). Since the n-th power map F_{t}^{∗} → F_{t}^{∗} is surjective, we have for the n-torsion in the tame Brauer group, _{n}TBr(F_{nr}) ≅ H^{2}(G′, μ_{n}(F_{t})), where G′ = G(F_{t}/F_{nr}) ≅ G((HGF)_{t}/(HGF)_{nr}). (Note that |μ_{n}(F_{t})| = n since F_{t} is Henselian and |μ_{n}(\overline{F_{t}})| = n, which holds as \overline{F_{t}} is separably closed.) Because μ_{n}(F_{t}) maps isomorphically onto μ_{n}((HGF)_{t}), the map _{n}TBr(F_{nr}) → _{n}TBr((HGF_{nr})) is an isomorphism. Since this is true for all n prime to \overline{p}, the right vertical map in (4.11) is an isomorphism.

The scalar extension map TBr(F) → TBr(F_{nr}) is in general not onto. In fact, we claim that the image of TBr(F) → TBr(F_{nr}) equals ⋃_{m} _{m}TBr(F_{nr}) as m ranges over the positive integers prime to \overline{p} such that |μ_{m}(\overline{F})| = m. For, by [TW, Th. 4.3, Prop. 4.2] every class in TBr(F_{nr}) is represented by a division algebra D which is tame and totally ramified over F, and D is determined up to isomorphism by its associated nondegenerate symplectic pairing β_{D}: Γ_{D}/Γ_{F_{nr}} ×Γ_{D}/Γ_{F_{nr}} → μ(\overline{F_{nr}}) (given by (δ + Γ_{F_{nr}}, ε + Γ_{F_{nr}}) ↦ \overline{(ded^{−1}e^{−1})} for any d, e ∈ D^{∗} with v(d) = δ and v(e) = ε). Moreover, if ℓ = exp(Γ_{D}/Γ_{F}) then ℓ is prime to \overline{p}, im(β_{D}) = μ_{ℓ}(\overline{F_{nr}}), and [D] has exponent ℓ in TBr(F_{nr}). If |μ_{ℓ}(F)| = ℓ (which occurs iff |μ_{ℓ}(\overline{F})| = ℓ as F is Henselian), then one can easily construct a tame totally ramified division algebra over F as a tensor product of symbol algebras which will have the same value group and pairing as D; this algebra will map to [D] in TBr(F_{nr}). Suppose, on the other hand, that |μ_{ℓ}(\overline{F})| < ℓ. By [JW, Lemma 6.2], any tame central division algebra E over F is representable as E ∼ S ⊗_{F} T in TBr(F) where S is inertially split and T is a tame and totally ramified division algebra over F. Then in TBr(F_{nr}), E ⊗_{F} F_{nr} ∼ T ⊗_{F} F_{nr}, and T ⊗_{F} F_{nr} has the same value group and pairing as T, so the image of the pairing must lie in μ(\overline{F}). So, T ⊗_{F} F_{nr} ≅ D̸, since their canonical pairings have different images. Thus, when |μ_{ℓ}(F)| < ℓ, then [D] cannot lie in the image of TBr(F), proving the claim. Since \overline{F} ≅ \overline{HGF}, so they have the same roots of unity, the claim shows the right vertical map of (4.11) restricts to an isomorphism of the images of the maps TBr(F) → TBr(F_{nr}) and TBr(HGF) → TBr((HGF)_{nr}). Thus, the 5-lemma can be applied to diagram (4.11) to see that γ is an isomorphism. □

Corollary 4.5. Suppose F is a Henselian valued field, and K is a finite degree tame Galois extension field of F. Take any 2-cocycle f ∈ Z^{2}(G(K/F), K^{∗}) such that f(σ, τ) ∈ 1 + M_{K} for all σ, τ ∈ G(K/F). Then the crossed product algebra (K/F, G(K/F), f) is split.

Proof. The group homomorphism F_{t}^{∗} → G(F_{t})^{∗} → (HGF)_{t}^{∗} has kernel 1 + M_{F_{t}}. Hence, γ[(K/F, G(K/F), f)] = 1 in TBr(HGF). So, Th. 4.4 shows that (K/F, G(K/F), f) must be split. □

§5 Isomorphisms between GBr and TBr

Let R be a graded field with Γ_{R} totally ordered. We have seen that for any GCSA A over R there is a CSA HA = HR ⊗_{R} A over the Henselian field HR. The map [A]_{g} ↦ [HA] gives a well-defined group homomorphism GBr(R) → Br(HR), since it is the composition of the forgetful homomorphism GBr(R) → Br(R) with the scalar extension map Br(R) → Br(HR). Also, A ≅_{g} M_{n}(E)(d), where E is a GCDA over R with [A]_{g} = [E]_{g}, and by (4.6), Prop. 2.3, and Prop. 4.3, HE is a tame CDA over HR with [HE : HR] = [E : R]. Thus, our map to Br(HR) actually lands in TBr(HR), and we have an index-preserving group homomorphism β: GBr(R) → TBr(HR). Now, the Henselian field HR has an associated graded field GHR, and by Prop. 4.3 there is a map δ: TBr(HR) → GBr(GHR) taking [D] ↦ [GD]_{g} for any tame CDA D over HR. Also, let Y be the maximal tame graded field extension of R, so we have isomorphisms (HR)_{t} ≅ HY and GHY ≅ G((HR)_{t}) ≅ (GHR)_{t}, by [HW, Prop. 5.1]. Let G = G(Y/R), which we identify with G((HR)_{t}/HR) and G(GHY/GHR) in view of [HW, Prop. 5.1]. The continuous G-module homomorphisms Y^{∗} → (HR)_{t}^{∗} and (HR)_{t}^{∗} → (GHY)^{∗} lead to homomorphisms β′: H^{2}(G, Y^{∗}) → H^{2}(G, (HR)_{t}^{∗}) and δ′: H^{2}(G, (HR)_{t}^{∗}) → H^{2}(G, GHY^{∗}). These maps fit together into a diagram

