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VALUE FUNCTIONS AND ASSOCIATED GRADED RINGS FOR SEMISIMPLE

ALGEBRAS

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

Abstract. We introduce a type of value function y called a gauge on a finite-dimensional semisimple

algebra A over a field F with valuation v. The filtration on A induced by y yields an associated graded

ring gry(A) which is a graded algebra over the graded field grv(F). Key requirements for y to be a gauge

are that gry(A) be graded semisimple and that dimgrv(F)(gry(A)) = dimF(A). It is shown that gauges

behave well with respect to scalar extensions and tensor products. When v is Henselian and A is central

simple over F, it is shown that gry(A) is simple and graded Brauer equivalent to grw(D) where D is the

division algebra Brauer equivalent to A and w is the valuation on D extending v on F. The utility of

having a good notion of value function for central simple algebras, not just division algebras, and with

good functorial properties, is demonstrated by giving new and greatly simplified proofs of some difficult

earlier results on valued division algebras.

Introduction

Valuation theory is a time-honored subject, which has undergone a robust development for non-

commutative 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 [W4] 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 carr´ ees of Bruhat and Tits

[BT] (see Rem. 1.21). 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 (1.16). 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 [Bl1], [Bl2], and [HW2]. But they have not been used

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.

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

in the earlier work with value functions on central simple algebras in [BT], nor with the value functions

associated to Dubrovin valuation rings in [M2]. (The relation between the value functions considered

here and Morandi’s value functions in [M2] 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 gry(A) is then a finite dimensional algebra over the graded field grv(F). If A is central

simple over F, there are typically many different v-gauges y on A; it turns out that gry(A) is always a

graded simple algebra (i.e., there are no nontrivial homogeneous ideals), and that the class of gry(A) is

uniquely determined in the graded Brauer group of its center, see Cor. 3.7.

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 = EndD(M), where M is a finite dimensional right D-vector space, we

prove in Th. 3.1 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 EndD(M) induced by α on M as described in §1.3. It follows

that gry(A) is isomorphic as a graded ring to Endgrw(D)(grα(M)); furthermore, the graded Brauer class

of gry(A) is the same as that of grw(D), and gry(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 gry(A) is grv(F), then

gry(A) is a graded central simple algebra over grv(F) with the same Schur index as A. We may then

consider its Brauer class [gry(A)] in the graded Brauer group GBr(grv(F)). The map [A] ?→ [gry(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(grv(F)). That Ψ is

an isomorphism was proved earlier in [HW2]; without the use of gauges the proof in [HW2] that Ψ is

a group homomorphism was particularly involved and arduous. The proof given here in Th. 3.8 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. 3.3 that for any v-

gauge on A the simple components of gry(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 vhon field Fhis 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 vh-gauge yhon

A⊗FFh, and gry(A) is graded isomorphic to gryh(A⊗FFh). 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 [W3] by much more complicated arguments.

The organization of the paper is as follows: §1 gives the definition of gauges and describes various ex-

amples on division algebras, endomorphism algebras, and tensor products. In §2 we review some results

on graded central simple algebras, complementing the discussion in [HW2] 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,

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VALUE FUNCTIONS AND ASSOCIATED GRADED RINGS FOR SEMISIMPLE ALGEBRAS3

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;

w(cd) = w(c) + w(d);

w(c + d) ≥ min?w(c),w(d)?.

(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)?.)

units of D, i.e., D×= D − {0}; its valuation ring VD= {d ∈ D| w(d) ≥ 0}; the unique maximal left

(and right) ideal MDof VD, MD= {d ∈ D| w(d) > 0}; and the residue division ring D = VD/MD.

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 Dγ= D≥γ/D>γ. The associated graded

ring of D with respect to w is grw(D) =

γ∈Γ

a well-defined multiplication Dγ× Dδ→ Dγ+δgiven by (c + D>γ) · (d + D>δ) = cd + D>γ+δ. This

multiplication is extended biadditively to all of grw(D), making grw(D) into a graded ring. When w is

clear, we write gr(D) for grw(D). The grade group of gr(D), denoted Γgr(D), is {γ ∈ Γ| Dγ?= 0}; note

that Γgr(D)= ΓD. Also, for the degree 0 component of gr(D), we have D0= D≥0/D>0= VD/MD= D.

For d ∈ D×, we write d′for the image of d in gr(D), i.e., d′= d + D>w(d)∈ Dw(d). The homogeneous

elements of gr(D) are those in

γ∈Γ

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

(1.1a)

(1.1b)

(1.1c)

Associated to the valuation on D, we have its value group ΓD = w(D×), where D×is the group of

?

