VALUE FUNCTIONS AND ASSOCIATED GRADED RINGS FOR SEMISIMPLE ALGEBRAS

J.-P. TIGNOL AND A. R. WADSWORTH

Introduction

Valuation theory is a time-honored subject, which has undergone a robust development for noncommutative division rings in the last two decades, spurred by its applications to the constructions of noncrossed products and of counterexamples to the Kneser–Tits conjecture: see [W_{4}] for a recent and fairly comprehensive survey. However, results that relate valuations with Brauer-group properties have been particularly difficult to establish; a major source of problems is that valuations are defined only on division algebras and not on central simple algebras with zero divisors. The purpose of this work is to introduce a more flexible tool, which we call gauge, inspired by the normes carrées of Bruhat and Tits [BT] (see Rem. 1.19). Gauges are valuation-like maps defined on finite-dimensional semisimple algebras over valued fields with arbitrary value group.

With any valuation there is an associated filtration of the ring, which yields an associated graded ring. Such filtrations and associated graded rings are actually defined not just for valuations, but also for more general value functions: the surmultiplicative value functions defined in (1.4) below, which are sometimes called pseudo-valuations. The gauges we consider here are the surmultiplicative value functions for which the associated graded algebra is semisimple, and which also satisfy a defectlessness condition, see Def. 1.4. It turns out that gauges exist in abundance and have good behavior with respect to tensor products, but that they still have sufficient uniqueness to reflect the structure of the algebras they are defined on.

Valuation theory typically derives information on fields or division algebras from properties of the residue field or algebra and of the ordered group of values. In a noncommutative setting, these structures interact since the value group acts naturally on the center of the residue algebra, see (1.13). It is therefore reasonable to consider the graded algebra associated with the valuation filtration, which encapsulates information on the residue algebra, the value group, and their interaction. This paper shows how fruitful it can be to work with the graded structures. Associated graded algebras have previously been studied for valuations on division algebras, as in [Bl_{1}], [Bl_{2}], and [HW_{2}]. But they have not been used in the earlier work with value functions on central simple algebras in [BT], nor with the value functions associated to Dubrovin valuation rings in [M_{2}]. (The relation between the value functions considered here and Morandi’s value functions in [M_{2}] is described in Prop. 2.5.)

For a given semisimple algebra A over a field F, we fix a valuation v on F and consider gauges y on A which restrict to v on F, which we call v-gauges or (when v is understood) F -gauges. The associated graded ring gr_{y}(A) is then a finite dimensional algebra over the graded field gr_{v}(F). If A is central simple over F, there are typically many different v-gauges y on A; it turns out that gr_{y}(A) is always a graded simple algebra (i.e., there are no nontrivial homogeneous ideals), and that the class of gr_{y}(A) is uniquely determined in the graded Brauer group of its center, see Cor. 3.8.

The first author is partially supported by the National Fund for Scientific Research (Belgium) and by the European Community under contract HPRN-CT-2002-00287, KTAGS. The second author would like to thank the first author and UCL for their hospitality while the work for this paper was carried out.

We get the strongest information when the valuation on F is Henselian. For any finite-dimensional division algebra D over F, it is well-known that the Henselian valuation v on F has a unique extension to a valuation w on D. For A = End_{D}(M), where M is a finite dimensional right D-vector space, we prove in Th. 3.3 that for any v-gauge y on A there is a norm α (a kind of value function) on M such that up to isomorphism y is the gauge on End_{D}(M) induced by α on M as described in §1.3. It follows that gr_{y}(A) is isomorphic as a graded ring to End_{gr_{w}(D)}(gr_{α}(M)); furthermore, the graded Brauer class of gr_{y}(A) is the same as that of gr_{w}(D), and gr_{y}(A) has the same matrix size as A. In particular, if A is central simple over F and the gauge is tame, in the sense that the center of gr_{y}(A) is gr_{v}(F), then gr_{y}(A) is a graded central simple algebra over gr_{v}(F) with the same Schur index as A. We may then consider its Brauer class [gr_{y}(A)] in the graded Brauer group GBr(gr_{v}(F)). The map [A] ↦ [gr_{y}(A)] defines an index-preserving group isomorphism Ψ from the tame Brauer group TBr(F), which is the subgroup of Br(F) split by the maximal tamely ramified extension of F, onto GBr(gr_{v}(F)). That Ψ is an isomorphism was proved earlier in [HW_{2}]; without the use of gauges the proof in [HW_{2}] that Ψ is a group homomorphism was particularly involved and arduous. The proof given here in Th. 3.9 is much easier and more natural, because we can work with central simple algebras, not just with division algebras, and because gauges work well with tensor products. The map Ψ should be compared with a similar map for Witt groups defined in [TW] to generalize Springer’s theorem on quadratic forms over complete discretely valued fields.

When v is Henselian and A is assumed just to be semisimple, we show in Th. 3.1 that for any vgauge on A the simple components of gr_{y}(A) are the graded algebras for the restrictions of y to the simple components of A. Thus, the results described above apply component-by-component. Also, the information obtained in the Henselian case can be extrapolated to gauges with respect to non-Henselian valuations v. For, if the valuation v^{h} on field F^{h} is the Henselization of a valuation v on F, and y is any v-gauge on a semisimple F -algebra A, then there is a canonical extension of y to a v^{h}-gauge y^{h} on A ⊗_{F} F^{h}, and gr_{y}(A) is graded isomorphic to gr_{y^{h}}(A ⊗_{F} F^{h}). Thus, any v-gauge on A gives insight into what happens with A on passage to the Henselization of v.

In the last section, we apply gauges to obtain information on the division algebra Brauer-equivalent to a crossed product or to a tensor product of symbol algebras over valued fields. The idea is that, since we are now freed from the constraint to deal with division algebras, we may easily define gauges on these central simple algebras, and use the associated graded structure to derive properties of their Brauer-equivalent division algebras. We thus easily recover in a straightforward way several results that were previously obtained in [JW] and [W_{3}] by much more complicated arguments.

The organization of the paper is as follows: §1 gives the definition of gauges and describes various examples on division algebras, endomorphism algebras, and tensor products. In §2 we review some results on graded central simple algebras, complementing the discussion in [HW_{2}] with a result characterizing the graded group of the Brauer-equivalent graded division algebra. The main results quoted above, relating semisimple algebras with a gauge over a Henselian field to their associated graded algebras, are given in §3. This section also contains the definition of the map Ψ: TBr(F) → GBr(gr(F)). The applications to crossed products and tensor products of symbols are in §4.

1. Value functions, norms, and gauges

Let D be a division ring finite-dimensional over its center. Let Γ be a divisible totally ordered abelian group. Let ∞ be an element of a set strictly containing Γ; extend the ordering on Γ to Γ ∪ {∞} by requiring that γ < ∞ for each γ ∈ Γ. Further set γ + ∞ = ∞ + ∞ = ∞ for all γ ∈ Γ. A valuation on D is a function w: D → Γ ∪ {∞} satisfying, for all c, d ∈ D,

w(d) = ∞ iff d = 0;  (1.1a)
w(cd) = w(c) + w(d);  (1.1b)
w(c + d) ≥ min(w(c), w(d)).  (1.1c)

(It follows that w(1) = w(−1) = 0 and if w(c) ≠ w(d) then w(c + d) = w(c − d) = min(w(c), w(d)).) Associated to the valuation on D, we have its value group Γ_{D} = w(D^{×}), where D^{×} is the group of units of D, i.e., D^{×} = D − {0}; its valuation ring V_{D} = { d ∈ D | w(d) ≥ 0 }; the unique maximal left (and right) ideal M_{D} of V_{D}, M_{D} = \{ d ∈ D | w(d) > 0 \}; and the residue division ring \overline{D} = V_{D}/M_{D}. Another key structure is the associated graded ring: for γ ∈ Γ, set D^{≥γ} = { d ∈ D | w(d) ≥ γ } and D^{>γ} = { d ∈ D | w(d) > γ }, which is a subgroup of D^{≥γ}; let D_{γ} = D^{≥γ}/D^{>γ}. The associated graded ring of D with respect to w is gr_{w}(D) = ⨁_{γ∈Γ} D_{γ}. For each γ, δ ∈ Γ, the multiplication in D induces

a well-defined multiplication D_{γ} × D_{δ} → D_{γ+δ} given by (c + D^{>γ}) · (d + D^{>δ}) = cd + D^{>γ+δ}. This multiplication is extended biadditively to all of gr_{w}(D), making gr_{w}(D) into a graded ring. When w is clear, we write gr(D) for gr_{w}(D). The grade group of gr(D), denoted Γ_{gr(D)}, is { γ ∈ Γ | D_{γ} ≠ 0 }; note that Γ_{gr(D)} = Γ_{D}. Also, for the degree 0 component of gr(D), we have D_{0} = D^{≥0}/D^{>0} = V_{D}/M_{D} = \overline{D}. For d ∈ D^{×}, we write d′ for the image of d in gr(D), i.e., d′ = d + D^{>w(d)} ∈ D_{w(d)}. The homogeneous elements of gr(D) are those in ⋃_{γ∈Γ} D_{γ}. It follows from property (1.1b) that gr(D) is a graded division

ring, i.e., every nonzero homogeneous element of gr(D) is a unit.

Now, let M be a right D-vector space, where D has a valuation w. A function α: M → Γ ∪ {∞} is called a D-value function with respect to w (or a w-value function) if for all m, n ∈ M and d ∈ D,

α(m) = ∞ iff m = 0;  (1.2a)
α(md) = α(m) + w(d);  (1.2b)
α(m + n) ≥ min(α(m), α(n)).  (1.2c)

Given such an α on M, we can form the associated graded module gr_{α}(M) just as before: for γ ∈ Γ, let M^{≥γ} = { m ∈ M | α(m) ≥ γ } and M^{>γ} = { m ∈ M | α(m) > γ }; then set M_{γ} = M^{≥γ}/M^{>γ}. Define gr(M) = gr_{α}(M) = ⨁_{γ∈Γ} M_{γ}. For nonzero m ∈ M, let m′ denote the image m + M^{>α(m)} of m in gr(M);

for 0 ∈ M, let 0′ = 0 ∈ gr(M). For γ, δ ∈ Γ there is a well-defined multiplication M_{γ} × D_{δ} → M_{γ+δ} given by (m + M^{>γ}) · (d + D^{>δ}) = (md) + M^{>γ+δ}. This is extended distributively to yield an operation gr(M)×gr(D) → gr(M) which makes gr(M) into a graded right gr(D)-module. It is well-known and easy to prove by a slight variation of the ungraded argument that every graded module over a graded division ring is a free module with a homogeneous base, and every two bases have the same cardinality. Thus, graded modules over graded division rings are called graded vector spaces; we write dim_{gr(D)}(gr(M)) for the cardinality of any gr(D)-module base of gr(M). If N = ⨁_{γ∈Γ} N_{γ} is another graded right gr(D)-vector

space, we say that M and N are graded isomorphic, written M ≅_{g} N, if there is a gr(D)-vector space isomorphism f: M → N with f(M_{γ}) = N_{γ} for each γ ∈ Γ.

Now, suppose M is finite-dimensional. A right D-vector space base (m_{i})_{1≤i≤k} of M is called a splitting base with respect to α if for all d_{1}, …, d_{k} ∈ D,

α(∑_{i=1}^{k} m_{i}d_{i}) = min_{1≤i≤k}(α(m_{i}) + w(d_{i})).  (1.3)

If there is a splitting base for the D-value function α, we say that α is a D-norm (or a w-norm) on M. Note that it is easy to construct D-norms on M: take any D-vector space base (m_{i})_{1≤i≤k} of M, and take any γ_{1}, …, γ_{k} ∈ Γ. Define α(m_{i}) = γ_{i} for 1 ≤ i ≤ k, and then define α on all of M by formula (1.3). It is straightforward to check that such an α is a D-norm on M and (m_{i})_{1≤i≤k} is a splitting base for α.

Recall the following from [RTW, Prop. 2.2, Cor. 2.3, Prop. 2.5]:

Proposition 1.1. Let α be a D-value function on M. Take any m_{1}, …, m_{ℓ} ∈ M.

(i) m′_{1}, …, m′_{ℓ} are gr(D)-linearly independent in gr(M) iff α(∑_{i=1}^{ℓ} m_{i}d_{i}) = min_{1≤i≤ℓ}(α(m_{i}) + w(d_{i})) for

all d_{1}, …, d_{ℓ} ∈ D. When this occurs, m_{1}, …, m_{ℓ} are D-linearly independent in M. (ii) dim_{gr(D)}(gr(M)) ≤ dim_{D}(M). Equality holds iff α is a D-norm on M. (iii) Suppose α is a D-norm on M. Then, for any D-subspace N of M, α|_{N} is a norm on N.

We are interested here in value functions on algebras. Let F be a field with valuation v: F → Γ∪{∞}, and let A be a finite-dimensional F -algebra. A function y: A → Γ ∪ {∞} is called a surmultiplicative F -value function on A if for any a, b ∈ A,

y(1) = 0, and y(a) = ∞ iff a = 0;  (1.4a)
y(ca) = v(c) + y(a) for any c ∈ F;  (1.4b)
y(a + b) ≥ min(y(a), y(b));  (1.4c)
y(ab) ≥ y(a) + y(b).  (1.4d)

Note that for such a y, there is a corresponding “valuation ring” V_{A} = A^{≥0} = { a ∈ A | y(a) ≥ 0 }. There is also an associated graded ring gr(A) = gr_{y}(A) = ⨁_{γ∈Γ} A_{γ}, where A_{γ} = A^{≥γ}/A^{>γ}, as above, and the

multiplication in gr(A) is induced by that of A. Furthermore, gr_{y}(A) is clearly a graded gr_{v}(F)-algebra. Also, gr_{v}(F) is a graded field, i.e., a commutative graded ring in which every nonzero homogeneous element is a unit. Since axioms (1.4a) – (1.4c) show that y is an F -value function for A as an F -vector space, Prop. 1.1(ii) implies that dim_{gr(F)}(gr(A)) ≤ dim_{F} (A), with equality iff y is an F -norm on A. The following lemma is convenient for verifying when an F -norm on A is surmultiplicative:

Lemma 1.2. Suppose y: A → Γ ∪ {∞} is an F -norm on A such that y(1) = 0. Let (a_{i})_{1≤i≤k} be a splitting base of A. If y(a_{i}a_{j}) ≥ y(a_{i}) + y(a_{j}) for all i, j, then y is a surmultiplicative F -value function on A.

Proof. We need only to verify axiom (1.4d). For this, take any b_{1} = ∑ c_{i}a_{i} and b_{2} = ∑ d_{i}a_{i} in A with c_{i}, d_{i} ∈ F. Then,

y(b_{1}b_{2}) = y(∑_{i,j} c_{i}d_{j}a_{i}a_{j}) ≥ min_{i,j}(y(c_{i}d_{j}a_{i}a_{j})) ≥ min_{i,j}(v(c_{i}) + v(d_{j}) + y(a_{i}) + y(a_{j}))

≥ min_{i}(v(c_{i}) + y(a_{i})) + min_{j}(v(d_{j}) + y(a_{j})) = y(b_{1}) + y(b_{2}).

□

If A has a surmultiplicative value function y, then for nonzero a ∈ A, we write a′ for the image a + A^{>y(a)} of a in A_{y(a)}. For 0 in A, we write 0′ = 0 ∈ gr(A). The following immediate consequence of the definitions will be used repeatedly below: for nonzero a, b ∈ A,

a′b′ = \{ \begin{matrix}(ab)′, & if a′b′ ≠ 0′, iff y(ab) = y(a) + y(b); \\ 0′, & if y(ab) > y(a) + y(b) . \end{matrix}  (1.5)

With this, we can readily characterize the inverse image in A of the group of homogeneous units of gr(A):

Lemma 1.3. Let y be a surmultiplicative F -value function on a finite-dimensional F -algebra A. For any nonzero u ∈ A, the following conditions are equivalent:

(i) u′ ∈ gr(A)^{×}, the group of units of gr(A); (ii) y(au) = y(a) + y(u) for all a ∈ A; (ii′) y(ua) = y(u) + y(a) for all a ∈ A; (iii) u ∈ A^{×} and y(u) + y(u^{−1}) = 0.

Proof. (i) ⇒ (ii) Suppose u′ ∈ gr(A)^{×}. Then, for any nonzero a ∈ A, we have a′u′ ≠ 0′; hence, y(au) = y(a) + y(u) by (1.5). (ii) ⇒ (i) By (1.5), (ii) implies that a′u′ ≠ 0′ for every nonzero a ∈ A. Therefore, as gr(A) is a finite-dimensional graded algebra over the graded field gr(F), u′ ∈ gr(A)^{×}.

(i) ⇔ (ii′) is proved analogously. (ii) ⇒ (iii) Condition (ii) shows that u is not a zero divisor in the finite-dimensional algebra A. Therefore, u ∈ A^{×}. The formula in (iii) follows by setting a = u^{−1} in (ii). (iii) ⇒ (ii) For any a ∈ A, we have y(a) = y(auu^{−1}) ≥ y(au) + y(u^{−1}). Therefore, (iii) yields

y(au) ≤ y(a) − y(u^{−1}) = y(a) + y(u) ≤ y(au) ;

so equality holds throughout, proving (ii). □

It is easy to construct numerous surmultiplicative value functions y on A using Lemma 1.2. We next make further restrictions on y so as to be able to relate the structure of gr(A) to that of A.

If K is a graded field, then a finite-dimensional graded K-algebra B is said to be graded simple if B has no homogeneous two-sided ideals except B and {0}. We say that B is a graded semisimple K-algebra if B is a direct product of finitely many graded simple K-algebras. By a variation of the ungraded argument, this is equivalent to: B has no nonzero nilpotent homogeneous ideals.

If B is an algebra (resp. graded algebra) over a field (resp. graded field) K, we write [B:K] for dim_{K}(B). Throughout the paper, all semisimple (resp. graded semisimple) algebras are tacitly assumed to be finite-dimensional.

Definition 1.4. Let F be a field with a valuation v. Let y be a surmultiplicative value function on a finite-dimensional F -algebra A. We say that y is an F -gauge (or a v-gauge) on A if y is an F -norm on A (i.e., [gr(A):gr(F)] = [A:F]) and gr(A) is a graded semisimple gr(F)-algebra. Note that if A has an F -gauge then A must be semisimple. For, if A had a nonzero ideal N with N^{2} = {0}, then gr(N) would be a nonzero ideal of gr(A) with gr(N)^{2} = {0}.

For any ring R, let Z(R) denote the center of R.