GBr(R) ⟶^{β} TBr(HR) ⟶^{δ} GBr(GHR)
↑  ↑  ↑
H^{2}(G, Y^{∗}) ⟶^{β′} H^{2}(G, (HR)_{t}^{∗}) ⟶^{δ′} H^{2}(G, GHY^{∗})

(5.1)

Theorem 5.1. For any graded field R with Γ_{R} totally ordered, diagram (5.1) is commutative, and all the maps in it are group isomorphisms. The maps β and δ are index-preserving. Also, δ ∘ β and δ′ ∘ β′ coincide with the isomorphisms arising from the canonical isomorphism GHR ≅_{g} R.

Proof. The vertical maps in (5.1) are the isomorphisms of Prop. 3.3 for the graded fields R, GHR, and the standard Brauer group isomorphism for the valued field HR. It is clear from the definitions that γ ∘ β and γ′ ∘ β′ are the isomorphisms arising from the canonical map R ≅_{g} GHR. Hence, the outer rectangle in (5.1) is commutative, and β and β′ are injective. Also, it is clear from the definitions that the left square in (5.1) is commutative, and that β is a group homomorphism (since it is essentially a scalar extension map). We will show below that β′ is onto. Assume this for now. Then from the commutative left square, β is onto, hence an isomorphism. So, δ = (δ ∘ β) ∘ β^{−1} is a group isomorphism. (Note that it is not at all apparent from the definition or from direct calculations that δ is even a group homomorphism.) Likewise, δ′ = (δ′ ∘ β′) ∘ β^{′−1} is an isomorphism. The commutativity of the right square in (5.1) follows from the commutativity of the outer rectangle and the left square. We noted above that β is index-preserving. Because δ ∘ β is also index-preserving, so must be δ.

It remains to verify the surjectivity of β′. For this, consider the maps

H^{2}(G, Y^{∗}) ⟶^{β′} H^{2}(G, (HR)_{t}^{∗}) ⟶^{δ′} H^{2}(G, GHY^{∗}) ⟶^{β′′} H^{2}(G, (HGHR)_{t}^{∗}),

where β′, δ′ are as in (5.1) and β′′ corresponds to β′ when we start with GHR instead of R as ground graded field. The canonical gr-isomorphism GHR ≅_{g} R induces an isomorphism HGHR ≅ HR of Henselian valued fields, and the map β′′ ∘ δ′ is the corresponding isomorphism of cohomology groups. (It is also the isomorphism given by Th. 4.4.) Since β′′ ∘ δ′ is onto, so is β′′. But with respect to the gr-isomorphism GHR ≅_{g} R, β′′ corresponds to β′, so β′ must also be onto. □

Remark 5.2. There is a variation of Th. 5.1 which does not involve Henselization. For this, let R be a graded field (with Γ_{R} torsion-free) and let Y be the maximal tame graded field extension of R, as in [HW, Prop. 3.7]. Give some total ordering to Γ_{R}, use this to define a valuation on the quotient field QR of R, and let GQR be the associated graded field for the filtration on QR arising from the valuation. Let QY = QR ⊗_{R} Y, which is the quotient field of Y. Then, there are index-preserving group isomorphisms β̃ and δ̃ in a diagram

GBr(R) ⟶^{β̃} Br(QY/QR) ⟶^{δ̃} GBr(GQR)  (5.2)

such that δ̃ ∘ β̃ coincides with the isomorphism arising from the canonical graded isomorphism GQR ≅_{g} R. Also, there is commutative diagram like (5.1) (where G = G(QY/QR) ) in which all the maps are isomorphisms, and the middle column is H^{2}(G, QY) → Br(QY/QR). Here, β̃ is the map [E]_{g} ↦ [E ⊗_{R} QR], and δ̃ will be described below.

The properties asserted for the maps in (5.2) can be seen as follows: Let HR be the Henselization of QR with respect to our valuation on QR, and consider the diagram

GBr(R) ⟶^{β} TBr(HR) ⟶^{δ} GBr(GHR)
∥  ↑^{ε}  ↑^{c}
GBr(R) ⟶^{β̃} Br(QY/QR) ⟶^{δ̃} GBr(GQR),

(5.3)

where β, δ are the maps of (5.1), ε is the scalar extension map, and c is the isomorphism arising from the canonical gr-isomorphisms GQR ≅_{g} R ≅_{g} GHR. The left square of (5.3) is clearly commutative. Hence, β̃ is injective and index-preserving, since this is true for β by Th. 5.1. Moreover, β̃ is onto, since the corresponding homological map is onto, by arguing just as in the last paragraph of the proof of Th. 5.1. Hence, ε is an index-preserving isomorphism (and indeed maps into TBr(HR), not just into Br(HR) ). Take any CDA D over QR with D split by QY. Since β̃ is onto and index-preserving, there is a GCDA E over R with D ≅ E ⊗_{R} QR = QE so the valuation on QE induced by the grading on E yields a valuation on D extending the one on QR; this valuation on D is uniquely determined, by [W_{1}, Th.], and it is a tame valuation (in the sense of Prop. 4.3(iii)) by Prop. 2.3. The map δ̃: Br(QR/QR) → GBr(GQR) can now be defined by [D] ↦ [GD]_{g}, where GD is the associated graded division ring arising from the valuation on D. It is clear that the right square in (5.3) is commutative. Hence, δ̃ is an index-preserving isomorphism, since this is true for ε, δ, and c.

