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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.
1
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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
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J.-P. TIGNOL AND A. R. WADSWORTH
component Kiof gr(Z(A)), char(F) ∤ |ΓKi:ΓF|. It will be shown below (see Cor. 3.6) that whenever
char(F) = 0 every F-gauge is tame.
While gauges are defined for semisimple algebras, the next proposition shows that the study of gauges
can be reduced to the case of simple algebras.
Proposition 1.6. Let F be a field with valuation v, and let A be a semisimple F-algebra with an
F-gauge y. Suppose A is a direct product of F-subalgebras
A = B1× ... × Bk.
Then, y|Biis a gauge for each i, y(b1,...,bn) = min?y(b1),...,y(bn)?, and
gr(A)∼=g gr(B1) × ... × gr(Bn).
Furthermore, y is tame if and only if each y|Biis tame.
The proof of Prop. 1.6 will invoke the following easy but very useful lemma:
Lemma 1.7. Let F be a field with a valuation v, and let A be an F-algebra with a surmultiplicative
F-value function y. Suppose there is e ∈ A with e2= e and y(e) = 0. Then for any F-subspace N of A,
gr(eN) = e′gr(N)andgr(Ne) = gr(N)e′. (1.6)
Proof. For any s ∈ N, we have e′s′= (es)′or e′s′= 0, by (1.5). Hence, e′gr(N) ⊆ gr(eN). On the
other hand, y(e(es)) = y(es) = y(e) + y(es). Hence, e′(es)′= (e(es))′= (es)′, showing that gr(eN) =
e′gr(eN) ⊆ e′gr(N). This proves the first equality in (1.6), and the second is proved analogously.
Proof of Prop. 1.6. It suffices by induction to prove the case n = 2. Therefore, assume A = B ×C with
B and C nontrivial subalgebras of A. Let e = (1B,0) and f = (0,1C) in A. So, e and f are nonzero
orthogonal central idempotents of A with e + f = 1, and B = Ae = eA, C = fA = Af.
Consider e′∈ gr(A). Since e ∈ Z(A), we have e′∈ Z(gr(A)). Also, y(e) ≤ 0, as y(e) = y(e2) ≥
y(e) + y(e). If y(e) < 0, then y(e2) > y(e) + y(e), so e′2= 0. But then, as e′is central, e′gr(A) is
a nonzero homogeneous nilpotent ideal of gr(A), contradicting the semisimplicity of gr(A). Therefore,
we must have y(e) = 0; hence, as y(e2) = 0 = y(e) + y(e), we have e′2= (e2)′= e′. Thus, e′is a
nonzero homogeneous central idempotent of gr(A). Likewise, the same is true for f′. Furthermore, as
y(e) = y(f) = y(e + f) = 0 we have e′+ f′= 1′in gr(F). Therefore,
gr(A)∼=g e′gr(A) × f′gr(A).
By Lemma 1.7, e′gr(A) = gr(eA) = gr(B); likewise, f′gr(A) = gr(C). Thus, (1.7) becomes
gr(A)∼=ggr(B) × gr(C).
Since y is a norm on A, y|B and y|C are norms by Prop. 1.1(iii). From the direct decomposition
of gr(A), Prop. 1.1 also shows that B and C are splitting complements in A for y, i.e., y(b,c) =
min?y(b),y(c)?for all b ∈ B, c ∈ C. Furthermore, gr(B) and gr(C) must be graded semisimple, since
Z(A) = Z(B) × Z(C) and Z(gr(A))∼=g Z?gr(B)) × gr(C)?
definition, y is tame if and only if y|Band y|Care each 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.
?
(1.7)
they are direct factors of gr(A), which is graded semisimple. Thus, y|Band y|Care F-gauges. We have
= Z(gr(B)) × Z(gr(C)), so Z(gr(A)) is
separable over gr(F) if and only if Z(gr(B)) and Z(gr(C)) are each separable over gr(F). Thus, by the
?
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VALUE FUNCTIONS AND ASSOCIATED GRADED RINGS FOR SEMISIMPLE ALGEBRAS31
Note that w and ? w are group homomorphisms, since the commutators of elements of π−1(A) are roots
π(xa) = a. Then (xa)a∈Ais an F-base of A. Define an F-norm y: A → Γ ∪ {∞} by
y? n ?