Definition 1.5. An F -gauge y on a finite-dimensional semisimple F -algebra A is called a tame F -gauge if Z(gr(A)) = gr(Z(A)) and Z(gr(A)) is separable over gr(F). By [HW_{1}, Th. 3.11], the second condition holds if and only if Z(gr(A))_{0} is separable over gr(F)_{0} and char(gr(F)_{0}) ∤ |Γ_{Z(A)}:Γ_{F} |. Thus, the gauge is tame if and only if Z(gr(A)) = gr(Z(A)), \overline{Z(A)} is separable over \overline{F}, and char(\overline{F}) ∤ |Γ_{Z(A)}:Γ_{F} |. It will be shown below (see Cor. 3.7) that whenever char(\overline{F}) = 0 every F -gauge is tame.

The notion of gauge generalizes that of defectless valuation on division algebras, and tame gauge generalizes tame valuation. We make this point clear in §1.2, and give fundamental examples of gauges on endomorphism algebras and on tensor products in §§1.3 and 1.4. We start our discussion of examples with commutative semisimple algebras.

1.1. Gauges on commutative algebras. For a commutative finite-dimensional algebra A over a field F, semisimplicity is equivalent to the absence of nonzero elements x ∈ A such that x^{2} = 0. A similar observation holds for graded algebras. Thus, if F has a valuation v and A has a surmultiplicative v-value function y, the following conditions are equivalent:

(a) gr_{y}(A) is semisimple;

(b) (x′)^{2} ≠ 0 for all nonzero x ∈ A;

(c) (x′)^{n} ≠ 0 for all nonzero x ∈ A and for every positive integer n.

In view of (1.5), these conditions are also equivalent to:

(d) y(x^{2}) = 2y(x) for all x ∈ A;

(e) y(x^{n}) = ny(x) for all x ∈ A.

We first consider the case where A is a field.

Proposition 1.6. Let (F, v) be a valued field and let K/F be a finite-degree field extension. Suppose y: K → Γ ∪ {∞} is a surmultiplicative v-value function such that gr_{y}(K) is semisimple. Then, there exist valuations v_{1}, …, v_{n} on K extending v such that

y(x) = min_{1≤i≤n}(v_{i}(x))  for x ∈ K.  (1.6)

Moreover, there is a natural graded isomorphism of graded gr(F)-algebras

gr_{y}(K) ≅_{g} gr_{v_{1}}(K) × … × gr_{v_{n}}(K).

Proof. Let Γ_{K} = y(K^{×}) ⊆ Γ and Γ_{F} = v(F^{×}) = y(F^{×}) ⊆ Γ_{K}. If x_{1}, …, x_{r} ∈ K are such that y(x_{1}), …, y(x_{r}) belong to different cosets of Γ_{F} in Γ_{K}, then x′_{1}, …, x′_{r} ∈ gr(K) are linearly independent over gr(F), hence x_{1}, …, x_{r} are linearly independent over F, see Prop. 1.1(i). Since [K:F] is finite, it follows that the index |Γ_{K}:Γ_{F} | is finite, hence Γ_{K}/Γ_{F} is torsion.

Let V_{F} and M_{F} denote the valuation ring of F and its maximal ideal, and let

V_{y} = {x ∈ K | y(x) ≥ 0}  and  M_{y} = {x ∈ K | y(x) > 0}.

Clearly, V_{y} is a subring of K containing V_{F} and M_{y} is an ideal of V_{y} containing M_{F}. Since gr_{y}(K) is assumed to be semisimple, we have y(x^{n}) = ny(x) for all x ∈ K (see condition (e) above), hence the ideal M_{y} is radical. We may therefore find a set of prime ideals P_{λ} ⊆ V_{y} (indexed by some set Λ) such that M_{y} = ⋂_{λ∈Λ} P_{λ}. By Chevalley’s Extension Theorem [EP, Th. 3.1.1], we may find for each λ ∈ Λ a

valuation ring V_{λ} of K with maximal ideal M_{λ} such that V_{y} ⊆ V_{λ} and P_{λ} = V_{y} ∩ M_{λ}. Claim: The valuation v_{λ} corresponding to V_{λ} extends v. We have V_{F} ⊆ V_{y} ⊆ V_{λ}, hence V_{F} ⊆ V_{λ} ∩ F. Similarly, M_{F} ⊆ M_{y} ⊆ P_{λ} ⊆ M_{λ}, so M_{F} ⊆ M_{λ} ∩ F. Since V_{λ} ∩ F is a valuation ring of F with maximal ideal M_{λ} ∩ F, the inclusions V_{F} ⊆ V_{λ} ∩ F and M_{F} ⊆ M_{λ} ∩ F imply V_{F} = V_{λ} ∩ F, proving the claim. Hence, each value group Γ_{K,v_{i}} embeds canonically into the divisible group Γ.

Since there are only finitely many extensions of v to K, and since for λ, λ′ ∈ Λ the equality V_{λ} = V_{λ′} implies M_{λ} = M_{λ′}, hence P_{λ} = P_{λ′}, it follows that Λ is a finite set. Let Λ = {1, …, n} and, for x ∈ K, let

w(x) = min_{1≤i≤n}(v_{i}(x)).

Since V_{y} ⊆ ⋂_{i=1}^{n} V_{i}, we have

y(x) ≥ 0 ⇒ w(x) ≥ 0 for x ∈ K.

Similarly, since M_{y} = ⋂_{i=1}^{n} P_{i} = V_{y} ∩ ( ⋂_{i=1}^{n} M_{i}), we have

y(x) > 0 ⟺ (y(x) ≥ 0 and w(x) > 0).

It follows that

y(x) = 0 ⇒ w(x) = 0  for x ∈ K^{×}.  (1.7)

Now, fix some x ∈ K^{×}. Since Γ_{K}/Γ_{F} is a torsion group we may find an integer m > 0 and an element u ∈ F^{×} such that my(x) = v(u), hence y(x^{m}u^{−1}) = 0. By (1.7), we then have w(x^{m}u^{−1}) = 0. Because each v_{i} extends v, we have w(x^{m}u^{−1}) = w(x^{m}) − v(u). Hence,

mw(x) = w(x^{m}) = v(u) = my(x).

Since Γ has no torsion, it follows that w(x) = y(x), which proves (1.6).

For i = 1, …, n we have y(x) ≤ v_{i}(x) for all x ∈ K, hence the identity map on K induces a map gr_{y}(K) → gr_{v_{i}}(K). Combining these maps, we obtain a graded homomorphism of graded gr(F)-algebras

gr_{y}(K) → gr_{v_{1}}(K) × … × gr_{v_{n}}(K).  (1.8)

This map is injective since for every x ∈ K^{×} there is some index i such that y(x) = v_{i}(x). For surjectivity, fix any k, 1 ≤ k ≤ n and any b ∈ K^{×}. Claim: There is c ∈ K^{×} with v_{k}(c) = v_{k}(b) and v_{i}(c) ≥ v_{k}(b) for i ≠ k. This will be proved below.

Now, since the valuations v_{1}, …, v_{n} are incomparable and V_{y} = ⋂_{i=1}^{n} V_{i} by (1.6), the map V_{y} → ∏_{i=1}^{n} V_{i}/M_{i}

is surjective by [EP, Th. 3.2.7(3)]. Therefore, there is d ∈ V_{y} with d ∈ M_{i} for i ≠ k and d − bc^{−1} ∈ M_{k}. Let a = cd. Then, v_{i}(a) > v_{i}(c) ≥ v_{k}(b) for i ≠ k, and v_{k}(a) = v_{k}(b) with a′ = b′ in gr_{v_{k}}(K). Hence, y(a) = v_{k}(b) and a′ ∈ gr_{y}(K) maps to (0, …, 0, b′, 0 …, 0) ∈ gr_{v_{1}}(K) × … × gr_{v_{n}}(K) (b′ in the k-th position). Since these n-tuples span gr_{v_{1}}(K) × … × gr_{v_{n}}(K), it follows that the natural map (1.8) is onto. Proof of Claim. Assume for simplicity that Γ is the divisible hull of Γ_{F} (= Γ_{F,v}, the value group of v on F). For any valuation z on F which is coarser than v there is an associated convex subgroup Δ_{F} ⊆ Γ_{F}, which is the kernel of the canonical epimorphism Γ_{F} → Γ_{F,z}. Let Δ be the divisible hull of Δ_{F} in Γ, and let Λ = Γ/Δ, which is a divisible group with ordering inherited from Γ. Since Γ_{F} ∩Δ = Δ_{F}, the order-preserving inclusion Γ_{F} /Δ_{F} ↪ Λ identifies Γ_{F,z} canonically with a subgroup of the divisible group Λ. Likewise, the value group of every extension of z to K can be viewed as a subgroup of Λ.

For each pair of valuations v_{i}, v_{j} on K with i ≠ j there is a valuation v_{ij} on K which is the finest common coarsening of v_{i} and v_{j}. (v_{ij} is the valuation associated to the valuation ring V_{i}V_{j}.) Let Δ_{ij} ⊆ Γ be the divisible hull of the convex subgroup of Γ_{F} associated to the restriction of v_{ij} to F. Let Γ_{i} = Γ_{K,v_{i}} for 1 ≤ i ≤ n. We say that an n-tuple (β_{1}, …, β_{n}) ∈ Γ_{1} × … × Γ_{n} is compatible if β_{i} − β_{j} ∈ Δ_{ij} for all i ≠ j. (This is equivalent to the definition of compatibility in [R, p. 127], though stated a little differently.) For our fixed b ∈ K^{×}, let γ_{i} = v_{i}(b) ∈ Γ_{i}, and note that (γ_{1}, …, γ_{n}) is compatible since v_{i}(b) and v_{j}(b) have the same image v_{ij}(b) in Γ/Δ_{ij}. For our fixed k and for each i, let ϵ_{i} = γ_{k} − γ_{i}. So, for all i ≠ j, ϵ_{i} − ϵ_{j} = γ_{j} − γ_{i} ∈ Δ_{ij}. Since 0 ≤ ||ϵ_{i}| − |ϵ_{j}|| ≤ |ϵ_{i} − ϵ_{j}| and Δ_{ij} is convex, we have |ϵ_{i}| − |ϵ_{j}| ∈ Δ_{ij}. Because each Γ/Γ_{i} is a torsion group, there is a positive integer m such that m|ϵ_{i}| ∈ Γ_{i} for each i. Let δ_{i} = γ_{i} + m|ϵ_{i}| ∈ Γ_{i}. Then δ_{k} = γ_{k}, and for each i we have

δ_{i} = γ_{i} + m|ϵ_{i}| ≥ γ_{i} + ϵ_{i} = γ_{k} .

Note that (δ_{1}, …, δ_{n}) is compatible, since (γ_{1}, …, γ_{n}) and (m|ϵ_{1}|, …, m|ϵ_{n}|) are compatible. Since the valuations v_{1}, …, v_{n} are incomparable, by [R, Th. 1, p. 135] there is c ∈ K^{×} with v_{i}(c) = δ_{i} for each i. This c has the properties of the claim. □

Let v_{1}, …, v_{r} be all the extensions of v to K. For i = 1, …, r, let e_{i} = e(v_{i}/v) be the ramification index and f_{i} = f(v_{i}/v) be the residue degree. We say that the fundamental equality holds for K/F if

[K:F] = ∑_{i=1}^{r} e_{i}f_{i}.

Corollary 1.7. There is an F -gauge on K if and only if the fundamental equality holds for K/F. When that condition holds, the F -gauge on K is unique and is defined by

y(x) = min_{1≤i≤r}(v_{i}(x))  for x ∈ K,  (1.9)

where v_{1}, …, v_{r} are all the extensions of v to K.

Proof. Suppose y is an F -gauge on K. By Prop. 1.6, we may find some extensions v_{1}, …, v_{n} of v to K such that gr_{y}(K) ≅_{g} gr_{v_{1}}(K) × … × gr_{v_{n}}(K). Now, [gr_{v_{i}}(K):gr(F)] = e_{i}f_{i} by [Bl_{1}, Cor. 2], and [gr_{y}(K):gr(F)] = [K:F] since y is an F -norm, so

[K:F] = ∑_{i=1}^{n} e_{i}f_{i}.

This implies v_{1}, …, v_{n} is the set of all extensions of v to K by [EP, Th. 3.3.4], and the fundamental equality holds. Conversely, if the fundamental equality holds, then formula (1.9) defines an F -gauge on K. □

The following special case will be used in §1.2:

Corollary 1.8. Let K/F be a finite-degree field extension and let v be a valuation on F. Suppose char(\overline{F}) ∤ [K:F]. If v extends uniquely to K, this extension is an F -gauge on K.

Proof. In view of Cor. 1.7, it suffices to show that

[K:F] = [\overline{K}:\overline{F}] |Γ_{K}:Γ_{F} |,  (1.10)

which may be regarded as a (weak) version of Ostrowski’s theorem. We include a proof for lack of a convenient reference. When the equality in (1.10) holds, we say that K/F is defectless.

Let N be a Galois closure of K/F, let z be the extension of v to K, and let w be an extension of z to N. Denote by D the decomposition group of w/v, by R the ramification group, and by N^{D} and N^{R} the corresponding fixed subfields of N. Since

[N^{D}:F] = number of extensions of v to N

= number of extensions of z to N = [KN^{D}:K] ,

we have [K:F] = [KN^{D}:N^{D}]. Also, N^{D}/F and KN^{D}/K are immediate extensions by [EP, Cor. 5.3.8(0)]. Therefore, by substituting N^{D} for F and KN^{D} for K, we may assume that N^{D} = F. Now, N^{R}/F is Galois and [N:N^{R}] is a power of char(\overline{F}) (N = N^{R} if char(\overline{F}) = 0), by [EP, Th. 5.3.3]. Since char(\overline{F}) ∤ [K:F], and [KN^{R}:N^{R}] = [K:K ∩ N^{R}] by Galois theory, we must have K ⊆ N^{R}. By [EP, Cor. 5.3.8], N^{R}/F is defectless, hence K/F is defectless. □

We now turn to the general type of commutative semisimple algebras.

Proposition 1.9. Let K_{1}, …, K_{m} be finite-degree field extensions of a field F and A = K_{1} ×…×K_{m}. Let v be a valuation on F, let y be a surmultiplicative v-value function on A, and let y_{i} = y|_{K_{i}} for i = 1, …, m. If gr_{y}(A) is graded semisimple, then each y_{i} has the form (1.6) and, for a = (a_{1}, …, a_{n}) ∈ A,

y(a) = min_{1≤i≤m}(y_{i}(a_{i})).  (1.11)

Moreover, there is a canonical graded isomorphism of graded gr(F)-algebras

gr_{y}(A) ≅_{g} gr_{y_{1}}(K_{1}) × … × gr_{y_{m}}(K_{m}).  (1.12)

There is a v-gauge on A if and only if the fundamental equality holds for each K_{i}/F. When that condition holds, there is a unique v-gauge y on A, defined by (1.11) where each y_{i} is the unique v-gauge on K_{i} as in Cor. 1.7.

Proof. Let e_{1}, …, e_{m} be the primitive idempotents of A, such that e_{i}A = K_{i}. Since gr_{y}(A) is graded semisimple, we have y(e_{i}^{2}) = 2y(e_{i}), hence y(e_{i}) = 0 for i = 1, …, m. For a ∈ A, we have a = ae_{1} + … + ae_{m}, hence

y(a) ≥ min_{1≤i≤m}(y(ae_{i})).

On the other hand, the surmutliplicativity of y yields

y(ae_{i}) ≥ y(a) + y(e_{i}) = y(a) for i = 1, …, m,

hence

y(a) = min_{1≤i≤m}(y(ae_{i})),

proving (1.11). Since y(a^{2}) = 2y(a) for a ∈ A, we also have y_{i}(a^{2}) = 2y(a) for a ∈ K_{i}, hence gr_{y_{i}}(K_{i}) is semisimple. Therefore, Prop. 1.6 shows that y_{i} has the form (1.6) for i = 1, …, m. The isomorphism (1.12) is clear; it implies

[gr_{y}(A):gr(F)] = ∑_{i=1}^{m}[gr_{y_{i}}(K_{i}):gr(F)],

hence y is a v-gauge if and only if each y_{i} is a v-gauge. The last assertions then follow from Cor. 1.7. □

1.2. Gauges on division algebras. Consider a finite-dimensional (not necessarily central) division algebra D over F. Suppose w is a valuation on D which extends the valuation v on F, and consider the canonical homomorphism

θ_{D}: Γ_{D}/Γ_{F} → Aut(Z(\overline{D}))  (1.13)

which for d ∈ D^{×} maps w(d) + Γ_{F} to the automorphism \overline{z} ↦ \overline{dzd^{−1}} (see [JW, (1.6)]).

Proposition 1.10. With the notation above, the valuation w is an F -gauge on D if and only if

[D:F] = [\overline{D}:\overline{F}] |Γ_{D}:Γ_{F} |.  (1.14)

When this condition holds, the gauge w is tame if and only if Z(\overline{D}) is separable over \overline{F} and char(\overline{F}) ∤ |ker(θ_{D})|. If char(\overline{F}) ∤ [D:F], the following conditions are equivalent:

(i) w is an F -gauge; (ii) w is a tame F -gauge; (iii) v extends uniquely to Z(D).

Proof. Since gr(D) is a graded division algebra, it is graded semisimple. Therefore, w is an F -gauge if and only if [gr(D):gr(F)] = [D:F]. By an easy calculation, cf. [HW_{2}, (1.7)] or [Bl_{2}, p. 4278], we have

[gr(D):gr(F)] = [\overline{D}:\overline{F}] |Γ_{D}:Γ_{F} |.  (1.15)

The first statement follows.

To prove the second statement, assume w is a gauge, hence [gr(D):gr(F)] = [D:F]. Since

[gr(D):gr(F)] = [gr(D):gr(Z(D))] [gr(Z(D)):gr(F)] and [D:F] = [D:Z(D)] [Z(D):F],

and since, by Prop. 1.1,

[D:Z(D)] ≥ [gr(D):gr(Z(D))] and [Z(D):F] ≥ [gr(Z(D)):gr(F)],

it follows that

[D:Z(D)] = [gr(D):gr(Z(D))].  (1.16)

From the definition of θ_{D}, it is clear that Γ_{Z(D)}/Γ_{F} ⊆ ker(θ_{D}), hence there is an induced map

\overline{θ}_{D}: Γ_{D}/Γ_{Z(D)} → Aut(Z(\overline{D})).