We next prove our main result, which is a theorem like 5.1, but starting with a Henselian field instead of with a graded field. Fix a field F with Henselian valuation v, and let GF be the associated graded field. As in Th. 4.4, let HGF be the Henselization of QGF with respect to the valuation induced by the grading on GF using the total ordering on Γ_{GF} corresponding to the ordering on Γ_{F}. There is a map α: TBr(F) → GBr(GF) mapping [D] ↦ [GD]_{g}, for any tame CDA D over F. There is also a group homomorphism β: GBr(GF) → TBr(HGF) given by [A]_{g} ↦ [A ⊗_{GF} HGF]. Let G = G(F_{t}/F) ≅ G(G(F_{t})/GF) ≅ G((HGF)_{t}/HGF) (see before Th. 4.4 and [HW, Prop. 5.1]). We have homomorphisms α′: H^{2}(G, F_{t}^{∗}) → H^{2}(G, G(F_{t})^{∗}) and β′: H^{2}(G, G(F_{t})^{∗}) → H^{2}(G, (HGF)_{t}) induced by the G-module homomorphisms F_{t}^{∗} → G(F_{t})^{∗} and G(F_{t})^{∗} → (HGF)_{t}^{∗}. These fit into a diagram:

TBr(F) ⟶^{α} GBr(GF) ⟶^{β} TBr(HGF)
↑  ↑  ↑
H^{2}(G, F_{t}^{∗}) ⟶^{α′} H^{2}(G, G(F_{t})^{∗}) ⟶^{β′} H^{2}(G, (HGF)_{t}^{∗})

(5.4)

Theorem 5.3. For any Henselian valued field F, diagram (5.4) is commutative, and all its maps are group isomorphisms. Furthermore, α and β are index-preserving.

Before proving Th. 5.3, we recall some facts about Dubrovin valuation rings. If A is a CSA over a field L, then a subring B of A is called a Dubrovin valuation ring of A if B has an ideal J such that B/J is simple Artinian, and for each a ∈ A − B there exist b_{1}, b_{2} ∈ B such that b_{1}a, ab_{2} ∈ B − J, cf. [D_{1}], [D_{2}], [W_{2}]. It is known that J is the Jacobson radical of B, that B is a prime ring which is a left and right order in A, that J ∩ L is a valuation ring of L, and that the two-sided ideals of B are linearly ordered by inclusion. Let st(B) = {a ∈ A^{∗} | aBa^{−1} = B}, and let Γ_{B} = st(B)/B^{∗}, the “value group” of B. For each δ = sB^{∗} ∈ Γ_{B} there is an associated fractional ideal (= B-B sub-bimodule of A), I^{δ} = Bs = sB. The ideals I^{δ} are all the fractional ideals of B which are cyclic as left and as right B-modules. One has I^{δ} · I^{δ′} = I^{δ′} · I^{δ} = I^{δ+δ′}. The abelian group Γ_{B} is given a total ordering by δ ≤ δ′ iff I^{δ} ⊇ I^{δ′}. If we set I^{>δ} = ⋃_{δ′>δ}I^{δ′} then we

may form the associated graded ring of B with respect to the filtration by fractional ideals I^{δ},

GB = ⨁_{δ∈Γ_{B}}GB_{δ}, where GB_{δ} = I^{δ}/I^{>δ}.

If V is a given valuation ring of L, Dubrovin’s existence theorem ([D_{2}, §3, Th. 2], see [BG, Th. 3.8] for another proof) says that there is a Dubrovin valuation ring B of A with B ∩ L = V. Moreover, the conjugacy theorem ([W_{2}, Th. A], with a more direct proof given in [G, Th. 3.3]) says that if B′ is another Dubrovin valuation ring of A with B′ ∩ L = V, then B′ = aBa^{−1}, for some a ∈ A^{∗}; so clearly GB′ ≅_{g} GB. Since we are interested in ground fields L with a fixed valuation v on L, we will write GA for GB, for any Dubrovin valuation ring B of A with B ∩ L = V_{L}. So, GA is a graded GL-algebra, which is well-defined up to gr-isomorphism, with Γ_{GA} = Γ_{B} and GA_{0} = B/J. The notation GA is consistent with our previous usage. For, if D is a valued CDA over L with V_{D} ∩ L = V_{L}, then V_{D} is the unique Dubrovin valuation ring of D contracting to V_{L} in L, and then GD as just defined (i.e., G(V_{D})) is exactly the graded division ring GD defined in §4 above.

Also, for any natural number n, if B is a Dubrovin valuation ring of A, then M_{n}(B) is a Dubrovin valuation ring of M_{n}(A) with M_{n}(B) ∩ L = B ∩ L and Γ_{M_{n}(B)} ≅ Γ_{B} canonically (cf. [W_{2}, Cor. 3.5]). Consequently, G(M_{n}(A)) ≅_{g} M_{n}(GA).

One convenient way of building Dubrovin valuation rings is by using Morandi’s value functions introduced in [M_{2}]: Given a CSA A over L, suppose Γ is a totally ordered abelian group and w: A − {0} → Γ is a function such that for all a, b ∈ A − {0},

(i) w(a + b) ≥ min(w(a), w(b)), if b ≠ −a; (ii) w(ab) ≥ w(a) + w(b) ; (iii) B_{w}/J_{w} is a simple Artinian ring, where B_{w} = {a ∈ A − {0} | w(a) ≥ 0} ∪ {0} (a ring),

and J_{w} = {a ∈ A − {0} | w(a) > 0} ∪ {0} (an ideal of B_{w}); (iv) im(w) = w(st(w)), where st(w) = {a ∈ A^{∗} | w(a^{−1}) = −w(a)}.