The definition of y depends on A, but is independent of the choice of the xa.
Proposition 4.7. The F-norm y is a tame F-gauge on A and ΓAis determined by
of unity, hence elements in F×of value 0. Clearly, w|F× = v. For each a ∈ A, pick xa∈ A×such that
i=1
λaxa
?
= min
a∈A
?v(λa) + w(xa)?.
ΓA/ΓF = ? w(A).
The graded algebra gr(A) has an armature isometric to (A,βA). Moreover, every homogeneous com-
ponent of gr(A) contains an invertible element, hence the subgroup ∆gr(A)⊆ Γgr(A)defined in Sec. 2
coincides with ΓA, and the map θgr(A)of (2.5) is a homomorphism
θgr(A): ΓA/ΓF→ Aut(Z(A0)).
Proof. Note that y|π−1(A)= w. Hence, for all a, b ∈ A, we have y(xaxb) = y(xa) + y(xb); so for the
image x′aof xain gr(A), (xaxb)′= x′ax′
π−1(A) maps to a subgroup of gr(A)×. Furthermore, (x′a)a∈Ais a gr(F)-base of gr(A) by Prop. 1.1(i),
since (xa)a∈Ais 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 xaF×?→ x′agr(F)×is clearly a group
isomorphism and also an isometry between the armature pairings βAand βA′ when we identify µs(F)
with µs(F). The pairing βA′ 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.
b. It follows by Lemma 1.2 that y is surmultiplicative, and that
?
Our next goal is to describe the degree 0 component A0⊆ gr(A), which is a semisimple algebra over
F0= F. For this, we consider
B = ker(? w) ⊆ A,Z = B ∩ B⊥,
and denote by Z ⊆ A the subalgebra spanned by π−1(Z). Since βAis trivial on Z, the F-algebra Z is
commutative.
Proposition 4.8. The F0-algebra A0has an armature B0canonically isomorphic to B with armature
pairing βB0isometric to the restriction of βAto B. Its center Z(A0) is the degree 0 component of Z,
i.e. Z(A0) = Z0; it is an (A/Z⊥)-Galois F0-algebra. For the map ψ: A/Z⊥֒→ AutF0(Z(A0)) given by
the Galois action, the following diagram is commutative:
A
?
e w
− − − − →ΓA/ΓF
?θgr(A)
A/Z⊥
ψ
− − − − → AutF0(Z(A0)).
(4.10)
Proof. We first fix a convenient choice of xa∈ A×such that π(xa) = a ∈ A: for b ∈ B, we choose xb
such that w(xb) = 0. As observed in the proof of Prop. 4.7, (x′a)a∈Ais a homogeneous gr(F)-base of
gr(A). We have y(xa) = 0 if and only if a ∈ B, hence (x′
x′
b
c
b)b∈Bis an F0-base of A0. We have
bx′
cx′
−1x′
−1= βA(b,c) ∈ µs(F0),
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J.-P. TIGNOL AND A. R. WADSWORTH
hence B0= {x′
restriction of βAto B. It follows that Z(A0) is spanned by (x′z)z∈Z, hence Z(A0) = Z0. As in Prop. 4.5,
Z0is Hom(Z,µs)-Galois, for the action defined by
χ ∗ x′
Since βAis nondegenerate, the map A → Hom(Z,µs) that carries a ∈ A to the character χ defined by
χ(z) = βA(a,z)
is surjective, and its kernel is Z⊥. Therefore, A/Z⊥ ∼= Hom(Z,µs), and Z is (A/Z⊥)-Galois. For z ∈ Z
and a ∈ A, (4.11) yields
χ ∗ xz = βA(a,z)xz = xaxzx−1
hence the action of χ on Z is conjugation by xa; the induced action on Z0 is conjugation by a′, so
diagram (4.10) commutes.
bF×
0| b ∈ B} is an armature of the F0-algebra A0, with armature pairing isometric to the
z= χ(z)x′
z
for χ ∈ Hom(Z,µs) and z ∈ Z.
for z ∈ Z (4.11)
a,
?