Clearly, |ker(θ_{D})| = |ker(\overline{θ}_{D})| |Γ_{Z(D)}:Γ_{F} |. When (1.16) holds, Boulagouaz proved in [Bl_{2}, Cor. 4.4] that Z(gr(D)) = gr(Z(D)) if and only if Z(\overline{D})/\overline{Z(D)} is separable and char(\overline{F}) ∤ |ker(\overline{θ}_{D})|. Therefore, the following statements are equivalent:

(a) Z(gr(D)) = gr(Z(D)), \overline{Z(D)}/\overline{F} is separable and char(\overline{F}) ∤ |Γ_{Z(D)}:Γ_{F} | (i.e., the gauge w is tame);

(b) Z(\overline{D})/\overline{F} is separable and char(\overline{F}) ∤ |ker(θ_{D})|.

The second statement is thus proved.

If v does not extend uniquely to Z(D), then the fundamental inequality for extensions of valuations [B, VI.8.2, Th. 1] yields

[Z(D):F] > [\overline{Z(D)}:\overline{F}] |Γ_{Z(D)}:Γ_{F} |,

hence (1.14) does not hold, and w is not a gauge. This proves (i) ⇒ (iii) (without hypothesis on char(\overline{F})). For the rest of the proof, assume char(\overline{F}) ∤ [D:F]. Then Z(\overline{D})/\overline{F} is separable and char(\overline{F}) ∤ |ker(θ_{D})|, hence (i) ⟺ (ii). Finally, assume (iii). By Cor. 1.8, we have

[Z(D):F] = [\overline{Z(D)}:\overline{F}] |Γ_{Z(D)}:Γ_{F} |.

On the other hand, a noncommutative version of Ostrowski’s theorem [M_{1}, Th. 3] yields

[D:Z(D)] = [\overline{D}:\overline{Z(D)}] |Γ_{D}:Γ_{Z(D)}|.

Therefore, (1.14) holds, and (i) follows. □

In the case where D is central and v is Henselian, various other characterizations of tame F -gauges are given in the following proposition:

Proposition 1.11. With the same notation as above, suppose F = Z(D) and v is Henselian. The following conditions are equivalent:

(i) the valuation w is a tame F -gauge; (ii) D is split by the maximal tamely ramified extension of F; (iii) either char(\overline{F}) = 0 or the char(\overline{F})-primary component of D is split by the maximal unramified

extension of F; (iv) D has a maximal subfield which is tamely ramified over F.

Proof. By [HW_{2}, Prop. 4.3], conditions (ii)–(iv) above are equivalent to: [D:F] = [gr(D):gr(F)] and Z(gr(D)) = gr(F), hence also to (i). □

Definition 1.12. As in [HW_{2}], a central division algebra D over a Henselian valued field F is called tame if the equivalent conditions of Prop. 1.11 hold. Note that by Cor. 3.4 below, w is the unique F -gauge on D (if any exists).

Example 1.13. A non-tame gauge. Let F = ℚ(x, y), the field of rational fractions in two indeterminates over the rationals. The 2-adic valuation of ℚ extends to a valuation v on F with residue field 𝔽_{2}(x, y), see [EP, Cor. 2.2.2]. This valuation further extends to a valuation w on the quaternion algebra D = (x, y)_{F}, with residue division algebra \overline{D} = 𝔽_{2}(x, y)(\sqrt{x}, \sqrt{y}). The valuation w is an F -gauge on D which is not tame.

Example 1.14. Gauges that are not valuations. Let D be the quaternion division algebra (−1, −1)_{ℚ} over the field of rational numbers. This algebra is split by the field ℚ_{3} of 3-adic numbers, hence the 3-adic valuation v on ℚ does not extend to a valuation on D, by [C, Th. 1] or [M_{1}, proof of Th. 2]. Let (1, i, j, k) be the quaternion base of D with i^{2} = j^{2} = −1 and k = ij = −ji, and define a ℚ-norm y on D by

y(a_{0} + a_{1}i + a_{2}j + a_{3}k) = min(v(a_{0}), v(a_{1}), v(a_{2}), v(a_{3})).

(This y is in fact the armature gauge on D with respect to v on ℚ and the abelian subgroup of D^{×}/ℚ^{×} generated by the images of i and j, as described in §4.2 below. But we will not use the §4 results here.) Lemma 1.2 shows that y is a surmultiplicative value function on D. We have gr(ℚ) = 𝔽_{3}[t, t^{−1}], where the indeterminate t is the image of 3. With primes denoting images in gr(D), we have i^{′2} = −1′, j^{′2} = −1′, and i′j′ = k′ = −j′i′. Thus, 1′, i′, j′, k′ span a copy of the 4-dimensional graded simple graded quaternion algebra (−1, −1)_{gr(ℚ)}. Since [gr(D):gr(ℚ)] = [D:ℚ] = 4, we have

gr(D) = (−1, −1)_{gr(ℚ)} ≅ (−1, −1)_{𝔽_{3}} ⊗_{𝔽_{3}} gr(ℚ) ≅ M_{2}(𝔽_{3})[t, t^{−1}] .

Thus, y is a tame gauge on D with D_{0} ≅ M_{2}(𝔽_{3}) and Γ_{D} = Γ_{ℚ} = ℤ. One can obtain other gauges on D by conjugation, by using Prop. 1.15 below.

The residue ring D_{0} in the preceding example is a simple ring, though not a division ring. Here is an example where the residue is not simple: Let k be any field of chacteristic not 2, let F = k(x, z), where x and z are algebraically independent over k. Let Q be the quaternion division algebra (1 + x, z)_{F}. Let v be the valuation on F obtained by restriction from the standard Henselian valuation on k((x))((z)). So, Γ_{F} = ℤ × ℤ with right-to-left lexicographic ordering, with v(x) = (1, 0) and v(z) = (0, 1). Again let (1, i, j, k) denote the quaternion base of Q with i^{2} = 1 + x, j^{2} = y, and k = ij = −ji. Define an F -norm y on Q by, for a_{0}, …, a_{3} ∈ F,

y(a_{0} + a_{1}i + a_{2}j + a_{3}k) = min (v(a_{0}), v(a_{1}), v(a_{2}) + (0, \frac{1}{2}), v(a_{3}) + (0, \frac{1}{2})) .

This y is the armature gauge of Q with respect to v and the abelian subgroup of Q^{×}/F^{×} generated by the images of i and j. By using results in §4.2 or by easy direct calculations, one sees that y is a tame gauge on Q with gr(Q) graded isomorphic to the graded quaternion algebra (1, y′)_{gr(F)}, with Γ_{Q} = ℤ × \frac{1}{2}ℤ, and Q_{0} ≅ k × k, which is clearly not simple. let is

A gauge y on a division ring D is a valuation iff gr_{y}(D) is a graded division ring. When this occurs, the gauge is invariant under conjugation. But for a gauge which is not a valuation, the associated graded ring is not a graded division ring, so it has nonzero homogenous elements which are not units. Then, conjugation yields different gauges, on D, as the next proposition shows:

Proposition 1.15. Let (F, v) be a valued field and let y be a v-gauge on a central simple F -algebra A. For any unit u ∈ A^{×}, the following conditions are equivalent:

(i) u′ is a unit of gr(A); (ii) y(uxu^{−1}) = y(x) for all x ∈ A.

Proof. (i) ⇒ (ii) If u′ is a unit of gr(A), then u′ is not a zero divisor. So, u′(u^{−1})′ = (uu^{−1})′ = 1′ in gr(A) by (1.5). Hence, (u^{−1})′ = (u′)^{−1}, which is a unit of gr(A). It follows by Lemma 1.3 that y(u) + y(u^{−1}) = 0 and for any x ∈ A,

y(uxu^{−1}) = y(ux) + y(u^{−1}) = y(u) + y(x) + y(u^{−1}) = y(x) .

(ii) ⇒ (i) The surmultiplicativity of y yields

y(u) + y(u^{−1}) ≤ y(1) = 0.

If equality holds here, then (i) follows by Lemma 1.3. Therefore, if (ii) holds and (i) does not, then for all x ∈ A

y(uxu^{−1}) = y(x) > y(u) + y(x) + y(u^{−1}).

By (1.5), it follows that u′x′(u^{−1})′ = 0 for all x ∈ A, hence u′gr(A)(u^{−1})′ = {0} since gr(A) is spanned by its homogeneous elements. This equation shows that gr(A)u′gr(A) ≠ gr(A) since (u^{−1})′ ≠ 0. On the other hand, gr(A)u′gr(A) ≠ {0} since u′ ≠ 0, hence the 2-sided homogeneous ideal gr(A)u′gr(A) is not trivial, and gr(A) is not graded simple. This is a contradiction to Cor. 3.8 below. (Observe that Prop. 1.15 is not used in the sequel; thus, the argument is not circular.) □

1.3. Gauges on endomorphism algebras. Let D be a finite-dimensional division algebra over a field F, let M be a finite-dimensional right D-vector space, and let A = End_{D}(M). Suppose w: D → Γ ∪ {∞} is a valuation on D, and let v = w|_{F}. Let α be a D-norm on M.

Lemma 1.16. Let (m_{i})_{1≤i≤k} be a splitting base of M for α. For every f ∈ A and nonzero m ∈ M,

α(f(m)) − α(m) ≥ min_{1≤i≤k}(α(f(m_{i})) − α(m_{i})).

Proof. Let m = ∑_{i=1}^{k} m_{i}d_{i} with d_{i} ∈ D. Then

α(f(m)) = α(∑_{i=1}^{k} f(m_{i})d_{i}) ≥ min_{1≤i≤k}(α(f(m_{i})) + w(d_{i})).  (1.17)

Writing α(f(m_{i})) + w(d_{i}) = α(f(m_{i})) − α(m_{i}) + α(m_{i}) + w(d_{i}), we obtain

min_{1≤i≤k}(α(f(m_{i})) + w(d_{i})) ≥ min_{1≤i≤k}(α(f(m_{i})) − α(m_{i})) + min_{1≤i≤k}(α(m_{i}) + w(d_{i})) .  (1.18)

Since (m_{i})_{1≤i≤k} is a splitting base of M, the second term on the right side is α(m). The lemma then follows from (1.17) and (1.18). □

In view of the lemma, we may define a function y_{α}: A → Γ ∪ {∞} as follows: for f ∈ A,

y_{α}(f) = min_{m∈M, m≠0}(α(f(m)) − α(m)) .  (1.19)

Indeed, the lemma shows that y_{α}(f) = min_{1≤i≤k}(α(f(m_{i})) − α(m_{i})) for any splitting base (m_{i})_{1≤i≤k} of M.

If f(m_{j}) = ∑_{i=1}^{k} m_{i}d_{ij} with d_{ij} ∈ D, we have,

y_{α}(f) = min_{1≤i,j≤k}(α(m_{i}) + w(d_{ij}) − α(m_{j})) .  (1.20)

Now, let E = End_{gr(D)}(gr(M)). Recall that E is graded as follows: for γ ∈ Γ,

E_{γ} = {f ∈ E | f(M_{δ}) ⊆ M_{δ+γ} for all δ ∈ Γ}.

Proposition 1.17. The map y_{α} of (1.19) is a surmultiplicative F -value function on A, and there is a canonical gr(F)-algebra isomorphism gr(A) ≅_{g} E. Moreover, y_{α} is an F -gauge (resp. a tame F -gauge) on A if and only if w is an F -gauge (resp. a tame F -gauge) on D.

Proof. We omit the easy proof that y_{α} is a surmultiplicative F -value function on A. To define the canonical isomorphism gr(A) → E, take any f ∈ A with f ≠ 0, and let γ = y_{α}(f) ∈ Γ. The definition of y_{α} says that α(f(m)) ≥ α(m) + γ for all m ∈ M, and equality holds for some nonzero m. For any δ ∈ Γ, this shows f maps M^{≥δ} into M^{≥δ+γ} and M^{>δ} into M^{>δ+γ}; so, f induces a well-defined additive group homomorphism

f̃_{δ}: M_{δ} → M_{δ+γ}, m′ ↦ f(m) + M^{>δ+γ} for all m ∈ M with α(m) = δ.

Define f̃ = ⨁_{δ∈Γ} f̃_{δ}: gr(M) → gr(M), an additive group homomorphism which shifts graded components

by γ. For any d ∈ D^{×} and nonzero m ∈ M, we have

f̃_{α(m)+w(d)}(m′ · d′) = f̃_{α}(m′) · d′ ∈ M_{α(m)+w(d)+γ}.

Thus, f̃ ∈ E_{γ}, and the definition of y_{α} assures that f̃ ≠ 0. Since for any g ∈ A^{>γ} we have { \atop f + g} = f̃, there is a well-defined injective map A_{γ} → E_{γ} given by f + A^{>γ} ↦ f̃. The direct sum of these maps is a graded (i.e., grade-preserving) gr(F)-algebra homomorphism ρ: gr(A) → E; this map is injective, since it is injective on each homogeneous component of gr(A). To prove ρ is onto, let (m_{i})_{1≤i≤k} be a splitting base of M with respect to α; so, (m′_{i})_{1≤i≤k} is a homogeneous gr(D)-base of gr(M). If we fix any i, j with 1 ≤ i, j ≤ k and any d ∈ D^{×}, and define g ∈ A by g(m_{j}) = m_{i}d and g(m_{ℓ}) = 0 for ℓ ≠ j, then y_{α}(g) = w(d) + α(m_{i}) − α(m_{j}); so, g̃(m′_{j}) = (m_{i}d) + A^{>w(d)+α(m_{i})−α(m_{j})+α(m_{j})} = m′_{i} · d′ ∈ M_{w(d)+α(m_{i})} and g̃(m′_{ℓ}) = 0 for ℓ ≠ j. Since such maps generate E as an additive group, ρ is onto, hence an isomorphism, as desired.

It follows from the isomorphism gr(A) ≅_{g} E that gr(A) is a graded simple gr(F)-algebra. Thus, y_{α} is an F -gauge on A iff y_{α} is an F -norm on A, iff, by Prop. 1.1(ii), [gr(A):gr(F)] = [A:F]. We have

[A:F] = k^{2}[D:F]  and  [gr(A):gr(F)] = [E:gr(F)] = k^{2}[gr(D):gr(F)].

Thus, y_{α} is an F -norm on A iff [gr(D):gr(F)] = [D:F] iff, by (1.15) and Prop. 1.10, w is an F -gauge on D.

Since Z(A) = Z(D) (up to canonical isomorphism) with y_{α}|_{Z(A)} = w|_{Z(D)} and Z(gr(A)) ≅_{g} Z(gr(D)) (a gr(F)-algebra isomorphism), y_{α} is a tame F -gauge on A iff w is a tame F -gauge on D. □

We now compare the “gauge ring” V_{A} = {f ∈ A | y_{α}(f) ≥ 0} with the valuation rings V_{D} and V_{F}.

Lemma 1.18. The following conditions are equivalent:

(i) V_{A} is integral over V_{F}; (ii) V_{D} is integral over V_{F}; (iii) the valuation v on F has a unique extension to Z(D).

In particular, these conditions all hold if v is Henselian.

Proof. (i) ⟺ (ii) Let (m_{i})_{1≤i≤k} be a splitting base of M with respect to α, and let f ∈ A. If

f(m_{j}) = ∑_{i=1}^{k} m_{i}d_{ij}, eq. (1.20) shows that f ∈ V_{A} iff w(d_{ij}) ≥ α(m_{i}) − α(m_{j}) for all i, j. Let e_{1}, …, e_{k} be

the orthogonal idempotents of A for the splitting base (m_{i})_{1≤i≤k} of M, i.e., e_{i}(m_{j}) = δ_{ij}m_{i} (Kronecker delta). We have y_{α}(e_{i}) = 0, so each e_{i} ∈ V_{A}. For f with matrix (d_{ij}) as above, e_{i}fe_{i} has matrix with i i-entry d_{ii} and all other entries 0. Hence, e_{i}V_{A}e_{i} ≅ V_{D}, a V_{F} -algebra isomorphism. If V_{A} is integral over V_{F}, then V_{D} ≅ e_{1}V_{A}e_{1} must also be integral over V_{F}. Conversely, suppose V_{D} is integral over V_{F}. Then, each e_{i}V_{A}e_{i} is integral over V_{F}, since it is isomorphic to V_{D}. For j ≠ i, each element of e_{i}V_{A}e_{j}

has square 0, so is integral over V_{F}. Since ∑_{i=1}^{k} e_{i} = 1, we have V_{A} = ∑_{1≤i,j≤k} e_{i}V_{A}e_{j}, with the elements

of each summand integral over V_{F}. Because A is a p.i.-algebra over F, the theorem [AS, Th. 2.3] of Amitsur and Small shows that V_{A} is integral over V_{F}.

(ii) ⟺ (iii) Let Z = Z(D), the center of D, and let V_{Z} be the valuation ring of w|_{Z}. Then V_{D} is always integral over V_{Z}, by [W_{1}, Cor.], and V_{Z} is integral over V_{F} iff w|_{Z} is the unique extension of the valuation v to Z, see [B, Ch. VI, §8.3, Remark] or [EP, Cor. 3.1.4]. □

Remark 1.19. The value function y_{α} on End_{D}(M) associated to a norm α on M is defined by Bruhat and Tits in [BT, 1.11, 1.13], where it is denoted by End α and called a norme carrée. In [BT, 2.13, Cor.] (see also the Appendix of [BT]), Bruhat and Tits establish a bijection between the set of normes carrées on End_{D}(M) and the building of GL(M), when the rank of the ordered group Γ is 1.

The following result is in [BT, 1.13]:

Proposition 1.20. For D-norms α, β on M, the following conditions are equivalent:

(i) α − β is constant, i.e. there exists γ ∈ Γ such that α(m) = β(m) + γ for all m ∈ M; (ii) y_{α} = y_{β}.

Proof. (i) ⇒ (ii) is clear from the definition (1.19). To prove (ii) ⇒ (i), choose any m, n ∈ M with m ≠ 0 and let A(m, n) = {f ∈ A | f(m) = n}. By definition of y_{α}, we have

y_{α}(f) ≤ α(n) − α(m) for f ∈ A(m, n).

On the other hand, we may choose a splitting base (m_{i})_{1≤i≤k} of M with m_{1} = m (see [RTW, Prop. 2.5]) and define g ∈ A(m, n) by g(m_{1}) = n, g(m_{i}) = 0 for i > 1; then y_{α}(g) = α(n) − α(m). Therefore,

α(n) − α(m) = max_{g∈A(m,n)}(y_{α}(g)).  (1.21)

If (ii) holds, it follows from (1.21) that

α(n) − α(m) = β(n) − β(m) for all m, n ∈ M with m ≠ 0.

Condition (i) readily follows. □

1.4. Gauges on tensor products. If P and Q are two graded vector spaces over a graded field K, then the grading on P ⊗_{K} Q is given by (P ⊗_{K} Q)_{γ} = ∑_{δ∈Γ} P_{δ} ⊗_{K_{0}} Q_{γ−δ}.