Then, by [M_{2}, Th. 2.4, Cor. 2.5], B_{w} is a Dubrovin valuation ring with B_{w} integral over B_{w} ∩ L. Further, st(B_{w}) = st(w) and Γ_{B_{w}} ≅ im(w). Also, for δ ∈ Γ_{B_{w}}, I^{δ} = {a ∈ A−{0} | w(a) ≥ δ}∪{0} and I^{>δ} = {a ∈ A − {0} | w(a) > δ} ∪ {0}.

Lemma 5.4. Let F be a Henselian field, let K be a tame finite degree Galois extension field of F, and set H = G(K/F). Let f ∈ Z^{2}(H, K^{∗}) be a normalized 2-cocycle, and let A = (K/F, H, f) = ⨁_{σ∈H} Kx_{σ}. There is a unique function w: {x_{σ} | σ ∈ H} → ℚ ⊗_{ℤ} Γ_{F} such that, for all σ, τ ∈ H,

w(x_{σ}) + w(x_{τ}) = v(f(σ, τ)) + w(x_{στ})  (5.5)

(where v denotes the valuation on K extending the given valuation on F). Extend w to A − {0} by defining

w( ∑_{σ∈H} c_{σ}x_{σ}) = min{v(c_{σ}) + w(x_{σ}) | c_{σ} ≠ 0}.  (5.6)

Suppose B_{w}/J_{w} is simple Artinian (for B_{w}, J_{w} defined as in (iii) above). Then GA ≅_{g} (GK/GF, H, \overline{f}), where \overline{f}(σ, τ) = image of f(σ, τ) in GK^{h}.

Proof. Let Δ_{F} = ℚ ⊗_{ℤ} Γ_{F} ⊇ Γ_{K}. The valuation v: K^{∗} → Δ_{F} is H-equivariant (with H acting trivially on Δ_{F}), so it induces a map in cohomology v^{∗}: H^{2}(H, K^{∗}) → H^{2}(H, Δ_{F}). Because H is finite and Δ_{F} is uniquely divisible, H^{2}(H, Δ_{F}) = H^{1}(H, Δ_{F}) = (0). The existence of {w(x_{σ})} satisfying (5.5) is a restatement of v^{∗}[f] = 0, and the uniqueness of {w(x_{σ})} follows from H^{1}(H, Δ_{F}) = (0). See the proof of Lemma 3.1.

Now consider the function w on Λ − {0} defined by (5.6). We check that w satisfies conditions (i)–(iv) above for a value function. (i) is clear, and (ii) follows because by (5.5), for any c, d ∈ K^{∗}, σ, τ ∈ H, w((cx_{σ})(dx_{τ})) = w(cx_{σ}) + w(dx_{τ}). Property (iii) holds by hypothesis. The previous equation shows {cx_{σ} | c ∈ K^{∗}, σ ∈ H} ⊆ st(w), so (iv) holds. Therefore, B_{w} is a Dubrovin valuation ring of A with B_{w} ∩ F = V_{F}, so GA = GB_{w}.

We show that GB_{w} is the desired crossed product. Since w restricts to v on K, GA contains GK as a graded subring. Further, if y_{σ} is the image of x_{σ} in GB_{w}, then y_{σ}y_{τ} = \overline{f}(σ, τ)y_{στ}; also, each y_{σ} ∈ (GB_{w})^{∗} and y_{σ}cy_{σ}^{−1} = σ(c) for any c ∈ GK^{h}, hence for any c ∈ GK. Because K is tame over F, the elements of H induce distinct graded automorphisms of GK, and GK is H-Galois over GF. The sum ∑_{σ∈H} GKy_{σ} in GB_{w} is actually a direct sum, as one can see by the usual argument (conjugating a homogeneous sum ∑ c_{σ}y_{σ} = 0 with the minimal number of nonzero c_{σ} ∈ GK by an element of GK^{h} − {0}, and subtracting to get a contradiction). Finally, to see ∑ GKy_{σ} is all of GB_{w}, take any δ ∈ Γ_{GB_{w}} = Γ_{B_{w}} and any b ∈ (GB_{w})_{δ}, b ≠ 0. Then, b is the image of some a = ∑_{σ∈H} c_{σ}x_{σ}, c_{σ} ∈ K, with w(a) = δ. So, each w(c_{σ}x_{σ}) ≥ δ and some

w(c_{τ} x_{τ}) = δ. If we let a′ = ∑_{σ∈S} c_{σ}x_{σ}, where S = {σ ∈ H | w(c_{σ}x_{σ}) = δ}, then a′ ≡ a (mod I^{>δ}),

so if we let bar denote the image in GA, b = \overline{a} = \overline{a′} = ∑_{σ∈S} \overline{c_{σ}x_{σ}} = ∑_{σ∈S} \overline{c_{σ}}y_{σ}. Thus, we have

GA = GB_{w} = ∑_{σ∈H} GKy_{σ} = ⨁_{σ∈H} GKy_{σ} = (GK/GF, H, \overline{f}), as desired. □

Proof of Th. 5.3. The vertical maps in (5.4) are isomorphisms (by Prop. 3.3 for the middle map). We know from Th. 5.1 that the right square of (5.4) is commutative and that β and β′ are isomorphisms with β index-preserving. Also, β′ ∘ α′ is the isomorphism γ of Th. 4.4. Hence, α′ = β^{′−1} ∘ (β′ ∘ α′) is also an isomorphism. Since α maps tame CDA’s over F to GCDA’s over GF, it is index-preserving. It remains only to prove that the left square of (5.4) is commutative. (Note that it is not apparent at this point even that α is a group homomorphism. We would like to complete the proof by invoking the fact that the δ of (5.1) is an isomorphism but we do not know how to carry out such an argument.)