The arguments above also show that {x′zF×
isomorphic to Z.
Prop. 4.5: consider the map
0| z ∈ Z } ⊆ Z×
0/F×
0
is an armature of Z0 which is
We may use this armature to determine the primitive idempotents of Z0 as in
ρ0: Z → F×
0/F×s
0
given by z ?→ x′
z
sF×s
0.
Let K0 = ker(ρ0), L0 = im(ρ0), and r0 = |K0|. Let also E be the graded division gr(F)-algebra
Brauer-equivalent to gr(A).
Proposition 4.9. The F0-algebra Z(A0) contains r0 primitive idempotents, which are conjugate in
gr(A). Letting t denote the index of any simple component of A0, we have
ms(gr(A)) = r0t−1?
Moreover, Z(E0) is the s-Kummer extension of F0associated with L0, and ΓE/ΓF= ? w(K⊥
Proof. Prop. 4.5 shows that Z0 contains r0 primitive idempotents, which are conjugate under the
Hom(Z,µs)-Galois action, and whose isotropy subgroup is the orthogonal of K0in Hom(Z,µs). On the
other hand, Prop. 4.8 shows that the Hom(Z,µs)-Galois action is also realized by inner automorphisms
of gr(A), and yields an isomorphism Hom(Z,µs)∼= A/Z⊥(see (4.11)) carrying the orthogonal of K0
in the character group to K⊥
Prop. 2.2 and 4.8 show that the inverse image of ΓE/ΓF in A is K⊥
The center Z(E0) is isomorphic to the simple components of Z0= Z(A0) (see Prop. 2.2), and hence
also to the s-Kummer extension of F0associated with L0, by Prop. 4.5.
Finally, we compute the matrix size of gr(A). First, note that ? w: K⊥
hence
|ΓE:ΓF| =
r0|B|
On the other hand,
[E0:F0] = t2[Z(E0):F0] = t2|L0| = t2|Z|r−1
Since [E:gr(F)] = [E0:F0]|ΓE:ΓF|, it follows that
t2|Z|[gr(A):gr(F)]
r2
0|B|
Since ms(gr(A)) =
|B:Z|. (4.12)
0).
0/Z⊥. Therefore, the primitive idempotents of Z0are conjugate in gr(A).
0; hence, ΓE/ΓF= ? w(K⊥
0).
0→ ΓE/ΓF is surjective with
kernel B, hence |ΓE:ΓF| = |K⊥
0||B|−1. Since the pairing βAis nondegenerate, we have |K⊥
0||K0| = |A|,
|A|
=
[gr(A):gr(F)]
r0|B|
.
0.
[E:gr(F)] =
=
t2[gr(A):gr(F)]
r2
0|B:Z|
. (4.13)
?[gr(A):gr(F)][E:gr(F)]−1, formula (4.12) follows.
?
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VALUE FUNCTIONS AND ASSOCIATED GRADED RINGS FOR SEMISIMPLE ALGEBRAS33
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) = r0t−1?|B:Z|.
When this occurs, D = E0and ΓD= ΓE.
Note that when it exists the valuation on D is necessarily tame since char(F) ∤ s = exp(A) while
deg(D) | deg(A) =?|A|.
Finally, we consider the case where v extends to a valuation vDon D that is totally ramified over F.
Recall from [TW, Sec. 3] that in this case there is a canonical pairing CD: ΓD/ΓF× ΓD/ΓF → µ(F)
defined by CD(γ1+ ΓF,γ2+ ΓF) = x1x2x−1
2
for any x1, x2∈ D×with vD(xi) = γifor 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) =
isometry from B⊥/Z with the nondegenerate pairing induced by βAonto ΓD/ΓF with the pairing CD.
1x−1
?|B⊥:Z|. When this occurs, we have ΓD/ΓF= ? w(B⊥) and ? w defines an
Proof. Since [gr(A):gr(F)] = |A| = |B||B⊥|, equation (4.13) yields
[E:gr(F)] = t2|B⊥||Z|r−2
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 = |L0| = 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 r0= |Z|, and obtain ms(A) =
[A:F] = [D:F]ms(A)2and [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)∼=gE, and r0= |Z|, hence K0= Z and ΓD/ΓF= ΓE/ΓF= ? w(Z⊥), by Prop. 4.9.
the canonical pairing CDwith the pairing C on ΓE/ΓF given by
0
= t2|B⊥:Z||Z|2r−2
0
= t2|L0|2|B⊥:Z|.