Proposition 1.21. Let (F, v) be a valued field, and let M and N be F -vector spaces such that M has an F -norm α and N has an F -value function β. There is a unique F -value function t on M ⊗_{F} N such that there is a graded isomorphism of gr(F)-vector spaces

gr_{t}(M ⊗_{F} N) ≅_{g} gr_{α}(M) ⊗_{gr(F)} gr_{β}(N)

satisfying (m⊗n)′ ↦ m′ ⊗n′ for all m ∈ M and n ∈ N. So, t(m⊗n) = α(m)+β(n). If β is an F -norm, then t is also an F -norm.

The unique value function t on M ⊗_{F} N satisfying the condition in the proposition will be denoted α ⊗ β, and the canonical isomorphism of graded vector spaces will be viewed as an identification.

The following lemma will be used in the proof of Prop. 1.21.

Lemma 1.22. Let D be a division ring with a valuation w, and let P be a right D-vector space. Let u and t be D-value functions P → Γ ∪ {∞} such that t(p) ≤ u(p) for all p ∈ P. Then, there is a canonical induced gr(D)-vector space homomorphism χ_{t,u}: gr_{t}(P) → gr_{u}(P) which is injective iff t = u.

Proof. Clear. □

Proof of Prop. 1.21. Let (m_{i})_{1≤i≤k} be a splitting base of M with respect to α. Define t: M ⊗_{F} N → Γ ∪ {∞} by

t(∑_{i=1}^{k} m_{i} ⊗ n_{i}) = min_{1≤i≤k}(α(m_{i}) + β(n_{i}))  for any n_{i} ∈ N.  (1.22)

Clearly, t is an F -value function on M ⊗_{F} N, and

(M ⊗_{F} N)^{≥γ} = ⨁_{i=1}^{k} m_{i} ⊗ N^{≥γ−α(m_{i})} for γ ∈ Γ.

Also, since (m′_{i})_{1≤i≤k} is a homogeneous gr(F)-base of gr(M), we have

(gr(M) ⊗_{gr(F)} gr(N))_{γ} = ⨁_{i=1}^{k} m′_{i} ⊗ N_{γ−α(m_{i})}.

For γ ∈ Γ, let π_{γ}: N^{≥γ} → N_{γ} be the canonical map. Define a surjective map ψ_{γ}: (M ⊗_{F} N)^{≥γ} → (gr(M) ⊗_{gr(F)} gr(N))_{γ} as follows:

ψ_{γ}(∑_{i=1}^{k} m_{i} ⊗ n_{i}) = ∑_{i=1}^{k} m′_{i} ⊗ π_{γ−α(m_{i})}(n_{i}) for n_{i} ∈ N^{≥γ−α(m_{i})}.

This ψ_{γ} is clearly an additive group homomorphism. Moreover, ψ_{γ}(∑ m_{i}⊗n_{i}) = 0 iff each n_{i} ∈ N^{>γ−α(m_{i})}, iff ∑ m_{i}⊗n_{i} ∈ (M⊗_{F} N)^{>γ}. Thus, ψ_{γ} induces a group isomorphism φ_{γ}: (M ⊗_{F} N)_{γ} → (gr(M) ⊗_{gr(F)} gr(N))_{γ} and φ = ⊕_{γ∈Γ} φ_{γ} is a graded gr(F)-vector space isomorphism gr_{t}(M ⊗_{F} N) → gr_{α}(M) ⊗_{gr_{v}(F)} gr_{β}(N).

To see that φ((m ⊗ n)′) = m′ ⊗ n′ for m ∈ M and n ∈ N, let m = ∑_{i=1}^{k} m_{i}r_{i} with r_{i} ∈ F. Then

m ⊗ n = ∑_{i=1}^{k} m_{i} ⊗ r_{i}n and α(m) = min{α(m_{i}) + v(r_{i})}, so

t(m ⊗ n) = min_{1≤i≤k}(α(m_{i}) + β(n) + v(r_{i})) = α(m) + β(n)

and

φ((m ⊗ n)′) = ψ_{α(m)+β(n)}(∑_{i=1}^{k} m_{i} ⊗ r_{i}n) = ∑_{i=1}^{k} m′_{i} ⊗ π_{α(m)−α(m_{i})+β(n)}(r_{i}n).

Note that

π_{α(m)−α(m_{i})+β(n)}(r_{i}n) = \begin{cases}r′_{i}n′ & if α(m_{i}) + v(r_{i}) = α(m), \\ 0 & if α(m_{i}) + v(r_{i}) > α(m). \end{cases}

Changing the indexing if necessary, we may assume α(m_{i}) + v(r_{i}) = α(m) for i = 1, …, ℓ and

α(m_{i}) + v(r_{i}) > α(m) for i > ℓ. Then m′ = ∑_{i=1}^{ℓ} m′_{i}r′_{i} and

φ((m ⊗ n)′) = ∑_{i=1}^{ℓ} m′_{i} ⊗ r′_{i}n′ = m′ ⊗ n′ ,

as desired.

Now, suppose u: M ⊗_{F} N → Γ ∪ {∞} is an F -value function with the same property. For m ∈ M and n ∈ N we have u(m ⊗ n) = α(m) + β(n), since the degree of (m ⊗ n)′ in gr_{u}(M ⊗ N) is the same as the degree of m′ ⊗ n′ in gr_{α}(M) ⊗ gr_{β}(N). Therefore, for n_{i} ∈ N we have

u(∑_{i=1}^{k} m_{i} ⊗ n_{i}) ≥ min_{1≤i≤k}(u(m_{i} ⊗ n_{i})) = min_{1≤i≤k}(α(m_{i}) + β(n_{i})) = t(∑_{i=1}^{k} m_{i} ⊗ n_{i}) .  (1.23)

We have graded gr(F)-vector space isomorphisms φ_{t}: gr_{t}(M ⊗_{F} N) → gr_{α}(M) ⊗_{gr(F)} gr_{β}(N) and φ_{u}: gr_{u}(M ⊗_{F} N) → gr_{α}(M) ⊗_{gr(F)} gr_{β}(N). Because of the inequality in (1.23), Lemma 1.22 yields a canonical gr(F)-vector space homomorphism χ_{t,u}: gr_{t}(M ⊗_{F} N) → gr_{u}(M ⊗_{F} N). Our hypotheses on φ_{t} and φ_{u} imply that φ_{u} ∘ χ_{t,u} and φ_{t} agree on (m ⊗ n)′ for all m ∈ M and n ∈ N. Since such (m ⊗ n)′ form a generating set for gr_{t}(M ⊗_{F} N), we have φ_{u} ∘ χ_{t,u} = φ_{t}. Then, χ_{t,u} is an isomorphism, since φ_{t} and φ_{u} are each isomorphisms. Lemma 1.22 then shows that u = t, proving the desired uniqueness of t.

If β is an F -norm on N, say with splitting base (n_{i})_{1≤i≤ℓ}, then it follows easily from (1.22) that (m_{i} ⊗ n_{j})\begin{matrix}1≤i≤k \\ 1≤j≤ℓ \end{matrix} is a splitting base for t on M ⊗_{F} N, so t is an F -norm for M ⊗_{F} N. □

Remark 1.23. A basis-free description of α ⊗ β is stated in [BT, p. 269]: for s ∈ M ⊗ N,

α ⊗ β(s) = sup{ min_{1≤j≤ℓ}(α(p_{j}) + β(q_{j})) | s = ∑_{j=1}^{ℓ} p_{j} ⊗ q_{j} } .

To see this equality, for s ∈ M ⊗_{F} N, let u(s) = sup{ min_{1≤j≤ℓ}(α(p_{j}) + β(q_{j})) | s = ∑_{j=1}^{ℓ} p_{j} ⊗ q_{j} }. Take any

representation for s as ∑_{j=1}^{ℓ} p_{j} ⊗q_{j}. Write each p_{j} = ∑_{i=1}^{k} m_{i}r_{ij} with r_{ij} ∈ F. Then, s = ∑_{i=1}^{k} m_{i}⊗( ∑_{j=1}^{ℓ} r_{ij}q_{j}),

and

min_{1≤j≤ℓ}(α(p_{j}) + β(q_{j})) = min_{1≤j≤ℓ}(α(∑_{i=1}^{k} m_{i}r_{ij}) + β(q_{j})) = min_{1≤j≤ℓ}(( min_{1≤i≤k}α(m_{i}) + v(r_{ij})) + β(q_{j}))

= min_{i, j}(α(m_{i}) + v(r_{ij}) + β(q_{j})) = min_{1≤i≤k}(α(m_{i}) + min_{1≤j≤ℓ}(v(r_{ij}) + β(q_{j}))

≤ min_{1≤i≤k}(α(m_{i}) + β( ∑_{j=1}^{ℓ} r_{ij}q_{j})) = α ⊗ β( ∑_{i=1}^{k} m_{i} ⊗ ( ∑_{j=1}^{ℓ} r_{ij}q_{j})) = α ⊗ β(s) .

So, α ⊗ β(s) is an upper bound for the quantities in the description of u(s). But, by using the repre-

sentation s = ∑_{i=1}^{k} m_{i} ⊗ ( ∑_{j=1}^{ℓ} r_{ij}q_{j}), we see that α ⊗ β(s) is one of those quantities. Hence, the sup exists,

and u(s) = α ⊗ β(s).

Corollary 1.24. Suppose (F, v) is a valued field and A is a semisimple F -algebra with an F -gauge y. Let (L, w) be any valued field extending (F, v). Then, y ⊗ w is a surmultiplicative L-value function on A ⊗_{F} L and the canonical isomorphism

gr(A ⊗_{F} L) ≅_{g} gr(A) ⊗_{gr(F)} gr(L)

is an isomorphism of gr(L)-algebras. Moreover, y ⊗ w is an L-gauge iff gr(A) ⊗_{gr(F)} gr(L) is semisimple, iff Z(gr(A)) ⊗_{gr(F)} gr(L) is a direct sum of graded fields. Furthermore, y ⊗ w is a tame L-gauge iff y is a tame F -gauge.

Proof. Let z = y ⊗ w. Prop. 1.21 shows that z is a well-defined F -value function on A ⊗_{F} L with z(a ⊗ ℓ) = y(a) + w(ℓ) for all a ∈ A, ℓ ∈ L. This equation shows that z is actually an L-value function. Moreover, if (a_{i})_{1≤i≤k} is an F -splitting base of A with respect to y, then formula (1.22) shows that (a_{i} ⊗ 1)_{1≤i≤k} is an L-splitting base of A ⊗_{F} L with respect to z. Since

z((a_{i} ⊗ 1)(a_{j} ⊗ 1)) = z(a_{i}a_{j} ⊗ 1) = y(a_{i}a_{j}) ≥ y(a_{i}) + y(a_{j}) = z(a_{i} ⊗ 1) + z(a_{j} ⊗ 1) ,

Lemma 1.2 shows that z is surmultiplicative. The value-function compatible F -algebra homomorphism A → A ⊗_{F} L induces a graded gr(F)-algebra homomorphism gr(A) → gr(A ⊗_{F} L), and hence a graded gr(L)-algebra homomorphism φ: gr(A)⊗_{gr(F)} gr(L) → gr(A⊗_{F} L). This φ is bijective, since it coincides with the gr(F)-vector space isomorphism of Prop. 1.21. Since z satisfies all the other conditions for an L-gauge, z is an L-gauge iff gr(A ⊗_{F} L) is a graded semisimple gr(L)-algebra, iff gr(A) ⊗_{gr(F)} gr(L) is a graded semisimple gr(L)-algebra. We have gr(A)⊗_{gr(F)}gr(L) ≅_{g} gr(A)⊗_{Z(gr(A))}(Z(gr(A))⊗_{gr(F)}gr(L)). Since gr(A) is semisimple, gr(A) ⊗_{gr(F)} gr(L) is semisimple iff Z(gr(A)) ⊗_{gr(F)} gr(L) is a direct sum of graded fields. This is justified just as in the ungraded analogue by reducing to the simple case and using the fact [HW_{2}, Prop. 1.1] that if B is a graded central simple algebra over a graded field K and M is any graded field extension of K, then B ⊗_{K} M is a graded central simple algebra over M.

Applying the first part of the corollary to Z(A) with the gauge y|_{Z(A)}, we obtain

gr(Z(A ⊗_{F} L)) = gr(Z(A) ⊗_{F} L) ≅_{g} gr(Z(A)) ⊗_{gr(F)} gr(L).

On the other hand,

Z(gr(A ⊗_{F} L)) ≅_{g} Z(gr(A) ⊗_{gr(F)} gr(L)) = Z(gr(A)) ⊗_{gr(F)} gr(L),  (1.24)

hence

[gr(Z(A ⊗ L)):gr(L)] = [gr(Z(A)):gr(F)] and [Z(gr(A ⊗ L)):gr(L)] = [Z(gr(A)):gr(F)]. (1.25)

Since gr(Z(A ⊗ L)) ⊆ Z(gr(A ⊗ L)), these finite-dimensional gr(L)-algebras coincide iff they have the same dimension. Similarly, gr(Z(A)) = Z(gr(A)) iff [gr(Z(A)):gr(F)] = [Z(gr(A)):gr(F)]. Therefore, (1.25) shows that

gr(Z(A ⊗ L)) = Z(gr(A ⊗ L)) iff gr(Z(A)) = Z(gr(A)).

Moreover, since gr(L) is a free gr(F)-module, it follows from (1.24) that Z(gr(A ⊗ L)) is separable over gr(L) iff Z(gr(A)) is separable over gr(F), by [KO, Ch. III, Prop. 2.1 and 22]. Therefore, y ⊗ w is a tame L-gauge iff y is a tame F -gauge. □

Remark 1.25. In the context of Cor. 1.24, since gr(A) is semisimple, Z(gr(A)) is a direct sum of graded fields finite-dimensional over gr(F). If each of these graded fields is separable over gr(F), or if gr(L) is separable over gr(F), then Z(gr(A)) ⊗_{gr(F)} gr(L) is a direct sum of graded fields. Recall from [HW_{1}, Th. 3.11, Def. 3.4] that if [gr(L):gr(F)] < ∞, then gr(L) is separable over gr(F) iff \overline{L} is separable over \overline{F} and char(\overline{F}) ∤ |Γ_{L}:Γ_{F} |.

Corollary 1.26. Suppose (F, v) is a valued field and A and B are semisimple F -algebras with respective F -gauges y and z. If gr(A) ⊗_{gr(F)} gr(B) is graded semisimple, then y ⊗ z is an F -gauge on A ⊗_{F} B, and the canonical isomorphism

gr(A ⊗_{F} B) ≅_{g} gr(A) ⊗_{gr(F)} gr(B)

is an isomorphism of graded gr(F)-algebras. Moreover gr(A) ⊗_{gr(F)} gr(B) is graded semisimple iff Z(gr(A)) ⊗_{gr(F)} Z(gr(B)) is a direct sum of graded fields. If A and B are central simple and y, z are tame gauges, then y ⊗ z is a tame gauge.

Proof. Since y is an F -norm on A, say with splitting base (a_{i})_{1≤i≤k}, and z is an F -norm on B, say with splitting base (b_{j})_{1≤j≤ℓ}, we saw in the proof of Prop. 1.21 that (a_{i} ⊗ b_{j})\begin{matrix}1≤i≤k \\ 1≤j≤ℓ \end{matrix} is a splitting base for the

F -norm y ⊗ z on A ⊗_{F} B. For any a_{i}, a_{p}, b_{j}, b_{q}, we have, by Prop. 1.21,

(y ⊗ z)((a_{i} ⊗ b_{j}) · (a_{p} ⊗ b_{q})) = (y ⊗ z)(a_{i}a_{p} ⊗ b_{j}b_{q}) = y(a_{i}a_{p}) + z(b_{j}b_{q}) ≥ y(a_{i}) + y(a_{p}) + z(b_{j}) + z(b_{q}) = (y ⊗ z)(a_{i} ⊗ b_{j}) + (y ⊗ z)(a_{p} ⊗ b_{q}) .

Therefore, Lemma 1.2 shows that y ⊗ z is surmultiplicative. We have gr(A ⊗_{F} B) ≅_{g} gr(A) ⊗_{gr(F)} gr(B) by Prop. 1.21. Thus, y ⊗ z is an F -gauge on A ⊗_{F} B iff gr(A) ⊗_{gr(F)} gr(B) is graded semisimple. Since gr(A) ⊗_{gr(F)} gr(B) ≅_{g} gr(A) ⊗_{Z(gr(A))} [Z(gr(A)) ⊗_{gr(F)} Z(gr(B))] ⊗_{Z(gr(B))} gr(B), the desired graded semisimplicity holds iff Z(gr(A)) ⊗_{gr(F)} Z(gr(B)) is a direct sum of graded fields. This follows just as in the ungraded case, using [HW_{1}, Prop. 1.1].

The last statement is immediate since a graded tensor product of graded central simple gr(F)-algebras is graded central simple over gr(F), by [HW_{2}, Prop. 1.1]. □

Note that the constructions in §§1.3 and 1.4 could be done in more generality. For instance, the valuation w in §1.3 could be replaced by a gauge, and in §1.4 the tensor products could be taken over division algebras instead of fields. Moreover, given a norm α on a right D-vector space M one can define a dual norm α^{∗} on the left D-vector space M^{∗} = Hom_{D}(M, D) and check that the tensor product α ⊗α^{∗} corresponds to the gauge y_{α} under the canonical isomorphism M ⊗_{D} M^{∗} ≅ End_{D}(M). (See [BT, §1].)

2. Graded central simple algebras

Let K be a graded field, and let B be a (finite-dimensional) graded central simple K-algebra. By the graded version of Wedderburn’s theorem, see, e.g., [HW_{2}, Prop. 1.3], there is a (finite-dimensional) graded central K-division algebra E and a finite-dimensional graded right E-vector space N such that B ≅_{g} End_{E}(N). We identify B with End_{E}(N). The grading on B is given as follows: for any ϵ ∈ Γ,

B_{ϵ} = {f ∈ B | f(N_{δ}) ⊆ N_{δ+ϵ} for all δ ∈ Γ}.

Since E is a graded division ring, E_{0} is a division ring, and for each γ ∈ Γ_{E}, E_{γ} is a 1-dimensional left and right E_{0}-vector space. The grade set of N, Γ_{N} = { γ ∈ Γ | N_{γ} ≠ {0} }, need not be a group, but it is clearly a union Γ_{N} = Γ_{1} ∪ … ∪ Γ_{k} where each Γ_{i} is a (non-empty) coset of the group Γ_{E}. Then, N has a canonical direct sum decomposition into graded E-vector subspaces

N = ⨁_{i=1}^{k} N_{Γ_{i}}  where N_{Γ_{i}} = ⨁_{γ∈Γ_{i}} N_{γ}.  (2.1)

For each coset Γ_{i}, choose and fix a representative γ_{i} ∈ Γ_{i}.