Take any finite tame Galois extension field K of F, set H = G(K/F), and take any f ∈ Z^{2}(H, K^{∗}). If f satisfies the hypothesis of Lemma 5.4 (i.e., B_{w}/J_{w} is simple Artinian), then that lemma shows the left square of (5.4) is commutative for the image of f in H^{2}(G, F_{t}^{∗}). Note the following two cases where this applies. First, whenever K is unramified over F, it was shown in [MW, Th. 2.3] that the hypotheses of Lemma 5.4 hold for any f ∈ Z^{2}(H, K^{∗}). Second, if T is any division algebra tame and totally ramified over F, then (as T is isomorphic to a tensor product of symbol algebras, see [Dr, Th. 1]) T has a maximal subfield L which is Galois (and necessarily tame and totally ramified) over F. So T is a crossed product, say T = ⨁_{σ∈G(L/F)} Lz_{σ},

with z_{σ}z_{τ} = g(σ, τ)z_{στ}. Then, for the unique w of (5.5), we must have w(z_{σ}) = v(z_{σ}), where v is the valuation on T. Furthermore, the v(z_{σ}) are distinct mod Γ_{L}. For, if v(z_{σ}) ≡ v(z_{τ}) (mod Γ_{L}), then v(z_{ρ}) ∈ Γ_{L} for ρ = στ^{−1}. Since the canonical pairing β_{T} on Γ_{T} /Γ_{F} is trivial on Γ_{L}/Γ_{F} as L is commutative, we have 1 = \overline{z_{ρ}ℓz_{ρ}^{−1}ℓ^{−1}} = \overline{ρ(ℓ)ℓ^{−1}} ∈ μ(\overline{F}) for all ℓ ∈ L^{∗}. From the nondegenerate pairing Γ_{L}/Γ_{F} × G(L/F) → μ(\overline{F}) for a tame totally ramified Galois field extension, see [TW, Prop. 1.4(i)], it follows that ρ = id_{L}; so σ = τ whenever v(z_{σ}) ≡ v(z_{τ}) (mod Γ_{L}). Hence, the function w on T − {0} defined by (5.6) from {w(z_{σ})} coincides with v on all of T − {0}. So, B_{w} = V_{T}, J_{w} = M_{T}, and B_{w}/J_{w} = \overline{T} ≅ \overline{F}, so the hypotheses of Lemma 5.4 are satisfied for the cocycle g.

Now, let ψ: H^{2}(G, F_{t}^{∗}) → TBr(F) be the left vertical map of (5.4) (an isomorphism), and take any [A] ∈ TBr(F). By [JW, Lemma 6.2], [A] = [S] + [T] in TBr(F), for some CDA’s S and T with S inertially split and T tame and totally ramified over F. By the preceding paragraph, the left square of (5.4) is commutative for ψ^{−1}[S] and ψ^{−1}[T]. But it was shown by Boulagouaz in [B_{2}, Prop. 9.4], using value functions and [MW, Th. 2.1] that α[A] = α[S] + α[T]. Consequently, the left square of (5.4) is commutative for ψ^{−1}[A], hence for all of H^{2}(G, F_{t}^{∗}), as ψ^{−1} is onto. This completes the proof. □

Remark 5.5. One interesting fact that Th. 5.3 makes clear is that if F is any Henselian valued field, then the structure of TBr(F) is independent of the ordering on Γ_{F} (though it certainly depends on Γ_{F} as an abstract group). For, TBr(F) ≅ GBr(GF), and the graded Brauer group is independent of the ordering on Γ_{F}.

Remark 5.6. It was proved in [B_{2}, Th. 10.3] that if F is a Henselian field and D is a tame CDA over F, then exp(GD) = exp(D), where exp(GD) denotes the order of [GD] in the abelian group GBr(GF). Observe that this follows immediately from the α of (5.4) being a monomorphism, which we proved in Th. 5.3.

Corollary 5.7. Let F ⊆ K be Henselian fields (with the valuation on K extending the one on F). Then, there is a commutative diagram:

TBr(F) ⟶^{α_{F}} GBr(GF) ⟶^{β_{F}} TBr(HGF)
↓  ↓  ↓
TBr(K) ⟶^{α_{K}} GBr(GK) ⟶^{β_{K}} TBr(HGK)

(5.7)

Proof. The vertical maps in (5.7) are the canonical scalar extension maps. It is clear from the description of K_{t} as the ramification field for the Galois extension K_{s}/K that F_{t} ⊆ K_{t}. So, the scalar extension maps Br(F) → Br(K) maps TBr(F) → TBr(K). Likewise, since we are assuming the Henselization HGK of QGK has been chosen to contain HGF, we have (HGF)_{t} ⊆ (HGK)_{t}, so TBr(HGF) maps to TBr(HGK). The right inner square in (5.7) is commutative because β_{F} and β_{K} are essentially scalar extension maps. Also, it is clear from the homological definition of the tame Brauer group, and Th. 5.3, that the outer rectangle in (5.7) is commutative. Hence, by Th. 5.3 and the fact that β_{F} and β_{K} are isomorphisms, it follows that the left inner square in (5.7) is also commutative. □

Corollary 5.8. Let F ⊆ K Henselian fields as in Cor. 5.7, and let A be a tame CSA over F. Then, K splits A iff GK splits GA, iff HGK splits HGA.