?|B||Z|. Since
Since Z = B ∩ B⊥, we have Z⊥= B + B⊥, hence ? w(Z⊥) = ? w(B⊥) since B = ker(? w). We may identify
C(γ + ΓF,δ + ΓF) = ξηξ−1η−1
for any nonzero ξ ∈ Eγ, η ∈ Eδ.
In order to relate C to βA, we identify a copy of E in gr(A). First, we choose for each a ∈ A an element
xa∈ A×such that π(xa) = a. As in the proof of Prop. 4.8, we choose xbsuch that w(xb) = 0 for b ∈ B.
Note that Z0= {x′zF×
Z0is the kernel of the s-power map Z0→ F×
scaling xzfor z ∈ Z by suitable units in F×we may assume x′z1x′z2= x′z1z2for z1, z2∈ Z.
As in the proof of Prop. 4.5, we consider e =
|Z|
that ex′z= e for z ∈ Z. For a ∈ A, we have
x′
a
|Z|
0| z ∈ Z } is an armature of Z0which is isomorphic to Z. Since Z = ker(ρ0),
0/F×s
0. Therefore, the proof of Prop. 4.5 shows that after
1
?
z∈Zx′z, which is a primitive idempotent in Z0such
aex′
−1=
1
?
z∈Z
βA(a,z)x′z,
which is e if a ∈ Z⊥, and is another primitive idempotent of Z0if a / ∈ Z⊥. Thus, ex′ae = ex′a= x′ae if
a ∈ Z⊥, and e(x′aex′a
If a1, ..., ar ∈ Z⊥are in different cosets modulo Z, then ex′a1, ..., ex′arare linearly independent
since each ex′ailies in the span of (x′zai)z∈Z. Let n = |B:Z|, m = |B⊥:Z|, and let b1,...,bn∈ B
(resp. c1,...,cm∈ B⊥) be representatives of the various cosets of B (resp. B⊥) modulo Z.
Z⊥= B + B⊥and Z = B∩B⊥, we have Z⊥/Z = (B/Z)⊕(B⊥/Z), hence {bicj| 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
ξi = ex′
bie ∈ eA0 ⊆ egr(A)e
−1) = 0, hence ex′ae = 0, if a / ∈ Z⊥. Therefore, egr(A)e is spanned by (ex′a)a∈Z⊥.
Since
bi= x′
andηj = ex′
cj= x′
cje ∈ egr(A)e.
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J.-P. TIGNOL AND A. R. WADSWORTH
Then (ξiηj| 1 ≤ i ≤ n, 1 ≤ j ≤ m) is a gr(F)-base of egr(A)e. Moreover, ξiηj= ηjξisince βA(bi,cj) = 1.
Therefore, the graded subalgebras B, B′⊆ egr(A)e spanned respectively by ξ1, ..., ξnand by η1,...,ηm
centralize each other, and
egr(A)e∼=g B ⊗gr(F)B′.
The degree of each ξiis 0 since bi∈ B = ker(? w), hence B = B0⊗F0gr(F). On the other hand, the
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 ΓE/ΓF
coincides with the pairing on B⊥/Z induced by βA.
Remarks 4.12. (i) The description of D and ΓDin Cor. 4.10 (with additional information from Prop. 4.8
and 4.9) were given in [W3, 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≥0is a Dubrovin valuation ring) if and only if |B0| = 1, i.e., if and only if
A≥0has a unique maximal two-sided ideal.
degree of ηjis 0 if and only if ηj∈ eF0since B⊥∩ B = Z. Therefore, eA0= B0. This algebra is split
?
(ii) Suppose the valuation v on F is strictly Henselian, i.e., v is Henselian and F is separably closed.
Then, in the setting of Prop. 4.11 with char(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 Math´ ematique Pure et Appliqu´ ee, Universit´ e 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