Proposition 2.1. The grade set of B is

Γ_{B} = ⋃_{i,j=1}^{k}(γ_{i} − γ_{j}) + Γ_{E}  (2.2)

and there is a canonical isomorphism of K_{0}-algebras

B_{0} ≅ ∏_{i=1}^{k} End_{E_{0}}(N_{γ_{i}}).  (2.3)

Moreover,

dim_{E}(N) = ∑_{i=1}^{k} dim_{E_{0}}(N_{γ_{i}}).  (2.4)

Proof. Eq. (2.2) readily follows from the description of the grading on B, and (2.4) from (2.1), since N_{Γ_{i}} = N_{γ_{i}} ⊗_{E_{0}} E. To prove (2.3), observe that every element in B_{0} maps each N_{Γ_{i}} to itself, so

B_{0} = ∏_{i=1}^{k} End_{E}(N_{Γ_{i}})_{0}.

Since N_{Γ_{i}} = N_{γ_{i}} ⊗_{E_{0}} E, restriction to N_{γ_{i}} defines a canonical isomorphism of K_{0}-algebras End_{E}(N_{Γ_{i}})_{0} ≅ End_{E_{0}}(N_{γ_{i}}). The proof is thus complete. □

This proposition shows that B_{0} is in general semisimple but not simple; however all its simple components are equivalent to E_{0} in the Brauer group Br(Z(E_{0})). Also, the grade set Γ_{B} is in general not a group. We next show how Γ_{E} can be detected within Γ_{B}.

Let H_{B} be the multiplicative group of homogeneous units of B and let Δ_{B} ⊆ Γ_{B} be the image of H_{B} under the grade homomorphism mapping each nonempty H_{B} ∩ B_{γ} to γ. The group action of H_{B} by conjugation on B preserves the grading, so sends B_{0}, hence also Z(B_{0}), to itself. If b, c ∈ H_{B} ∩ B_{γ}, then b^{−1}c ∈ B_{0}^{×} centralizes Z(B_{0}), hence b and c have the same action on Z(B_{0}). Therefore, there is a well-defined homomorphism

θ_{B}: Δ_{B}/Γ_{K} → Aut(Z(B_{0})/K_{0}),  (2.5)

which maps γ + Γ_{K} to z ↦ bzb^{−1} for z ∈ Z(B_{0}) and b ∈ H_{B} ∩ B_{γ}. Of course, if B = E then Δ_{B} = Γ_{E}. The homomorphism θ_{B} then coincides with the homomorphism θ_{E} defined in [HW_{2}, (2.2)].

Proposition 2.2. Let e be any primitive idempotent of Z(B_{0}). Then

Γ_{E} = {γ ∈ Δ_{B} | θ_{B}(γ + Γ_{K})(e) = e}.

Moreover, the following diagram is commutative:

Γ_{E}/Γ_{K}  ⟶^{i} Δ_{B}/Γ_{K}
θ_{E} ↓  ↓^{θ_{B}}
Aut(Z(E_{0})) ⟶^{d} Aut(Z(B_{0}))

(2.6)

where i is induced by the inclusion Γ_{E} ⊆ Δ_{B} and d is the diagonal map, letting an automorphism of Z(E_{0}) act on each component of Z(B_{0}) ≅ Z(E_{0}) × … × Z(E_{0}).

Proof. From the description of B_{0} in Prop. 2.1, it follows that the primitive idempotents of Z(B_{0}) are the maps e_{1}, …, e_{k} such that e_{i}|_{N_{Γ_{i}}} = id and e_{i}|_{N_{Γ_{j}}} = 0 for i ≠ j. Suppose γ ∈ Δ_{B} is such that θ_{B}(γ + Γ_{K}) fixes some e_{i}, and let h ∈ H_{B} ∩ B_{γ}; then he_{i} = e_{i}h, hence h(N_{Γ_{i}}) = N_{Γ_{i}} and therefore γ ∈ Γ_{E}.

Conversely, suppose γ ∈ Γ_{E}. Take any nonzero c ∈ E_{γ} and any homogeneous E-vector space base of N built from bases of the N_{Γ_{i}}; let f ∈ B be defined by mapping each base vector n to nc. Then fe_{i} = e_{i}f for all i and f ∈ H_{B} ∩ B_{γ}, so θ_{B}(γ + Γ_{K}) fixes each e_{1}, …, e_{k}. Moreover, θ_{B}(γ + Γ_{K}) induces on each component e_{i}Z(B_{0}) ≅ Z(E_{0}) of Z(B_{0}) the automorphism of conjugation by c, which is θ_{E}(γ + Γ_{K}). Therefore, diagram (2.6) commutes. □

Corollary 2.3. If B_{0} is simple, then Γ_{B} = Δ_{B} = Γ_{E} and θ_{B} = θ_{E}. Moreover, B and B_{0} have the same matrix size, and [B:K] = [B_{0}:K_{0}]|Γ_{B}:Γ_{K}|.

Proof. If B_{0} is simple, Prop. 2.1 yields Γ_{B} = Γ_{E}. It also yields B_{0} ≅ End_{E_{0}}(N_{γ_{1}}) where dim_{E_{0}}(N_{γ_{1}}) = dim_{E}(N), hence B and B_{0} have the same matrix size. The equalities for Γ_{B} and θ_{B} follow from Prop. 2.2. Since B ≅_{g} End_{E}(N), we have

[B:K] = (dim_{E}(N))^{2} [E:K] = (dim_{E_{0}}(N_{γ_{1}}))^{2} [E_{0}:K_{0}] |Γ_{E}:Γ_{K}| = [B_{0}:K_{0}] |Γ_{B}:Γ_{K}| .
□

Example 2.4. With the notation of Prop. 2.1, if the dimensions dim_{E_{0}}(N_{γ_{1}}), …, dim_{E_{0}}(N_{γ_{k}}) are all different then every invertible homogeneous element in B has grade in Γ_{E}; therefore, Δ_{B} = Γ_{E} and θ_{B} = θ_{E}.

We can now see how the gauges considered here are related to Morandi value functions. The main earlier approach to value functions for central simple algebras is that of P. Morandi in [M_{2}] and [MW]. Let A be a central simple algebra over a field F with a valuation v. Let y: A → Γ ∪ {∞} be a surmultiplicative v-value function, and let

st(y) = { a ∈ A^{×} | y(a^{−1}) = −y(a) } ,

a subgroup of A^{×}, cf. Lemma 1.3. Let V_{y} = A^{≥0}, the “valuation ring” of y, and let A_{0} = A^{≥0}/A^{>0}, the degree 0 part of gr_{y}(A). Then, y is a Morandi value function if

(i) A_{0} is a simple ring; (ii) y(st(y)) = Γ_{A}.

When this occurs, it is known that V_{y} is a Dubrovin valuation ring integral over its center, which is V_{v}, and y is completely determined by V_{y}. Conversely, to every Dubrovin valuation ring B of A with B integral over its center, there is a canonically associated Morandi value function y_{B} with V_{y_{B}} = B. (For the theory of Dubrovin valuation rings, see [MMU], [W_{2}], or [G], and the references given there. In particular, it is known that for every central simple algebra A over a field F and every valuation v on F there is a Dubrovin valuation ring B of A with B ∩ F = V_{v}, and such a B is unique up to conjugacy, so unique up to isomorphism. See [W_{2}, Th. F] for characterizations of when B is integral over V_{v}.) For a Morandi value function y on A, the defect δ(y) (an integer) is defined by

δ(y) = [A:F] / [A_{0}:F_{0}] |Γ_{A}:Γ_{F} | .

If (F^{h}, v^{h}) is a Henselization of (F, v), and D^{h} is the division algebra Brauer-equivalent to A ⊗_{F} F^{h}, it is known (see. [W_{2}, Th. C]) that δ(y) coincides with the defect of the valuation on D^{h} extending v^{h} on F^{h}. Hence, δ(y) = 1 if char(\overline{F}) = 0 and δ(y) is a power of char(\overline{F}) otherwise. The requirement of integrality of V_{y} over V_{v} has been a significant limitation in applying the machinery of Dubrovin valuation rings in connection with value functions on central simple algebras.

Proposition 2.5. Let A be a central simple algebra over a field F with a valuation v, and let y be a surmultiplicative v-value function on A. Then, y is a Morandi value function on A with δ(y) = 1 if and only if y is a gauge on A with A_{0} simple.

Proof. ⇒ Suppose y is a Morandi value function on A with δ(y) = 1. For any γ ∈ Γ_{A} = y(st(y)), there is u_{γ} ∈ st(y) with y(u_{γ}) = γ. Then, u′_{γ} is a homogeneous unit of gr(A), by Lemma 1.3. Take any nonzero homogeneous ideal I of gr(A). Then, there is a γ ∈ Γ_{A} with I ∩ A_{γ} ≠ {0}. So, for the ideal I ∩ A_{0} of A_{0} we have I ∩ A_{0} ⊇ u_{γ}^{′−1}(I ∩ A_{γ}) ≠ {0}. Because A_{0} is a simple ring, we must have I ∩ A_{0} = A_{0}, so 1 ∈ I, so I = gr(A). Thus, gr(A) is a simple ring, which is finite-dimensional over gr(F) by Prop. 1.1(ii). Furthermore, as δ(y) = 1, Cor. 2.3 yields

[A:F] = [A_{0}:F_{0}] |Γ_{A}:Γ_{F} | = [gr(A):gr(F)] .

Hence, y is a gauge on A. By hypothesis, A_{0} is simple.

⇐ Suppose y is a gauge on A with A_{0} simple. Since gr(A) is graded semisimple, it is a graded direct product of graded simple rings. But, if gr(A) has nontrivial graded direct product decomposition gr(A) = C × D, then A_{0} = C_{0} × D_{0} with C_{0} and D_{0} nontrivial. This cannot occur as A_{0} is simple. Hence, gr(A) is graded simple. (This also follows from the simplicity of A. See Cor. 3.8 below.) Since A_{0} is simple, Cor. 2.3 applies with B = gr(A). The equality Γ_{gr(A)} = Δ_{gr(A)} from Cor. 2.3 shows that for each γ ∈ Γ_{A} = Γ_{gr(A)}, there is a homogeneous unit u in A_{γ}. Pick any a ∈ A with y(a) = γ and a′ = u. Then, a ∈ st(y) by Lemma 1.3. Hence, y(st(y)) = Γ_{A}, proving that y is a Morandi value function. Furthermore, as y is a norm on A, by Cor. 2.3 and Prop. 1.1(ii) we have

δ(y) = [A:F] / [A_{0}:F_{0}] |Γ_{A}:Γ_{F} | = [A:F] / [gr(A):gr(F)] = 1 .
□

3. Gauges over Henselian fields

We write ms(B) for the matrix size of a simple (or graded simple) algebra B.

Theorem 3.1. Let F be a field with a Henselian valuation v, and let A be a semisimple F -algebra with an F -gauge y. Let A_{1}, …, A_{n} be the simple components of A,

A = A_{1} × … × A_{n}.

For i = 1, …, n, the restriction y|_{A_{i}} is a gauge on A_{i}, the graded algebra gr_{y|_{A_{i}}} (A_{i}) is graded simple with ms(A_{i}) = ms(gr(A_{i})), and gr(A_{1}), …, gr(A_{n}) are the graded simple components of gr(A),

gr(A) = gr(A_{1}) × … × gr(A_{n}).

For a = (a_{1}, …, a_{n}) ∈ A_{1} × … × A_{n},

y(a) = min_{1≤i≤n}(y|_{A_{i}}(a_{i})).

Moreover, the gauge y is tame if and only if each y|_{A_{i}} is tame.

A major tool in the proof is the following lemma, which allows us to lift idempotents from gr(A):

Lemma 3.2. With the same hypotheses as in Th. 3.1, let ẽ_{1}, …, ẽ_{k} be a family of nonzero homogeneous orthogonal idempotents in gr(A) such that ẽ_{1}+…+ẽ_{k} = 1. There is a family of orthogonal idempotents e_{1}, …, e_{k} in A such that e′_{i} = ẽ_{i} for i = 1, …, k, e_{1} + … + e_{k} = 1, and

gr(e_{i}Ae_{j}) = e′_{i}gr(A)e′_{j},  gr(e_{i}A) = e′_{i}gr(A),  gr(Ae_{j}) = gr(A)e′_{j} for i, j = 1, …, k.

Proof. Let C = End_{F} (A), and let z = z_{y}: C → Γ ∪ {∞} be the gauge on C arising from the norm y on A, as in §1.3. Take any a ∈ V_{A}. For each b ∈ A, we have

y(ab) − y(b) ≥ y(a) + y(b) − y(b) ≥ 0.

This shows that for the left multiplication map λ_{a} ∈ C given by b ↦ ab, we have z(λ_{a}) ≥ 0. That is, λ_{a} ∈ V_{C}. Lemma 1.18 shows that V_{C} is integral over V_{F} hence λ_{a} is integral over V_{F}, and so is a. This shows that V_{A} is integral over V_{F}. Because V_{F} is Henselian, the integrality implies that we can lift families of orthogonal idempotents to V_{A} from any homomorphic image of it, by [MMU, Th. A.18, p. 180]. The homogeneous idempotents ẽ_{i} are necessarily of grade 0; they lie in A_{0} = V_{A}/A^{>0}. We may thus find orthogonal idempotents e_{1}, …, e_{k−1} of V_{A} with each e′_{i} = ẽ_{i} in A_{0}; let e_{k} = 1−(e_{1}+…+e_{k−1}). Then, e_{k}^{2} = e_{k}, e_{k}e_{i} = e_{i}e_{k} = 0 for i < k; also, e′_{k} = 1′ −(e_{1} +…+e_{k−1})′ = 1′ −(ẽ_{1} +…+ẽ_{k−1}) = ẽ_{k}. So, e_{1}, …, e_{k} are pairwise orthogonal idempotents with each y(e_{i}) = 0 and e′_{i} = ẽ_{i}, and e_{1} + … + e_{k} = 1.

To simplify notation, for any F -subspace N of A, we let N′ = gr(N), the associated graded gr(F)- vector space determined by y|_{N}; we view N′ as an F′ = gr(F)-subspace of A′ = gr(A). Since y is a norm on A, we have dim_{F′}(N′) = dim_{F} (N), by Prop. 1.1(iii) and (ii). For 1 ≤ i, j ≤ k, we claim that

(e_{i}Ae_{j})′ = e′_{i}A′e′_{j} and dim_{F} (e_{i}Ae_{j}) = dim_{F′}(e′_{i}A′e′_{j}) .  (3.1)

For, we have e′_{i}A′e′_{j} ⊆ (e_{i}Ae_{j})′ (cf. (1.5)). Since A = ⨁_{1≤i,j≤k} e_{i}Ae_{j} and A′ = ⨁_{1≤i,j≤k} e′_{i}A′e′_{j}, we have

dim_{F} (A) = dim_{F′}(A′) = ∑_{i,j} dim_{F′}(e′_{i}A′e′_{j}) ≤ ∑_{i,j} dim_{F′}((e_{i}Ae_{j})′) = ∑_{i,j} dim_{F} (e_{i}Ae_{j}) = dim_{F} (A) .

Thus, equality must hold throughout, so that dim_{F′}(e′_{i}A′e′_{j}) = dim_{F′}((e_{i}Ae_{j})′) = dim_{F} (e_{i}Ae_{j}), for all i, j; this shows that e′_{i}A′e′_{j} = (e_{i}Ae_{j})′, proving (3.1). A similar calculation shows that for all i,

(e_{i}A)′ = e′_{i}A′ and (Ae_{i})′ = A′e′_{i} .
□

We now consider the special case of Th. 3.1 where gr(A) is graded simple, and show that all gauges in this case are obtained from the construction in §1.3.

Theorem 3.3. Let F be a field with a Henselian valuation v. Let A be a semisimple F -algebra with an F -gauge y such that gr(A) is graded simple. Then, A is simple. Let D be the division algebra Brauerequivalent to A, and let w be the valuation on D extending v on F. Then w is defectless. Furthermore, there is a finite-dimensional right D-vector space M with a w-norm α such that

A ≅ End_{D}(M), gr(A) ≅_{g} End_{gr(D)}(gr(M)),

and y on A corresponds to the F -gauge y_{α} on End_{D}(M) induced by α as in §1.3. In particular, ms(A) = ms(gr(A)), and the gauge y is tame if and only w is a tame F -gauge.

Proof. To simplify notation, let A′ = gr(A). Let ẽ_{1}, …, ẽ_{n} be a family of orthogonal primitive idempotents of the semisimple ring A_{0} with ẽ_{1} + … + ẽ_{n} = 1. The ẽ_{i} are primitive homogeneous idempotents of gr(A), since all homogeneous idempotents in gr(A) have grade 0. Lift these idempotents to a family of orthogonal primitive idempotents e_{1}, …, e_{n} of V_{A} as in Lemma 3.2. Let e = e_{1} and let D = eAe. Then gr(D) = e′A′e′. Because e′ = ẽ_{1} is a primitive homogeneous idempotent of A′, we have e′A′e′ is a graded division ring. Therefore,

y(cd) = y(c) + y(d) for all c, d ∈ D^{×}.  (3.2)

It follows that D has no zero divisors; since D is finite-dimensional over F, it must be a division ring. Furthermore, (3.2) shows that y|_{D} is a valuation on D which restricts to v on F (when we identify F with eF ⊆ D). Thus, y|_{D} = w, since there is only one extension of v to D. Note that since y is a gauge, its restriction to any subspace of A is a norm, by Prop. 1.1(iii). Hence [gr(D):gr(F)] = [D:F], which means the valuation w is also an F -gauge on D, so D is defectless over F.

Let M = Ae, which is a right D-vector space. By Lemma 3.2, we have gr(M) = A′e′. This is a graded right gr(D)-vector space, hence by (1.5),

y(md) = y(m) + y(d) for all m ∈ M and d ∈ D.

Thus, y|_{M} is a D-value function on M with respect to w. Furthermore,

dim_{gr(D)}(gr(M)) = dim_{gr(F)}(gr(M))/dim_{gr(F)}(gr(D)) = dim_{F} (M)/dim_{F} (D) = dim_{D}(M).

Hence, y|_{M} is actually a D-norm on M. We let α = y|_{M}.