Proof. This is immediate from the commutativity of diagram (5.7), since the horizontal maps are isomorphisms by Th. 5.3. □

Theorem 5.9. Let D be a tame CDA over a Henselian field F, and let A be a tame CDA over a field K ⊇ F such that [K : F] < ∞ and K is defectless over F. Then A is F -isomorphic to an F -subalgebra of D iff GA is GF -gr-isomorphic to a graded GF -subalgebra of GD, iff HGA is HGF -isomorphic to an HGF -subalgebra of HGD.

Proof. Let k = [K : F] and let a^{2} = [A : K]. Recall (see, e.g., [MiW, Prop. 2.1(b)]) that ind(D)/ind(D ⊗_{F} A^{op}) ≤ ak, with equality holding iff A embeds F -isomorphically into D.

Clearly, any F -monomorphism of A into D induces a graded GF -monomorphism of GA into GD. Suppose next that GA is graded GF -isomorphic to a subalgebra of GD. Since K is defectless over F, GK ⊗_{GF} HGF = HGK. Also, HGF is flat over GF, since it is a direct sum of copies of the localization QGF of GF. Hence HGA, which equals GA⊗_{GK} HGK ≅ GA⊗_{GF} HGF, embeds in HGD = GD ⊗_{GF} HGF.

Now, suppose HGA embeds in HGD over HGF. Since HGK = Z(HGA) and also [HGA : HGK] = [A : K] = a^{2} and [HGK : HGF] = [K : F] = k (as K/F is defectless), the formula in the first paragraph gives

ind(HGD)/ind(HGD ⊗_{HGF} (HGA)^{op}) = ak.  (5.8)

Let γ_{F} = β_{F} ∘ α_{F}, where β_{F} and α_{F} are the maps of (5.4) for F; likewise, let γ_{K} = β_{K} ∘ α_{K}. Also, let res denote the scalar extension map. Then, in TBr(HGK) we have

[HGD ⊗_{HGF} (HGA)^{op}] = [HGD ⊗_{HGF} HGK] + [(HGA)^{op}]
= res_{HGK/HGF} ∘ γ_{F} [D] + γ_{K}[A^{op}]
= γ_{K}[D ⊗_{F} K] + γ_{K}[A^{op}] = γ_{K}[D ⊗_{F} A^{op}],

where the second equality uses Th. 5.3 twice and the third equality is by Cor. 5.7. Since γ_{K} and γ_{F} are index-preserving by Th. 5.3, this yields

ind(D)/ind(D ⊗_{F} A^{op}) = ind(HGD)/ind(HGD ⊗_{HGF} (HGA)^{op}) = ak.

It follows by the first paragraph that there is an F -algebra monomorphism of A into D. □

Remark 5.10. It follows from Th. 5.9 that results on defectless subfields of division algebras over Henselian fields can be carried over completely to results on graded subfields of graded division algebras. This applies, for example, to the work of Morandi and Sethuraman in [MS] on Kummer subfields, and that of Brussel in [Br] on totally ramified subfields, as well as many of the results in [JW]. Also, there are of course analogues to Cor. 5.7, 5.8, Th. 5.9 where we start with a graded field instead of a Henselian field, and use Th. 5.1 instead of Th. 5.3.

§6 Compatibility with corestriction

We have shown that the maps between tame Brauer groups of Henselian valued fields and graded Brauer groups of graded fields are compatible with scalar extensions. We now show that they are compatible with the corestriction.

Let R ⊆ S be graded fields (with Γ_{R} torsion-free, as always), with [S : R] = k < ∞ and S tame over R. Then, by [HW, Th. 3.9, Th. 3.11] there is a graded field U ⊇ S with [U : R] < ∞ and U Galois over R. Let G = G(U/R) = G(QU/QR), and let H = G(QU/QS) ⊆ G. Then, as S = U ∩QS (see [HW, Cor. 2.5(b)]), for the elements of U fixed by H, we have U^{H} = U ∩ QU^{H} = S; so by [Gr, Th. 2.2, p. 7], U is H-Galois over S, and |G : H| = [QS : QR] = [S : R] = k, by (3.1). Note that if N is any finite-dimensional graded U-vector space, and G acts on N by graded R-automorphisms and the action is semilinear (i.e. σ(un) = σ(u) · σ(n) for all σ ∈ G, u ∈ U, n ∈ N), then

N^{G} is a graded R-vector space with dim_{R}(N^{G}) = dim_{U}(N), and the map
U ⊗_{R} N^{G} → N given by u ⊗ n ↦ u · n is a graded U-vector space isomorphism.

(6.1)

That N^{G} is graded is clear; the other assertions in (6.1) follow by noting that QR ⊗_{R} N^{G} = (QU ⊗_{U} N)^{G}, and applying the corresponding properties for semilinear group actions on vector spaces (since G = G(QU/QR) ).

We first describe the corestriction of a finite dimensional graded S-vector space M. Let τ_{1}, …, τ_{k} be a set of representatives of the left cosets of H in G. The left action of G on the cosets {τ_{1}H_{1} … τ_{n}H} associates to each σ ∈ G a permutation σ̃ of {1, 2, …, k} defined by στ_{i}H = τ_{σ̃(i)}H. For each i, let M_{i} = U ⊗_{S,τ_{i}} M, i.e., the scalar extension of M from S to U, with U treated as an S-algebra via τ_{i}: S → U. That is, M_{i} satisfies the middle linearity rule u ⊗ sm = uτ_{i}(s) ⊗ m, for all u ∈ U, s ∈ S, m ∈ M. Since τ_{i} is a gr-isomorphism of U, M_{i} is a graded U-vector space. Note that for each σ ∈ G there is a (well-defined!) σ-semilinear graded R-vector space isomorphism \overline{σ} : M_{i} → M_{σ̃(i)} given by u⊗m ↦ σ(u)⊗m; clearly, \overline{ρσ} = \overline{ρ}∘\overline{σ} for ρ, σ ∈ G. Then, observe that there is a graded semilinear action of G on M_{1} ⊗_{U} … ⊗_{U} M_{k} given by σ(m_{1} ⊗ … ⊗ m_{k}) = n_{1} ⊗ … ⊗ n_{k}, where n_{τ(i)} = \overline{σ}(m_{i}) for each i. Define cor_{S/R}(M) to be (M_{1} ⊗_{U} … ⊗_{U} M_{k})^{G}. So, cor_{S/R}(M) is a graded R-vector space, and (6.1) shows