Dγ. For each γ, δ ∈ Γ, the multiplication in D induces

?

Dγ. It follows from property (1.1b) that gr(D) is a graded division

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;

α(md) = α(m) + w(d);

α(m + n) ≥ min?α(m),α(n)?.

Given such an α on M, we can form the associated graded module grα(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 Mγ= M≥γ/M>γ. Define

gr(M) = grα(M) =?

for 0 ∈ M, let 0′= 0 ∈ gr(M). For γ, δ ∈ Γ there is a well-defined multiplication Mγ× Dδ→ Mγ+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 dimgr(D)(gr(M)) for

the cardinality of any gr(D)-module base of gr(M). If N =?

(1.2a)

(1.2b)

(1.2c)

γ∈Γ

Mγ. For nonzero m ∈ M, let m′denote the image m + M>α(m)of m in gr(M);

γ∈Γ

Nγis another graded right gr(D)-vector

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

space, we say that M and N are graded isomorphic, written M∼=gN, if there is a gr(D)-vector space

isomorphism f : M → N with f(Mγ) = Nγfor each γ ∈ Γ.

Now, suppose M is finite-dimensional. A right D-vector space base (mi)1≤i≤kof M is called a splitting

base with respect to α if for all d1, ..., dk∈ D,

α?k ?

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 (mi)1≤i≤kof M, and take

any γ1, ..., γk∈ Γ. Define α(mi) = γifor 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 (mi)1≤i≤kis a splitting base for α.

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

i=1

midi

?

= min

1≤i≤k

?α(mi) + w(di)?. (1.3)

Proposition 1.1. Let α be a D-value function on M. Take any m1, ..., mℓ∈ M.

ℓare gr(D)-linearly independent in gr(M) iff α? ℓ ?

all d1, ..., dℓ∈ D. When this occurs, m1, ..., mℓare D-linearly independent in M.

(ii) dimgr(D)(gr(M)) ≤ dimD(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|Nis 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;

y(ca) = v(c) + y(a) for any c ∈ F;

y(a + b) ≥ min?y(a),y(b)?;

y(ab) ≥ y(a) + y(b).

Note that for such a y, there is a corresponding “valuation ring” VA= A≥0= {a ∈ A| y(a) ≥ 0}. There

is also an associated graded ring gr(A) = gry(A) =

(i) m′

1, ..., m′

i=1

midi

?

= min

1≤i≤ℓ

?α(mi)+w(di)?for

(1.4a)

(1.4b)

(1.4c)

(1.4d)

?

γ∈Γ

Aγ, where Aγ= A≥γ?A>γ, as above, and the

multiplication in gr(A) is induced by that of A. Furthermore, gry(A) is clearly a graded grv(F)-algebra.

Also, grv(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 dimgr(F)(gr(A)) ≤ dimF(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 (ai)1≤i≤k be a

splitting base of A. If y(aiaj) ≥ y(ai) + y(aj) 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 b1=?ciaiand b2=?diaiin A with

y(b1b2) = y??

≥ min

ij

ci,di∈ F. Then,

i,j

cidjaiaj

?

≥ min

i,j

?y(cidjaiaj)?

+ min

≥ min

i,j

?v(ci) + v(dj) + y(ai) + y(aj)?

= y(b1) + y(b2).

?v(ci) + y(ai)?

?v(dj) + y(aj)?

?

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VALUE FUNCTIONS AND ASSOCIATED GRADED RINGS FOR SEMISIMPLE ALGEBRAS5

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 Ay(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,

?

0′,if y(ab) > y(a) + y(b).

a′b′=

(ab)′, if a′b′?= 0′, iff y(ab) = y(a) + y(b);

(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.

finite-dimensional algebra A. Therefore, u ∈ A×. The formula in (iii) follows by setting a = u−1in (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).

(ii) ⇒ (iii) Condition (ii) shows that u is not a zero divisor in the

?

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

dimK(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 N2= {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). Just as in the ungraded case,

Z(gr(A))∼=gK1× ... × Kn, where each Kiis a graded field which is the center of a simple component

of A. Also, Z(gr(A)) is separable over gr(F) if and only if each Ki is separable over F. By [HW1,

Th. 3.11], this holds if and only if Ki,0is separable over gr(F)0and char(gr(F)0) ∤ |ΓKi:ΓF|. Thus,

the gauge is tame if and only if Z(gr(A)) = gr(Z(A)), Z(A) is separable over F, and for each simple