We have an F -homomorphism β: A → End_{D}(M) given by β(a)(m) = am. Let y_{α}: End_{D}(M) → Γ ∪ {∞} be the F -gauge on End_{D}(M) induced by the D-norm α on M, as in §1.3. That is, for f ∈ End_{D}(M), we have y_{α}(f) = min_{m∈M, m≠0}(α(f(m)) − α(m)). We claim that for every nonzero a ∈ A,

y_{α}(β(a)) = y(a) .  (3.3)

For, we have y_{α}(β(a)) = min_{m∈M, m≠0}(y(am)−y(m)) ≥ y(a), since y(am) ≥ y(a)+y(m) for all m. Suppose

y_{α}(β(a)) > y(a). Then, y(am) − y(m) > y(a), for all m. Hence, a′ · m′ = 0 in gr(M) for all m ∈ M, showing that a′gr(M) = 0, since gr(M) is generated by its homogeneous elements. But, gr(M) = A′e′, by Lemma 3.2. Thus, 0 = a′gr(M) = a′A′e′. Because A′ is graded simple, we have A′e′A′ = A′, and hence 0 = a′A′e′A′ = a′A′, which shows that a′ = 0. This contradicts a ≠ 0, proving (3.3), and showing that β is injective. We have

[A:F] = [A′:gr(F)] = (dim_{gr(D)}gr(M))^{2}[gr(D):gr(F)] = (dim_{D}M)^{2}[D:F] = [End_{D}(M):F].

Therefore, the injective F -algebra map β is an isomorphism. Furthermore, from (3.3) and Prop. 1.17 we have

gr_{y}(A) ≅_{g} gr_{y_{α}}(End_{D}(M)) ≅_{g} End_{gr_{w}(D)}(gr_{α}(M)),

as desired. Note that ms(A) = dim_{D}(M) = dim_{gr(D)}(gr(M)) = ms(gr(A)). Moreover, Prop. 1.17 shows that the gauge y_{α} is tame if and only if w is a tame F -gauge. This completes the proof of Th. 3.3. □

Proof of Th. 3.1. Since y is a gauge, the graded algebra gr(A) is graded semisimple. Consider its decomposition into graded simple components:

gr(A) = Ã_{1} × … × Ã_{ℓ}.

Let ẽ_{1}, …, ẽ_{ℓ} be the homogeneous central idempotents of gr(A) which are the unity elements of Ã_{1}, …, Ã_{ℓ}. By Lemma 3.2, we may lift these idempotents to idempotents e_{1}, …, e_{ℓ} of A such that e′_{i} = ẽ_{i} for each i, and e_{1} + … + e_{ℓ} = 1. Moreover, for i ≠ j we have

gr(e_{i}Ae_{j}) = ẽ_{i}gr(A)ẽ_{j} = {0} ,

hence e_{i}Ae_{j} = {0}. For a ∈ A, we have

a = ae_{1} + … + ae_{ℓ} = e_{1}a + … + e_{ℓ}a.

Since e_{i}ae_{j} = 0 for i ≠ j, it follows by multiplying on the left or on the right by e_{i} that

e_{i}a = e_{i}ae_{i} = ae_{i} for i = 1, …, ℓ.

Therefore, e_{1}, …, e_{ℓ} are central idempotents in A. Let A_{i} = e_{i}A for i = 1, …, ℓ. Then,

A = A_{1} × … × A_{ℓ}.

Moreover, by Lemma 3.2 we have

gr_{y|_{A_{i}}} (A_{i}) = ẽ_{i}gr(A) = Ã_{i} for i = 1, …, ℓ.

Since Ã_{i} is graded simple, y|_{A_{i}} is an F -gauge on A_{i}, and it follows from Th. 3.3 that A_{i} is simple, with ms(A_{i}) = ms(Ã_{i}). Therefore, A_{1}, …, A_{ℓ} are the simple components of A.

For a = (a_{1}, …, a_{n}) ∈ A_{1} × … × A_{n} we have a_{i} = ae_{i} hence by the surmultiplicativity of y

y|_{A_{i}}(a_{i}) = y(ae_{i}) ≥ y(a) + y(e_{i}) = y(a).

On the other hand, since a = ae_{1} + … + ae_{n} we have y(a) ≥ min_{1≤i≤n}(y(ae_{i})), hence

y(a) = min_{1≤i≤n}(y(a_{i})).

To complete the proof, it remains to show that y is a tame gauge if and only if each y|_{A_{i}} is tame. We have

gr(Z(A)) = gr(Z(A)e_{1} ⊕ … ⊕ Z(A)e_{ℓ}) = gr(Z(A_{1})) × … × gr(Z(A_{ℓ}))

and

Z(gr(A)) = Z(gr(A_{1}) × … × gr(A_{ℓ})) = Z(gr(A_{1})) × … × Z(gr(A_{ℓ})).

Therefore, gr(Z(A)) = Z(gr(A)) if and only if gr(Z(A_{i})) = Z(gr(A_{i})) for all i, and Z(gr(A)) is separable over gr(F) if and only if each Z(gr(A_{i})) is separable over gr(F). □

Corollary 3.4. Let F be a field with a Henselian valuation v, let D_{1}, …, D_{n} be (finite-dimensional) division F -algebras and let

A = D_{1} × … × D_{n}.

For i = 1, …, n, let w_{i} be the unique valuation on D_{i} extending v. The F -algebra A carries a v-gauge if and only if each D_{i} is defectless (i.e. (1.14) holds). When this condition holds, there is a unique v-gauge on A, defined by

y(d_{1}, …, y_{n}) = min_{1≤i≤n}(w_{i}(d_{i}))  for d_{i} ∈ D_{i}.  (3.4)

Proof. If D_{i} is defectless for i = 1, …, n, then w_{i} is a v-gauge on D_{i} by Prop. 1.10 and it is straightforward to check that formula (3.4) defines a gauge on A. For the converse, suppose y is a gauge on A, and let y_{i} = y|_{D_{i}} for i = 1, …, n. By Th. 3.1, each y_{i} is a v-gauge on D_{i} and y(d_{1}, …, d_{n}) = min_{1≤i≤n}(y_{i}(d_{i})).

Moreover, ms(gr_{y_{i}}(D_{i})) = ms(D_{i}) hence gr_{y_{i}}(D_{i}) is a graded division ring. Therefore, for all a, b ∈ D_{i}^{×} we have (ab)′ = a′b′, hence y_{i}(ab) = y_{i}(a) + y_{i}(b). This shows that y_{i} is a valuation, hence y_{i} = w_{i}. However, w_{i} is a gauge only when D_{i} is defectless, by Prop. 1.10. The proof is thus complete. □

If the valuation v is not Henselian, we may still apply Th. 3.1 after scalar extension to a Henselization (F^{h}, v^{h}) of (F, v). If y is a v-gauge on the semisimple F -algebra A, then Cor. 1.24 shows that y ⊗ v^{h} is a v^{h}-gauge on A ⊗_{F} F^{h} with gr(A ⊗_{F} F^{h}) = gr(A), since gr(F^{h}) = gr(F). Let B_{1}, …, B_{n} be the simple components of A ⊗_{F} F^{h},

A ⊗_{F} F^{h} = B_{1} × … × B_{n}.

For i = 1, …, n, let D_{i} be the division algebra Brauer-equivalent to B_{i}.

Proposition 3.5. With the notation above, let ℓ be the number of simple components of A and ℓ′ be the number of graded simple components of gr(A). Then ℓ ≤ ℓ′ = n. Moreover, the following conditions are equivalent:

(i) y is a tame v-gauge; (ii) y ⊗ v^{h} is a tame v^{h}-gauge; (iii) for i = 1, …, n, the unique valuation w_{i} on D_{i} extending v^{h} is a tame v^{h}-gauge.

Proof. The equality ℓ′ = n follows from Th. 3.1 and the equality ℓ ≤ n is clear. The equivalence (i) ⟺ (ii) readily follows from Cor. 1.24. For i = 1, …, n, let y_{i} be the restriction of y ⊗ v^{h} to B_{i}. Th. 3.1 shows that y ⊗ v^{h} is a tame v^{h}-gauge if and only if each y_{i} is a tame v^{h}-gauge. By Th. 3.3, this condition is equivalent to (iii). □

Corollary 3.6. Suppose A is a simple algebra finite-dimensional over a field F with valuation v. If y is a v-gauge on A, then the number of simple components of gr(A) equals the number of extensions of v from F to Z(A).

Proof. Let (F^{h}, v^{h}) be a Henselization of (F, v), and let K = Z(A). Since F^{h} is separable over F, we can write K⊗_{F} F^{h} = L_{1}×…×L_{n}, where each L_{i} is a field. Then, A⊗_{F} F^{h} ≅ A⊗_{K}(K⊗_{F} F^{h}) ≅ B_{1}×…×B_{n}, where each B_{i} = A⊗_{K} L_{i}, a central simple L_{i}-algebra. With respect to the F^{h}-gauge y ⊗v^{h} on A⊗_{F} F^{h}, Th. 3.1 shows that

gr(A ⊗_{F} F^{h}) ≅_{g} gr(B_{1}) × … × gr(B_{n})  (3.5)

with each gr(B_{i}) simple. Now, y|_{K} is a surmultiplicative norm on K by Prop. 1.1(iii), and gr(K) is semisimple since it is a central subalgebra of the semisimple gr(F)-algebra gr(A); so, y|_{K} is a gauge. Hence, by Prop. 1.6 and Cor. 1.7, the number of simple summands of gr(K) equals the number of extensions of v to K. We have y|_{K} ⊗ v^{h} is a gauge on K ⊗_{F} F^{h}, and also,

gr(K ⊗_{F} F^{h}) ≅_{g} gr(L_{1}) × … × gr(L_{n})  (3.6)

by Prop. 1.9. Furthermore, each gr(L_{i}) is simple by Cor. 1.7 and Prop. 1.6 because the Henselian valuation v^{h} has a unique extension from F^{h} to L_{i}. Since gr(A) ≅_{g} gr(A ⊗_{F} F^{h}) and gr(K) ≅_{g} gr(K ⊗_{F} F^{h}) by Cor. 1.24, equations (3.5) and (3.6) show that

n = number of simple components of gr(A)

= number of simple components of gr(K)

= number of extensions of v to K .

□

Corollary 3.7. If char(\overline{F}) = 0, then every F -gauge on a semisimple F -algebra is tame.

Proof. Condition (iii) of Prop. 3.5 holds if char(\overline{F}) = 0, by Prop. 1.11. □

In the rest of this section, we consider the case where the semisimple F -algebra A is central simple. Recall from [HW_{2}] that the graded Brauer group GBr(E) of a graded field E can be defined on the same model as the classical Brauer group of fields. The elements of GBr(E) are graded isomorphism classes of graded division algebras with center E.

Corollary 3.8. Let (F, v) be a valued field and let (F^{h}, v^{h}) be a Henselization of (F, v). Let A be a central simple F -algebra, let D be the division algebra Brauer-equivalent to A ⊗_{F} F^{h}, and let w be the valuation on D extending v^{h}. Let y be any v-gauge on A. Then, the gr_{v}(F)-algebra gr_{y}(A) is graded simple and Brauer-equivalent to gr_{w}(D). Moreover, if y is tame then D is tame (see Def. 1.12) and gr_{y}(A) is central simple over gr_{v}(F).

Proof. Let A^{h} = A⊗_{F} F^{h} and let y^{h} = y⊗v^{h}. Th. 3.1 shows that gr_{y^{h}}(A^{h}) is graded simple, and Th. 3.3 yields gr_{y^{h}}(A^{h}) ≅_{g} End_{gr(D)}(gr(M)) for some finite-dimensional right D-vector space M, hence gr_{y^{h}}(A^{h}) is Brauer-equivalent to gr(D). These properties carry over to gr_{y}(A) because gr_{y}(A) = gr_{y^{h}}(A^{h}) by Cor. 1.24, since gr(F^{h}) = gr(F). When y is tame, it follows by definition that Z(gr(A)) = gr(F), and from Prop. 3.5 that D is tame. □

For any valued field (F, v), define the tame part of the Brauer group Br(F) (with respect to v) to be

TBr(F) = { [A] | A is a central simple F -algebra with a tame F -gauge } .

Cor. 3.8 shows that if y is a tame v-gauge on a central simple F -algebra A, then the Brauer class [gr_{y}(A)] in GBr(gr(F)) does not depend on the choice of the tame v-gauge but only on the Brauer class of A, since it coincides with the Brauer class [gr(D)] where D is the division algebra Brauer-equivalent to A ⊗_{F} F^{h}. Therefore, there is a well-defined map

Ψ: TBr(F) → GBr(gr(F)), [A] ↦ [gr_{y}(A)] for any v-gauge y on A.

By Cor. 1.26, TBr(F) is a subgroup of Br(F) and the map Ψ is a group homomorphism.

Theorem 3.9. The kernel of Ψ consists of the elements in TBr(F) which are split by any Henselization F^{h} of F with respect to v. Moreover, for any valued field (L, w) extending (F, v), there is a commutative diagram

TBr(F) ⟶^{Ψ_{F}} GBr(gr(F))
↓  ↓
TBr(L) ⟶^{Ψ_{L}} GBr(gr(L)).

(3.7)

If v is Henselian, then Ψ is an index-preserving group isomorphism.

Proof. Cor. 1.24 shows that the scalar extension map −⊗_{F} L sends TBr(F) to TBr(L), and that diagram (3.7) is commutative. When v is Henselian, Th. 3.3 shows that ms(gr_{y}(A)) = ms(A) for any central simple F -algebra, hence Ψ is index-preserving and injective. In this Henselian case, Ψ is also surjective, by [HW_{2}, Th. 5.3]. Another more direct proof of the surjectivity is possible, by showing that we can construct algebra classes of unramified algebras and inertially split cyclic algebras and tame totally ramified symbol algebras over F which map onto generators of GBr(F). No longer assuming that v is Henselian, take L = F^{h} in commutative diagram (3.7); since Ψ_{F^{h}} is bijective and gr(F^{h}) = gr(F), the kernel of Ψ_{F} is the kernel of the scalar extension map TBr(F) → TBr(F^{h}). □

This theorem generalizes [HW_{2}, Th. 5.3], which showed that Ψ_{F} is a group isomorphism when v is Henselian. The proof given here is vastly simpler than the one in [HW_{2}].

4. Applications

The utility of Theorems 3.1 and 3.3 depends on being able to construct gauges on algebras over valued fields. We give several examples where this can be done, obtaining as a result considerably simplified and more natural proofs of some earlier theorems. In each case, we use the following result, which can be viewed as a detection theorem: it allows one to use a gauge to determine whether the division algebra D Brauer-equivalent to a given central simple algebra A has a valuation extending a given valuation on the center, without first determining D.

Theorem 4.1. Let (F, v) be a valued field, and let A be a central simple F -algebra, and let D be the division algebra Brauer-equivalent to A. Suppose A has an F -gauge y. Then, gr(A) is simple, and ms(gr(A)) ≥ ms(A). Moreover, the following conditions are equivalent:

(i) v extends to a valuation on D; (ii) ms(gr(A)) = ms(A); (iii) [D:F] = [E:gr(F)], where E is the graded division algebra Brauer-equivalent to gr(A).

When these conditions hold, gr(D) ≅_{g} E and gr(A) ≅_{g} End_{gr(D)}(N) for some graded right gr(D)- vector space N. Hence, \overline{D} is Brauer equivalent to any simple component of A_{0}, and Γ_{D} and θ_{D} are determinable from gr(A), as described in Prop. 2.2.

Proof. Let (F^{h}, v^{h}) be a Henselization of (F, v). Let A^{h} = A ⊗_{F} F^{h} and D^{h} = D ⊗_{F} F^{h}. So, A^{h} ≅ M_{n}(D^{h}), where A ≅ M_{n}(D). As noted in the proof of Cor. 3.8, y ⊗ v^{h} is an F^{h}-gauge on A^{h} with gr(A^{h}) ≅_{g} gr(A). Since A^{h} is simple, Th. 3.1 shows that gr(A^{h}) (so also gr(A)) is graded simple, and that ms(gr(A)) = ms(A^{h}). Of course, ms(A^{h}) ≥ ms(A). Thus, ms(gr(A)) = ms(A) iff ms(A^{h}) = ms(A), iff D^{h} is a division ring, iff v extends to a valuation on D, by Morandi’s theorem [M_{1}, Th. 2].

Let D̃ be the division algebra Brauer-equivalent to A^{h}. By Th. 3.3, we have A^{h} ≅ EndD̃(N) for some right D̃-vector space N with a D̃-norm, and gr(A^{h}) ≅_{g} End_{gr(D̃)}(gr(M)). Hence, E ≅_{g} gr(D̃), by the uniqueness part of the graded Wedderburn theorem. Since Th. 3.3 also says that the valuation on D̃ extending v^{h} is an F^{h}-gauge, we have [E:gr(F)] = [gr(D̃):gr(F^{h})] = [D̃:F^{h}]. But, [D:F] = [D^{h}:F^{h}]. Hence, [E:gr(F)] = [D:F] iff [D̃:F^{h}] = [D^{h}:F^{h}] iff D^{h} is a division ring, iff (as above) v extends to D. When this occurs, since the valuation on D^{h} extends the one on D, we have an inclusion ι: gr(D) ↪ gr(D^{h}); but ι is actually an isomorphism, since \overline{D} ≅ \overline{D^{h}} and Γ_{D} = Γ_{D^{h}} by [M_{1}, Th. 2], hence D_{0} ≅ D_{0}^{h} and Γ_{gr(D)} = Γ_{gr(D^{h})}. Thus, E ≅_{g} gr(D̃) ≅_{g} gr(D^{h}) ≅_{g} gr(D), so that gr(A) ≅_{g} End_{gr(D)}(N). Hence the Brauer class of \overline{D} = D_{0} coincides with that of any simple component of A_{0}, and θ_{D} is determinable from gr(A) as described in Prop. 2.2. □

4.1. Crossed products. We now show how to construct tame gauges on crossed product algebras when the Galois extension is indecomposed and defectless with respect to the valuation.

Let K/F be a finite Galois extension of fields, and let G be the Galois group G(K/F). Let A be a crossed product algebra (K/F, G, f), where f is a 2-cocycle in Z^{2}(G, K^{×}), and assume for convenience that f is normalized. Explicitly, write

A = ⨁_{σ∈G} Kx_{σ},

where

x_{id} = 1, x_{σ}cx_{σ}^{−1} = σ(c) for all c ∈ K and x_{σ}x_{τ} = f(σ, τ)x_{στ} for all σ, τ ∈ G.

Assume v is a valuation on F which has a unique extension to a valuation w of K, and that w is defectless over v. Thus, w is a v-norm on K as an F -vector space, and every automorphism σ ∈ G(K/F) induces an automorphism σ′ of gr(K). Assume further that \overline{K} is separable over \overline{F} and that char(\overline{F}) ∤ |Γ_{K}:Γ_{F} |. Then gr(K) is Galois over gr(F) and the canonical map G(K/F) → G(gr(K)/gr(F)) given by σ ↦ σ′ is an isomorphism. Let G′ = G(gr(K)/gr(F)) and define f′: G′ → gr(K)^{×} by f′(σ′, τ′) = f(σ, τ)′; then f′ ∈ Z^{2}(G′, gr(K)^{×}) and we may consider the crossed product algebra (gr(K)/gr(F), G′, f′). This is a graded simple gr(F)-algebra, see [HW_{2}, Lemma 3.1].