dim_{R}(cor_{S/R}(M)) = [S : R] dim_{S}(M) and U ⊗_{R} cor_{S/R}(M) ≅_{g} M_{1} ⊗_{U} … ⊗_{U} M_{k}. (6.2)

Now, if A is a GCSA over S, then cor_{S/R}(A), as just defined but with A replacing M, is a graded R-algebra since G acts on A_{1}⊗_{U} …⊗_{U} A_{k} (where A_{i} = U ⊗_{S,τ_{i}}A) by algebra automorphisms. Furthermore, the graded U-algebra isomorphism U ⊗_{R} cor_{S/R}(A) ≅_{g} A_{1} ⊗_{U} … ⊗_{U} A_{k} shows that cor_{S/R}(A) must be a GCSA over R. The construction of cor_{S/R}(A) is clearly independent of the choice of coset representatives τ_{i} of H in G, and, by the usual argument, is also independent (up to graded isomorphism) of the choice of Galois graded field extension U of R containing S. If B is another GCSA over S, then clearly cor_{S/R}(A) ⊗_{R} cor_{S/R}(B) ≅_{g} cor_{S/R}(A ⊗_{S} B). Also, if M is a finite dimensional graded S-vector space, then clearly U ⊗_{S,τ_{i}} End_{S}(M) ≅_{g} End_{U}(U ⊗_{S,τ_{i}} M), with compatible G-actions. It follows easily that cor_{S/R}(End_{S}(M)) ≅_{g} End_{R}(cor_{S/R}(M)). Thus, the corestriction cor_{S/R} yields a well-defined group homomorphism cor_{S/R}: GBr(S) → GBr(R). It is clear from the definitions and a dimension count that

QR ⊗_{R} cor_{S/R}(A) ≅ cor_{QS/QR}(QS ⊗_{S} A).  (6.3)

Likewise, for any total ordering on Γ_{S} (with corresponding ordering on Γ_{R} ⊆ Γ_{S}), if HS is the Henselization of QS with respect to the valuation on QS induced by the ordering on Γ_{S}, then

HR ⊗_{R} cor_{S/R}(A) ≅ cor_{HS/HR}(HS ⊗_{S} A).  (6.4)

It follows from (6.3) and the injectivity of GBr(U/R) → Br(QU/QR) that we have a commutative diagram

H^{2}(H, U^{∗}) ⟶^{cor} H^{2}(G, U^{∗})
≅↓  ≅↓
GBr(U/S) ⟶_{cor_{S/R}} GBr(U/R)

(6.5)

where the top map in (6.5) is the cohomological corestriction.

Theorem 6.1. Let F be a field with Henselian valuation, and K a finite degree tame field extension of F. Then, there is a commutative diagram,

TBr(K) ⟶^{α_{K}} GBr(GK) ⟶^{β_{K}} TBr(HGK)
↓^{cor_{K/F}}  ↓^{cor_{GK/GF}}  ↓^{cor_{HGK/HGF}}
TBr(F) ⟶^{α_{K}} GBr(GF) ⟶^{β_{K}} TBr(HGF)

(6.6)

where α, β are the maps of (5.4).

Proof. Let γ_{F} = β_{F} ∘ α_{F} and γ_{K} = β_{k} ∘ α_{K}, which are isomorphisms by Th. 5.3. The outer rectangle of (6.6) is commutative since by Th. 5.3 γ_{F} and γ_{K} correspond to maps in cohomology, the cohomological corestriction is functorial with respect to G-module homomorphisms, and the cohomological corestriction is consistent with the algebra corestriction. The right inner square of (6.6) is commutative by (6.4). Since α_{K} = β_{K} ∘ γ_{K}^{−1} and α_{F} = β_{F} ∘ γ_{F}^{−1}, it follows that the left inner square of (6.6) is also commutative. □

Proposition 6.2. Let R ⊂ S be a finite degree tame extension of graded fields (with Γ_{R} torsionfree). Let E be a GCDA over S, and let A be the underlying GCDA of cor_{S/R}(E) (i.e., [A]_{g} = [cor_{S/R}(E)]_{g} in GBr(R)). Then,

Γ_{A} ⊆ Γ_{cor_{S/R}(E)} ⊆ Γ_{E}  (6.7)

and

Z(A_{0}) ⊆ N^{1/k},  (6.8)

where N is the normal closure of Z(E_{0}) over R_{0} and k = exp(ker(θ_{E})), where θ_{E} is the map of (2.2) .

Proof. The inclusions in (6.7) are evident from the definitions. For (6.8), let ≤ be any total ordering on Γ_{R}, and HR the Henselization of QR with respect to the valuation on QR determined by the ordering on Γ_{R}. Then HA (= HR ⊗_{R} A) is a CDA over HR (see after (4.4) above), and [HA] = [cor_{HS/HR}(HE)] in Br(HR) by (6.4). Since \overline{HE} ≅ E_{0} and \overline{HR} ≅ R_{0} and the map θ_{HE} of (4.8) corresponds to θ_{E} (so ker(θ_{HE}) ≅ ker(θ_{E}) ), it follows by [H, Th. 18] that Z(\overline{HA}) ⊆ N^{1/k}. Then (6.8) follows as A_{0} ≅ \overline{HA}. □

The value of k given in Prop. 6.2 can be improved by taking into account [S_{0} : R_{0}] and which roots of unity lie in R_{0}. See [H, Th. 18] for the full statement.

References

[B_{1}] M. Boulagouaz, The graded and tame extensions, pp. 27–46 in Commutative Ring Theory (Fès, 1992), P. J. Cahen, et. al., eds., Lecture Notes in Pure and Applied Math., No. 153, Marcel Dekker, New York, 1994.

[B_{2}] M. Boulagouaz, Le graduè d’une algèbre à division valuée, Comm. Alg., 23 (1995), 4275–4300.

[BG] H. H. Brungs and J. Gräter, Extensions of valuation rings in central simple algebras, Trans. Amer. Math. Soc., 317 (1990), 287–302.

[Br] E. Brussel, Division algebra subfields introduced by an indeterminate, J. Algebra, 188 (1997), 216–255.

[CvO] S. Caenepeel and F. van Oystaeyen, Brauer Groups and the Cohomology of Graded Rings, Marcel Dekker, New York, 1988.

[DI] F. DeMeyer and E. Ingraham, Separable Algebras over Commutative Rings, Lecture Notes in Math, No. 181, Springer, Berlin, 1971.

[Dr] P. Draxl, Ostrowski’s theorem for Henselian valued skew fields, J. Reine Angew. Math., 354 (1984) 213–218.

[D_{1}] N. I. Dubrovin, Noncommutative valuation rings, Trudy Moskov. Mat. Obshch., 45 (1982), 265–280; English trans.: Trans. Moscow Math. Soc., 45 (1984), 273–287.

[D_{2}] N. I. Dubrovin, Noncommutative valuation rings in simple finite-dimensional algebras over a field, Mat. Sb., 123 (165) (1984), 496–509; English trans.: Math. USSR Sb., 51 (1985), 493–505.

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

[G] J. Gräter, The “Defektsatz” for central simple algebras, Trans. Amer. Math. Soc., 330 (1992), 823–843.

[Gr] C. Greither, Cyclic Galois Extensions of Commutative Rings, Lecture Notes in Math., No. 1534, Springer, Berlin, 1992.

[HvO] L. Huishi and F. van Oystaeyen, Filtrations on simple Artinian rings, J. Algebra, 132 (1990), 361–376.

[H] Y.-S. Hwang, The corestriction of valued division algebras over Henselian fields, II, Pacific J. Math., 170 (1995), 83–103.

[HW] Y.-S. Hwang and A. R. Wadsworth, Algebraic extensions of graded and valued fields, preprint, UCSD, 1997.

[JW] B. Jacob and A. R. Wadsworth, Division algebras over Henselian fields, J. Algebra, 128 (1990), 126–179.

[L] S. Lang, Algebra, Addison-Wesley, Reading, Mass., 1971.

[MiW] J. Minác̆ and A. R. Wadsworth, The u-invariant for algebraic extensions, pp. 333–358 in K-Theory and Algebraic Geometry: Connections with Quadratic Forms and Division Algebras, eds. B. Jacob and A. Rosenberg, Proc. Symp. Pure Math., Vol. 58, Part 2, Amer. Math. Soc., Providence, R.I., 1995.

[M_{1}] P. J. Morandi, The Henselization of a valued division algebra, J. Algebra, 122 (1989),

232–243. [M_{2}] P. J. Morandi, Value functions on central simple algebras, Trans. Amer. Math. Soc.,

315 (1989), 605–622.

[MS] P. J. Morandi and B. A. Sethuraman, Kummer subfields of tame division algebras, J. Algebra, 172 (1995), 554–583.

[MW] P. J. Morandi and A. R. Wadsworth, Integral Dubrovin valuation rings, Trans. Amer. Math. Soc., 315 (1989), 623–640.

[NvO] C. Nǎstǎsescu and F. van Oystaeyen, Graded Ring Theory, North-Holland, Amsterdam,

1982.

[R] I. Reiner, Maximal Orders, Academic Press, London, 1975.

[Ri] M. Rieffel, A general Wedderburn theorem, Proc. Nat. Acad. Sci. U.S.A., 54 (1965), 1513.

[S] O. F. G. Schilling, The Theory of Valuations, Amer. Math. Soc., Providence, R.I., 1950.

[T] J.-P. Tignol, Algèbres à division et extensions de corps sauvagement ramifiées de degré

premier, J. Reine Angew. Math., 404 (1990), 1–40.

[TW] J.-P. Tignol and A. R. Wadsworth, Totally ramified valuations on finite-dimensional division algebras, Trans. Amer. Math. Soc., 302 (1987), 223–250.

[W_{1}] A. R. Wadsworth, Extending valuations to finite dimensional division algebras, Proc.

Amer. Math. Soc., 98 (1986), 20–22. [W_{2}] A. R. Wadsworth, Dubrovin valuation rings and Henselization, Math. Ann., 283 (1989),

301–328.

Department of Mathematics College of Science Korea University 5–1, Anam-dong, Sungbuk-ku Seoul 136–701 Korea

e-mail: yhwang@semi.korea.ac.kr

Department of Mathematics, 0112 University of California, San Diego 9500 Gilman Drive La Jolla, California 92093-0112 USA

                                e-mail: arwadsworth@ucsd.edu