Toward defining a v-gauge on A, we set for σ ∈ G

y(x_{σ}) = \frac{1}{|G|} ∑_{ρ∈G} w(f(σ, ρ)) .  (4.1)

We extend y to a value function on A by letting

y( ∑_{σ∈G} c_{σ}x_{σ}) = min_{σ∈G}(w(c_{σ}) + y(x_{σ})) .  (4.2)

Proposition 4.2. The value function y is a v-gauge on A, and there is a canonical isomorphism

gr(A) ≅_{g} (gr(K)/gr(F), G′, f′).

Proof. Let (a_{i})_{1≤i≤n} be any splitting base of w on K as an F -norm. Then, it follows from the definition in (4.2) that (a_{i}x_{σ} | 1 ≤ i ≤ n, σ ∈ G) is a splitting base for y, so that y is an F -norm on A. Note that w is invariant under the action of G on K, since w is the unique extension of v to K. Thus, when we apply w to the basic cocycle equation f(σ, τ)f(στ, ρ) = σ(f(τ, ρ))f(σ, τρ) and sum over all ρ ∈ G, we obtain in Γ,

|G|w(f(σ, τ)) + |G|y(x_{στ}) = |G|y(x_{τ}) + |G|y(x_{σ}) .

Since Γ is torsion-free this yields

y(x_{σ}) + y(x_{τ}) = w(f(σ, τ)) + y(x_{στ}) ,  (4.3)

for all σ, τ ∈ G. Therefore, for any i, j, σ, τ we have

y((a_{i}x_{σ})(a_{j}x_{τ})) = y(a_{i}σ(a_{j})f(σ, τ)x_{στ} ) = w(a_{i}) + w(a_{j}) + w(f(σ, τ)) + y(x_{στ})
= w(a_{i}) + w(a_{j}) + y(x_{σ}) + y(x_{τ}) = y(a_{i}x_{σ}) + y(a_{j}x_{τ}) .

By Lemma 1.2 it follows that y is surmultiplicative.

Now consider gr(A). Since (a′_{i})_{1≤i≤n} is a homogeneous gr(F)-base for gr(K) and ((a_{i}x_{σ})′ | 1 ≤ i ≤ n, σ ∈ G) is a homogeneous gr(F)-base of gr(A), and since (a_{i}x_{σ})′ = a′_{i}x′_{σ} by (4.2), we have gr(A) = ⨁_{σ∈G} gr(K)x′_{σ}. For any c ∈ K^{×} and σ ∈ G, we have y(x_{σ}c) = y(σ(c)x_{σ}) = w(c) + y(x_{σ}); hence,

in gr(A),

x′_{σ}c′ = (x_{σ}c)′ = (σ(c)x_{σ})′ = σ(c)′x′_{σ} = σ′(c′)x′_{σ} .

Moreover, formula (4.3) shows that x′_{σ}x′_{τ} = f(σ, τ)′x′_{στ} for all σ, τ ∈ G. Therefore, gr(A) ≅_{g} (gr(K)/gr(F), G′, f′). It follows that gr(A) is a graded central simple gr(F)-algebra, hence y is a tame F -gauge on A. □

We can describe A_{0} to some extent for this A. The value function y yields a map

λ: G → Γ/Γ_{K} given by σ ↦ y(x_{σ}) + Γ_{K},  (4.4)

and (4.3) shows that λ is a group homomorphism. Let H = ker(λ). Write A = ⨁_{σ∈G} Kz_{σ}, where

z_{σ} = d_{σ}x_{σ}, with the d_{σ} ∈ K^{×} chosen so that w(d_{σ}) = −y(x_{σ}) if σ ∈ H .  (4.5)

Thus, d_{σ}d_{τ} = g(σ, τ)d_{στ}, where g(σ, τ) = d_{σ}σ(d_{τ})d_{στ}^{−1}f(σ, τ), so g is a 2-cocycle cohomologous to f. We have y(z_{ρ}) = 0 for ρ ∈ H, and the analogue to (4.3) for g shows that w(g(ρ, τ)) = 0 for all ρ, τ ∈ H. For σ ∉ H, because y(z_{σ}) ∉ Γ_{K} the summand gr(K)z′_{σ} makes no contribution to A_{0}. Thus,

A_{0} = ⨁_{ρ∈H} K_{0}z′_{ρ} .

This A_{0} is semisimple, as gr(A) is simple, see (2.3), but it need not be a crossed product algebra, nor even simple, depending on how H acts on K_{0}. Recall that K_{0} = \overline{K}. Because the extension of v to K is indecomposed, each σ ∈ G induces an automorphism σ̃ of \overline{K} which coincides with the restriction of σ′ to K_{0}.

Proposition 4.3. In the situation just described where A = (K/F, G, f) = ⨁_{σ∈G} Kx_{σ} and H = ker(λ)

for λ as in (4.4), suppose that each ρ ∈ H induces a different automorphism of \overline{K}. Then, Z(A_{0}) = K_{0}^{H}, the subfield of K_{0} fixed by H, and A_{0} = ⨁_{ρ∈H} K_{0}z′_{ρ} is a crossed product algebra over Z(A_{0}). Let E be

the graded division algebra Brauer-equivalent to gr(A). Then, Γ_{E} = Γ_{gr(A)} = Γ_{K} + ⟨y(x_{σ}) | σ ∈ G⟩ and E_{0} is the division algebra Brauer-equivalent to A_{0}. The map θ_{gr(A)} : Γ_{E}/Γ_{F} → G(Z(A_{0})/F_{0}) induces θ̃: Γ_{gr(A)}/Γ_{K} → G(K_{0}^{H}/F_{0}), and we have a commutative diagram,

\begin{matrix}G & ⟶^{λ} & Γ_{gr(A)}/Γ_{K} \\ ↓ & & ↓θ̃ \\ G(\overline{K}/\overline{F}) & ⟶ & G(K_{0}^{H}/F_{0}) . \end{matrix}  (4.6)

Proof. Since each ρ ∈ H induces a different automorphism of K_{0} = \overline{K}, it is clear that A_{0} = ⨁_{ρ∈H} K_{0}z′_{ρ} is

a crossed product algebra over its center K_{0}^{H} = \overline{K}^{H}. Clearly, Γ_{gr(A)} = Γ_{K} + ⟨y(x_{σ}) | σ ∈ G⟩. Since A_{0} is simple, we have Γ_{E} = Γ_{gr(A)} and θ_{E} = θ_{gr(A)}, by Cor. 2.3. Furthermore, since A_{0} has only one simple component, for any γ ∈ Γ_{E}, θ_{E}(γ + Γ_{F}) is the automorphism of Z(A_{0}) induced by conjugation by any a ∈ A_{γ} ∩ A^{×}. If γ ∈ Γ_{K}, then a can be chosen in gr(K), and the conjugation is trivial, as Z(A_{0}) ⊆ K_{0}. Hence, θ_{gr(A)} induces θ̃. For each γ ∈ Γ_{gr(A)}, there is σ ∈ G with λ(z_{σ}) ≡ γ (mod Γ_{K}). Then θ̃(γ+Γ_{K}) is given by conjugation by z′_{σ} on K_{0}^{H}, which coincides with σ̃|\overline{K}_{H}. Therefore, diagram (4.6) is commutative, where the left map is σ ↦ σ̃ and the bottom map is restriction of the automorphism from \overline{K} = K_{0} to \overline{K}^{H} = K_{0}^{H}. □

Now consider the unramified case of the preceding discussion. Suppose the field K is Galois over F, and suppose F has a valuation v which has a unique and unramified (and defectless) extension to a valuation w of K. So, \overline{K} is Galois over \overline{F} and G(\overline{K}/\overline{F}) ≅ G(K/F). Let G = G(K/F). The short exact sequence of trivial G-modules

0 → Γ_{F} → Γ → Γ/Γ_{F} → 0

yields a connecting homomorphism ∂: H^{1}(G, Γ/Γ_{F}) → H^{2}(G, Γ_{F}). In fact, ∂ is an isomorphism, since for the divisible torsion-free group Γ we have H^{1}(G, Γ) = H^{2}(G, Γ) = {0}. Thus, we have a succession of maps

H^{2}(G, K^{×}) → H^{2}(G, Γ_{F}) ⟶^{∂^{−1}} H^{1}(G, Γ/Γ_{F}) = Hom(G, Γ/Γ_{F}) ,  (4.7)

where the left map is induced by the G-module homomorphism w: K^{×} → Γ_{K} = Γ_{F}.

Corollary 4.4. Let K be an unramified and defectless Galois extension field of F with respect to the valuation v on F. Let G = G(K/F), and take any f ∈ Z^{2}(G, K^{×}). Let A = (K/F, G, f) = ⨁_{σ∈G} Kx_{σ},

and let D be the division algebra Brauer-equivalent to A. Let y be the tame F -gauge on A defined in (4.2) above, and let λ be as defined in (4.4), let H = ker(λ), and the z_{σ} as defined in (4.5). So, for the graded simple ring gr_{y}(A), we have A_{0} = ⨁_{ρ∈H} K_{0}z′_{ρ}, as above, and A_{0} is a crossed product algebra over

\overline{K}^{H}. Furthermore,

(i) The map λ of (4.4) is the image of [f] ∈ H^{2}(G, K^{×}) under the maps of (4.7).

(ii) v extends to a valuation on D iff ms(A_{0}) = ms(A).

(iii) Suppose v extends to D. Then, Γ_{D}/Γ_{F} = im(λ); Z(\overline{D}) = \overline{K}^{H}; \overline{D} is the division algebra Brauerequivalent to A_{0}; and θ_{D}: Γ_{D}/Γ_{F} → G(Z(\overline{D})/\overline{F}) is the isomorphism which is the inverse to the composite map G(Z(\overline{D})/\overline{F}) ⟶^{≅} G/H ⟶^{≅} Γ_{D}/Γ_{F} induced by λ.

Proof. Note that Prop. 4.3 applies here, since the map G(K/F) → G(\overline{K}/\overline{F}) is injective. Hence, A_{0} is a crossed product with center \overline{K}^{H}.

(i) follows from (4.3) above and the definition of the connecting homomorphism in group cohomology. For (ii), we have ms(gr(A)) = ms(A_{0}), as A_{0} is simple, by Cor. 2.3. But, by Th. 4.1, v extends to D

iff ms(A) = ms(gr(A)). This proves (ii).

(iii) Suppose v extends to D. Let E = gr(D). By Th. 4.1, we have gr(A) ≅_{g} End_{E}(N) for some graded right E-vector space N. Then, using Cor. 2.3 and Prop. 4.3, we have Γ_{D}/Γ_{F} = Γ_{E}/Γ_{F} = Γ_{gr(A)}/Γ_{F} = im(λ). Also, \overline{D} = E_{0}, which is the division algebra Brauer-equivalent to A_{0}, so Z(\overline{D}) ≅ Z(A_{0}) = K_{0}^{H} = \overline{K}^{H}. Finally, again using Cor. 2.3, we have θ_{D} = θ_{E} = θ_{gr(A)}. In commutative diagram (4.6), θ̃ = θ_{gr(A)}, as Γ_{K} = Γ_{F}. The diagram shows that the surjective map θ̃, is also injective, as ker(λ) = H = ker(θ̃∘λ). So, θ̃ is an isomorphism, and the diagram shows that θ̃ is the inverse of the isomorphism induced by λ. □

In the context of Cor. 4.4, if the valuation v on F is Henselian, then v always extends to D, so Cor. 4.4 applies. It yields a new proof of [JW, Th. 5.6(b)] for inertially split division algebras over Henselian fields, which is significantly simpler and more direct than previous proofs. It does not use generalized crossed products, as in the proof in [JW], nor Dubrovin valuation rings, as in the proof in [MW, Cor. 3.7].

4.2. Tensor products of symbol algebras. Let A be a finite-dimensional algebra over an arbitrary field F. Recall from [TW, §2] (see also [T]) that an armature of A is an abelian subgroup A ⊂ A^{×}/F^{×} such that |A| = [A:F] and {a ∈ A | aF^{×} ∈ A} spans A as an F -vector space.

For example, suppose F contains a primitive n-th root of unity ω for some n ≥ 2, and A is a symbol algebra of degree n, i.e. an F -algebra generated by two elements i, j subject to the relations i^{n} ∈ F^{×}, j^{n} ∈ F^{×}, and ij = ωji. The images in A^{×}/F^{×} of the standard generators i, j generate an armature of A. More generally, in a tensor product of symbol algebras the images of the products of standard generators generate an armature. Tensor products of symbol algebras can actually be characterized by the existence of armatures of a certain type, see [TW, Prop. 2.7].

Although tensor products of symbol algebras are the main case of interest to us, we first consider commutative algebras. Let Z be an armature of a commutative F -algebra Z. Suppose F contains a primitive s-th root of unity for some multiple s of the exponent exp(Z), and let μ_{s} ⊆ F denote the group of s-th roots of unity. Since char(F) ∤ s, we have |Z| ≠ 0 in F. Let π: Z^{×} → Z^{×}/F^{×} be the canonical map and let

X = π^{−1}(Z) ⊆ Z^{×}.

Since exp(Z) divides s, we have x^{s} ∈ F^{×} for all x ∈ X. Therefore, there is a commutative diagram with exact rows:

\begin{aligned}1 & ⟶ & F^{×} & ⟶ & X & ⟶^{π} & Z & ⟶ & 1 \\ & & ^{s}↓ & & ↓^{s} & & ↓^{ρ} \\ 1 & ⟶ & F^{×s} & ⟶ & F^{×} & ⟶ & F^{×}/F^{×s} & ⟶ & 1. \end{aligned}  (4.8)

Let K = ker(ρ) and L = im(ρ), and let L = F (\{\sqrt[s]{c} | cF^{×s} ∈ L\}) be the s-Kummer extension field of F associated with L. Let also G = Hom(Z, μ_{s}), the character group of Z, and let H ⊆ G be the subgroup orthogonal to K,

H = {χ ∈ G | χ(k) = 1 for all k ∈ K}.

Let also r = |K| = |G:H|, and let K ⊆ Z be the subalgebra spanned by π^{−1}(K). The following proposition extends [TW, Lemma 2.9]:

Proposition 4.5. The F -algebra Z is G-Galois, and contains r primitive idempotents e_{1}, …, e_{r}, which form an F -base of K and are conjugate under the G-action. The isotropy subgroup of any e_{i} is H, and e_{i}Z ≅ L is a Galois extension of F with Galois group isomorphic to H. In particular, Z ≅ L^{r}, a direct product of r copies of L.

Proof. For each z ∈ Z, choose x_{z} ∈ X such that π(x_{z}) = z. By definition of an armature, (x_{z})_{z∈Z} is an F -base of Z. Let _{s}X = {x ∈ X | x^{s} = 1}. Applying the snake lemma to (4.8), we get the exact sequence

1 → μ_{s} → _{s}X ⟶^{π} K → 1.

Since μ_{s} is a cyclic group of order s in the finite abelian group _{s}X of exponent s, this exact sequence splits. Therefore, we may assume that the elements x_{z} satisfy

x_{k}x_{k′} = x_{kk′}  for k, k′ ∈ K.  (4.9)

In particular, x_{1} = 1. In the base (x_{z})_{z∈Z}, the matrix of multiplication by x_{z} is monomial, and the corresponding permutation is multiplication by z in Z. This permutation has no fixed point if z ≠ 1, hence the trace map T_{Z/F} : Z → F satisfies

T_{Z/F} (x_{1}) = |Z| ≠ 0  and  T_{Z/F} (x_{z}) = 0 for z ≠ 1.

It is then straightforward to check that the bilinear trace form on Z is not degenerate, hence Z is étale. An action of G on Z is defined by

χ ∗ x_{z} = χ(z)x_{z} for z ∈ Z.

If z ≠ 1, there exists χ ∈ G with χ(z) ≠ 1, hence F ⊆ Z is the set of fixed points under the G-action. Since |G| = |Z| = [Z:F], it follows that Z is a G-Galois F -algebra, see [KMRT, Sec. 18B].

Now, consider e = \frac{1}{r} ∑_{k∈K} x_{k}. In view of (4.9), we have ex_{k} = e for all k ∈ K, hence e^{2} = e. Let z_{1}, …, z_{m} ∈ Z be representatives of the cosets modulo K. Since x_{z_{i}}x_{k} ∈ x_{z_{i}k}F^{×}, the products x_{z_{i}}x_{k} for k ∈ K and i = 1, …, m form a base of Z. For i = 1, …, m the product ex_{z_{i}} is in the F -span of (x_{z_{i}}x_{k})_{k∈K}, hence ex_{z_{1}}, …, ex_{z_{m}} are linearly independent. These elements span eZ since ex_{z_{i}}x_{k} = ex_{z_{i}} for k ∈ K, hence they form a base of eZ. Let

eX = {ex | x ∈ X} = ⋃_{i=1}^{m}ex_{z_{i}}F^{×} ⊆ (eZ)^{×}.

Mapping ex to ρπ(x) ∈ L defines a surjective map eX → L with kernel eF^{×}, hence L may be identified with an armature of eZ. By [TW, Lemma 2.9], it follows that eZ ≅ L. Since L is a field, e is a primitive idempotent in Z. From the definition of e, it is clear that H ⊆ G is the subgroup of elements that leave e fixed, hence the orbit of e has r elements, which span K. The structure theorem of Galois algebras (see [KMRT, (18.18)]) shows that the primitive idempotents of Z are the conjugates of e, and that eZ is H-Galois. The proof is thus complete. □

Remark 4.6. The G-structure of Z can be made explicit by [KMRT, Prop. (18.18)]: it is an induced algebra Z = Ind_{H}^{G}(eZ).

For an armature A of an arbitrary finite-dimensional F -algebra A, there is an associated armature pairing

β_{A}: A × A → μ(F) given by (aF^{×}, bF^{×}) ↦ aba^{−1}b^{−1} ,

where μ(F) denotes the group of roots of unity in F. It is shown in [TW, §2] that β_{A} is a welldefined symplectic bimultiplicative pairing, and if β_{A} is nondegenerate, then A is isomorphic to a tensor product of symbol algebras. Conversely, in any tensor product of symbol algebras the standard generators generate an armature whose associated pairing is nondegenerate. For any subgroup B ⊆ A, we let

B^{⊥} = {a ∈ A | β_{A}(a, b) = 1 for all b ∈ B},

which is a subgroup of A. Note that when β_{A} is nondegenerate, i.e., A ∩ A^{⊥} = {1}, we have |B| |B^{⊥}| = |A|.

We now fix the setting we will consider for the rest of the paper. Let A be an F -algebra with an armature A such that β_{A} is nondegenerate. Let s = exp(A). The nondegeneracy of β_{A} implies that μ_{s} ⊆ F. We denote by π: A^{×} → A^{×}/F^{×} the canonical map. Let v: F → Γ ∪ {∞} be a valuation on F. Assume that char(\overline{F}) ∤ s. Hence, the group μ_{s} = μ_{s}(F) of s-th roots of unity in F maps bijectively to μ_{s}(\overline{F}). We build a tame F -gauge on A using the armature A. For this, define functions

w: π^{−1}(A) → Γ given by w(x) = \frac{1}{s}v(x^{s}), and w̃: A → Γ/Γ_{F} , the map induced by w .

Note that w and w̃ are group homomorphisms, since the commutators of elements of π^{−1}(A) are roots of unity, hence elements in F^{×} of value 0. Clearly, w|_{F^{×}} = v. For each a ∈ A, pick x_{a} ∈ A^{×} such that π(x_{a}) = a. Then (x_{a})_{a∈A} is an F -base of A. Define an F -norm y: A → Γ ∪ {∞} by

y( ∑_{i=1}^{n} λ_{a}x_{a}) = min_{a∈A}(v(λ_{a}) + w(x_{a})) .

The definition of y depends on A, but is independent of the choice of the x_{a}.

Proposition 4.7. The F -norm y is a tame F -gauge on A and Γ_{A} is determined by

Γ_{A}/Γ_{F} = w̃(A) .

The graded algebra gr(A) has an armature isometric to (A, β_{A}). Moreover, every homogeneous component of gr(A) contains an invertible element, hence the subgroup Δ_{gr(A)} ⊆ Γ_{gr(A)} = Γ_{A} defined in Sec. 2 coincides with Γ_{A}, and the map θ_{gr(A)} of (2.5) is a homomorphism

θ_{gr(A)} : Γ_{A}/Γ_{F} → Aut(Z(A_{0})).

Proof. Note that y|_{π^{−1}(A)} = w. Hence, for all a, b ∈ A, we have y(x_{a}x_{b}) = y(x_{a}) + y(x_{b}); so for the image x′_{a} of x_{a} in gr(A), (x_{a}x_{b})′ = x′_{a}x′_{b}. It follows by Lemma 1.2 that y is surmultiplicative, and that π^{−1}(A) maps to a subgroup of gr(A)^{×}. Furthermore, (x′_{a})_{a∈A} is a gr(F)-base of gr(A) by Prop. 1.1(i), since (x_{a})_{a∈A} is an F -splitting base of A. Thus, the image A′ of {x′_{a} | a ∈ A} in gr(A)^{×}/gr(F)^{×} could be called a graded armature for gr(A). The map A → A′ given by x_{a}F^{×} ↦ x′_{a}gr(F)^{×} is clearly a group isomorphism and also an isometry between the armature pairings β_{A} and β_{A′} when we identify μ_{s}(F) with μ_{s}(\overline{F}). The pairing β_{A′} is therefore nondegenerate, so an argument analogous to the ungraded one in [TW, Prop. 2.7] shows that gr(A) is isomorphic to a graded tensor product of graded symbol algebras over gr(F). Since it is easy to see that graded symbol algebras are graded central simple gr(F)-algebras (by a slight variation of the ungraded argument), it follows that gr(A) is graded central simple over gr(F). Thus, y is a tame F -gauge on A. □

Our next goal is to describe the degree 0 component A_{0} ⊆ gr(A), which is a semisimple algebra over F_{0} = \overline{F}. For this, we consider

B = ker(w̃) ⊆ A, Z = B ∩ B^{⊥},

and denote by Z ⊆ A the subalgebra spanned by π^{−1}(Z). Since β_{A} is trivial on Z, the F -algebra Z is commutative.

Proposition 4.8. The F_{0}-algebra A_{0} has an armature B_{0} canonically isomorphic to B with armature pairing β_{B_{0}} isometric to the restriction of β_{A} to B. Its center Z(A_{0}) is the degree 0 component of Z, i.e. Z(A_{0}) = Z_{0}; it is an (A/Z^{⊥})-Galois F_{0}-algebra. For the map ψ: A/Z^{⊥} ↪ Aut_{F_{0}}(Z(A_{0})) given by the Galois action, the following diagram is commutative:

\begin{matrix}A & \underrightarrow{w̃} & Γ_{A}/Γ_{F} \\ ↓ & & ↓^{θ_{gr(A)}} \\ A/Z^{⊥} & ⟶^{ψ} & Aut_{F_{0}}(Z(A_{0})). \end{matrix}  (4.10)

Proof. We first fix a convenient choice of x_{a} ∈ A^{×} such that π(x_{a}) = a ∈ A: for b ∈ B, we choose x_{b} such that w(x_{b}) = 0. As observed in the proof of Prop. 4.7, (x′_{a})_{a∈A} is a homogeneous gr(F)-base of gr(A). We have y(x_{a}) = 0 if and only if a ∈ B, hence (x′_{b})_{b∈B} is an F_{0}-base of A_{0}. We have

x′_{b}x′_{c}x′_{b}^{−1}x′_{c}^{−1} = \overline{β_{A}(b, c)} ∈ μ_{s}(F_{0}),

hence B_{0} = {x′_{b}F_{0}^{×} | b ∈ B} is an armature of the F_{0}-algebra A_{0}, with armature pairing isometric to the restriction of β_{A} to B. It follows that Z(A_{0}) is spanned by (x′_{z})_{z∈Z}, hence Z(A_{0}) = Z_{0}. As in Prop. 4.5, Z_{0} is Hom(Z, μ_{s})-Galois, for the action defined by

χ ∗ x′_{z} = χ(z)x′_{z} for χ ∈ Hom(Z, μ_{s}) and z ∈ Z.

Since β_{A} is nondegenerate, the map A → Hom(Z, μ_{s}) that carries a ∈ A to the character χ defined by

χ(z) = β_{A}(a, z)  for z ∈ Z  (4.11)

is surjective, and its kernel is Z^{⊥}. Therefore, A/Z^{⊥} ≅ Hom(Z, μ_{s}), and Z is (A/Z^{⊥})-Galois. For z ∈ Z and a ∈ A, (4.11) yields

χ ∗ x_{z} = β_{A}(a, z) x_{z} = x_{a}x_{z}x_{a}^{−1},

hence the action of χ on Z is conjugation by x_{a}; the induced action on Z_{0} is conjugation by a′, so diagram (4.10) commutes. □

The arguments above also show that { x′_{z}F_{0}^{×} | z ∈ Z } ⊆ Z_{0}^{×}/F_{0}^{×} is an armature of Z_{0} which is isomorphic to Z. We may use this armature to determine the primitive idempotents of Z_{0} as in Prop. 4.5: consider the map

ρ_{0} : Z → F_{0}^{×}/F_{0}^{×s} given by z ↦ x′_{z}^{s}F_{0}^{×s}.

Let K_{0} = ker(ρ_{0}), L_{0} = im(ρ_{0}), and r_{0} = |K_{0}|. Let also E be the graded division gr(F)-algebra Brauer-equivalent to gr(A).

Proposition 4.9. The F_{0}-algebra Z(A_{0}) contains r_{0} primitive idempotents, which are conjugate in gr(A). Letting t denote the index of any simple component of A_{0}, we have

ms(gr(A)) = r_{0}t^{−1}\sqrt{|B:Z|} .  (4.12)

Moreover, Z(E_{0}) is the s-Kummer extension of F_{0} associated with L_{0}, and Γ_{E}/Γ_{F} = w̃(K_{0}^{⊥}).

Proof. Prop. 4.5 shows that Z_{0} contains r_{0} primitive idempotents, which are conjugate under the Hom(Z, μ_{s})-Galois action, and whose isotropy subgroup is the orthogonal of K_{0} in Hom(Z, μ_{s}). On the other hand, Prop. 4.8 shows that the Hom(Z, μ_{s})-Galois action is also realized by inner automorphisms of gr(A), and yields an isomorphism Hom(Z, μ_{s}) ≅ A/Z^{⊥} (see (4.11)) carrying the orthogonal of K_{0} in the character group to K_{0}^{⊥}/Z^{⊥}. Therefore, the primitive idempotents of Z_{0} are conjugate in gr(A). Prop. 2.2 and 4.8 show that the inverse image of Γ_{E}/Γ_{F} in A is K_{0}^{⊥}; hence, Γ_{E}/Γ_{F} = w̃(K_{0}^{⊥}).

The center Z(E_{0}) is isomorphic to the simple components of Z_{0} = Z(A_{0}) (see Prop. 2.2), and hence also to the s-Kummer extension of F_{0} associated with L_{0}, by Prop. 4.5.

Finally, we compute the matrix size of gr(A). First, note that w̃: K_{0}^{⊥} → Γ_{E}/Γ_{F} is surjective with kernel B, hence |Γ_{E}:Γ_{F} | = |K_{0}^{⊥}| |B|^{−1}. Since the pairing β_{A} is nondegenerate, we have |K_{0}^{⊥}| |K_{0}| = |A|, hence

|Γ_{E}:Γ_{F} | = \frac{|A|}{r_{0}|B|} = \frac{[gr(A):gr(F)]}{r_{0}|B|}.

On the other hand,

[E_{0}:F_{0}] = t^{2}[Z(E_{0}):F_{0}] = t^{2}|L_{0}| = t^{2}|Z|r_{0}^{−1}.

Since [E:gr(F)] = [E_{0}:F_{0}] |Γ_{E}:Γ_{F} |, it follows that

[E:gr(F)] = \frac{t^{2}|Z| [gr(A):gr(F)]}{r_{0}^{2}|B|} = \frac{t^{2}[gr(A):gr(F)]}{r_{0}^{2}|B:Z|}.  (4.13)

Since ms(gr(A)) = \sqrt{[gr(A):gr(F)] [E:gr(F)]^{−1}}, formula (4.12) follows. □

Let D be the division F -algebra Brauer-equivalent to A. By combining Th. 4.1 and Prop. 4.9, we readily obtain a criterion for the extension of the valuation v on F to D:

Corollary 4.10. The valuation v on F extends to a valuation on D if and only if ms(A) = r_{0}t^{−1}\sqrt{|B:Z|}. When this occurs, \overline{D} = E_{0} and Γ_{D} = Γ_{E}.

Note that when it exists the valuation on D is necessarily tame since char(\overline{F}) ∤ s = exp(A) while deg(D) | deg(A) = \sqrt{|A|}.

Finally, we consider the case where v extends to a valuation v_{D} on D that is totally ramified over F. Recall from [TW, Sec. 3] that in this case there is a canonical pairing C_{D}: Γ_{D}/Γ_{F} × Γ_{D}/Γ_{F} → μ(\overline{F}) defined by C_{D}(γ_{1} + Γ_{F} , γ_{2} + Γ_{F}) = \overline{x_{1}x_{2}x_{1}^{−1}x_{2}^{−1}} for any x_{1}, x_{2} ∈ D^{×} with v_{D}(x_{i}) = γ_{i} for i = 1, 2.

Proposition 4.11. The valuation v on F extends to a valuation on D that is tamely and totally ramified over F if and only if deg(D) = \sqrt{|B^{⊥}:Z|}. When this occurs, we have Γ_{D}/Γ_{F} = w̃(B^{⊥}) and w̃ defines an isometry from B^{⊥}/Z with the nondegenerate pairing induced by β_{A} onto Γ_{D}/Γ_{F} with the pairing C_{D}.

Proof. Since [gr(A):gr(F)] = |A| = |B| |B^{⊥}|, equation (4.13) yields

[E:gr(F)] = t^{2}|B^{⊥}| |Z|r_{0}^{−2} = t^{2}|B^{⊥}:Z| |Z|^{2} r_{0}^{−2} = t^{2}|L_{0}|^{2} |B^{⊥}:Z|.

On the other hand, Th. 4.1 yields [D:F] ≥ [E:gr(F)]. Therefore, if [D:F] = |B^{⊥}:Z|, then we must have [D:F] = [E:gr(F)] and t = |L_{0}| = 1, hence v extends to valuation on D that is totally ramified over F.

For the converse, we apply Cor. 4.10 with t = 1 and r_{0} = |Z|, and obtain ms(A) = \sqrt{|B| |Z|}. Since [A:F] = [D:F]ms(A)^{2} and [A:F] = |A| = |B| |B^{⊥}|, it follows that [D:F] = |B^{⊥}:Z|.

For the rest of the proof, assume v extends to a valuation on D that is tamely and totally ramified over F. Then gr(D) ≅_{g} E, and r_{0} = |Z|, hence K_{0} = Z and Γ_{D}/Γ_{F} = Γ_{E}/Γ_{F} = w̃(Z^{⊥}), by Prop. 4.9. Since Z = B ∩ B^{⊥}, we have Z^{⊥} = B + B^{⊥}, hence w̃(Z^{⊥}) = w̃(B^{⊥}) since B = ker(w̃). We may identify the canonical pairing C_{D} with the pairing C on Γ_{E}/Γ_{F} given by

C(γ + Γ_{F} , δ + Γ_{F}) = ξηξ^{−1}η^{−1} for any nonzero ξ ∈ E_{γ}, η ∈ E_{δ}.

In order to relate C to β_{A}, we identify a copy of E in gr(A). First, we choose for each a ∈ A an element x_{a} ∈ A^{×} such that π(x_{a}) = a. As in the proof of Prop. 4.8, we choose x_{b} such that w(x_{b}) = 0 for b ∈ B. Note that Z_{0} = { x′_{z}F_{0}^{×} | z ∈ Z } is an armature of Z_{0} which is isomorphic to Z. Since Z = ker(ρ_{0}), Z_{0} is the kernel of the s-power map Z_{0} → F_{0}^{×}/F_{0}^{×s}. Therefore, the proof of Prop. 4.5 shows that after scaling x_{z} for z ∈ Z by suitable units in F^{×} we may assume x′_{z_{1}}x′_{z_{2}} = x′_{z_{1}z_{2}} for z_{1}, z_{2} ∈ Z.

As in the proof of Prop. 4.5, we consider e = \frac{1}{|Z|} ∑_{z∈Z} x′_{z}, which is a primitive idempotent in Z_{0} such that ex′_{z} = e for z ∈ Z. For a ∈ A, we have

x′_{a}ex′_{a}^{−1} = \frac{1}{|Z|} ∑_{z∈Z} \overline{β_{A}(a, z)}x′_{z},

which is e if a ∈ Z^{⊥}, and is another primitive idempotent of Z_{0} if a ∉ Z^{⊥}. Thus, ex′_{a}e = ex′_{a} = x′_{a}e if a ∈ Z^{⊥}, and e(x′_{a}ex′_{a}^{−1}) = 0, hence ex′_{a}e = 0, if a ∉ Z^{⊥}. Therefore, egr(A)e is spanned by (ex′_{a})_{a∈Z^{⊥}}. If a_{1}, …, a_{r} ∈ Z^{⊥} are in different cosets modulo Z, then ex′_{a_{1}}, …, ex′_{a_{r}} are linearly independent since each ex′_{a_{i}} lies in the span of (x′_{za_{i}})_{z∈Z}. Let n = |B:Z|, m = |B^{⊥}:Z|, and let b_{1}, …, b_{n} ∈ B (resp. c_{1}, …, c_{m} ∈ B^{⊥}) be representatives of the various cosets of B (resp. B^{⊥}) modulo Z. Since Z^{⊥} = B + B^{⊥} and Z = B ∩B^{⊥}, we have Z^{⊥}/Z = (B/Z)⊕(B^{⊥}/Z), hence {b_{i}c_{j} | 1 ≤ i ≤ n, 1 ≤ j ≤ m} is a set of representatives of the various cosets of Z^{⊥} modulo Z. For i = 1, …, n and j = 1, …, m, let

ξ_{i} = ex′_{b_{i}} = x′_{b_{i}}e ∈ eA_{0} ⊆ egr(A)e  and  η_{j} = ex′_{c_{j}} = x′_{c_{j}}e ∈ egr(A)e.

Then (ξ_{i}η_{j} | 1 ≤ i ≤ n, 1 ≤ j ≤ m) is a gr(F)-base of egr(A)e. Moreover, ξ_{i}η_{j} = η_{j}ξ_{i} since β_{A}(b_{i}, c_{j}) = 1. Therefore, the graded subalgebras B, B′ ⊆ egr(A)e spanned respectively by ξ_{1}, …, ξ_{n} and by η_{1}, …, η_{m} centralize each other, and

egr(A)e ≅_{g} B ⊗_{gr(F)} B′.

The degree of each ξ_{i} is 0 since b_{i} ∈ B = ker(w̃), hence B = B_{0} ⊗_{F_{0}} gr(F). On the other hand, the degree of η_{j} is 0 if and only if η_{j} ∈ eF_{0} since B^{⊥} ∩ B = Z. Therefore, eA_{0} = B_{0}. This algebra is split by hypothesis. Therefore, B is split and gr(A) is Brauer-equivalent to B′. Since [B′:gr(F)] = |B^{⊥}:Z| = [E:gr(F)], we may identify B′ with E. Clearly, under this identification the canonical pairing on Γ_{E}/Γ_{F} coincides with the pairing on B^{⊥}/Z induced by β_{A}. □

Remarks 4.12. (i) The description of \overline{D} and Γ_{D} in Cor. 4.10 (with additional information from Prop. 4.8 and 4.9) were given in [W_{3}, Th. 1], and proved using Morandi value functions. The proof given here is easier and more direct. By Prop. 2.5 the tame gauge y defined here is a Morandi value function (so the associated valuation ring A^{≥0} is a Dubrovin valuation ring) if and only if |B_{0}| = 1, i.e., if and only if A^{≥0} has a unique maximal two-sided ideal.

(ii) Suppose the valuation v on F is strictly Henselian, i.e., v is Henselian and \overline{F} is separably closed. Then, in the setting of Prop. 4.11 with char(\overline{F}) ∤ exp(A), v necessarily extends to a valuation on the division algebra D Brauer-equivalent to A, and D is totally and tamely ramified over F. In that situation, the description of the canonical pairing on D (which then determines D up to isomorphism by [TW, Prop. 4.2]) was given in [TW, Th. 4.3], with a more difficult proof.

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Institut de Mathématique Pure et Appliquée, Université catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium

   E-mail address: jean-pierre.tignol@uclouvain.be

Department of Mathematics, University of California, San Diego, La Jolla, CA-92093-0112, USA

   E-mail address: arwadsworth@ucsd.edu