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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 ALGEBRAS7
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 x2= 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) gry(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(x2) = 2y(x) for all x ∈ A;
(e) y(xn) = ny(x) for all x ∈ A.
We first consider the case where A is a field.
Proposition 1.8. 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 gry(K) is semisimple. Then, there
exist valuations v1, ..., vnon K extending v such that
y(x) = min
1≤i≤n(vi(x))for x ∈ K.(1.8)
Moreover, there is a natural graded isomorphism of graded gr(F)-algebras
gry(K)∼=g grv1(K) × ... × grvn(K).
Proof. Let ΓK= y(K×) ⊆ Γ and ΓF= v(F×) = y(F×) ⊆ ΓK. If x1, ..., xr∈ K are such that y(x1),
..., y(xr) belong to different cosets of ΓF in ΓK, then x′
over gr(F), hence x1, ..., xrare linearly independent over F, see Prop. 1.1(i). Since [K:F] is finite, it
follows that the index |ΓK:ΓF| is finite, hence ΓK/ΓF is torsion.
Let VF and MF denote the valuation ring of F and its maximal ideal, and let
1, ..., x′r∈ gr(K) are linearly independent
Vy = {x ∈ K | y(x) ≥ 0} andMy = {x ∈ K | y(x) > 0}.
Clearly, Vyis a subring of K containing VF and Myis an ideal of Vycontaining MF. Since gry(K) is
assumed to be semisimple, we have y(xn) = ny(x) for all x ∈ K (see condition (e) above), hence the
ideal Myis radical. We may therefore find a set of prime ideals Pλ⊆ Vy(indexed by some set Λ) such
that My=
λ∈Λ
valuation ring Vλof K with maximal ideal Mλsuch that Vy⊆ Vλand Pλ= Vy∩ Mλ.
Claim: The valuation vλcorresponding to Vλextends v. We have VF⊆ Vy⊆ Vλ, hence VF⊆ Vλ∩ F.
Similarly, MF⊆ My⊆ Pλ⊆ Mλ, so MF⊆ Mλ∩F. Since Vλ∩F is a valuation ring of F with maximal
ideal Mλ∩ F, the inclusions VF ⊆ Vλ∩ F and MF ⊆ Mλ∩ F imply VF = Vλ∩ F, proving the claim.
Hence, each value group ΓK,viembeds 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 Vλ= Vλ′
implies Mλ= Mλ′, hence Pλ= Pλ′, it follows that Λ is a finite set. Let Λ = {1,...,n} and, for x ∈ K,
let
?
Pλ. By Chevalley’s Extension Theorem [EP, Th. 3.1.1], we may find for each λ ∈ Λ a
w(x) = min
1≤i≤n
?vi(x)?.
Since Vy⊆
n ?
i=1
Vi, we have
y(x) ≥ 0 ⇒ w(x) ≥ 0for x ∈ K.
Page 8
8
J.-P. TIGNOL AND A. R. WADSWORTH
Similarly, since My=
n ?
i=1
Pi= Vy∩?n ?
y(x) > 0 ⇐⇒ (y(x) ≥ 0 and w(x) > 0).
i=1
Mi
?, we have
It follows that
y(x) = 0 ⇒ w(x) = 0for x ∈ K×. (1.9)
Now, fix some x ∈ K×. Since ΓK/ΓF is a torsion group we may find an integer m > 0 and an element
u ∈ F×such that my(x) = v(u), hence y(xmu−1) = 0. By (1.9), we then have w(xmu−1) = 0. Because
each viextends v, we have w(xmu−1) = w(xm) − v(u). Hence,
mw(x) = w(xm) = v(u) = my(x).
Since Γ has no torsion, it follows that w(x) = y(x), which proves (1.8).
For i = 1, ..., n we have y(x) ≤ vi(x) for all x ∈ K, hence the identity map on K induces a map
gry(K) → grvi(K). Combining these maps, we obtain a graded homomorphism of graded gr(F)-algebras
gry(K) → grv1(K) × ... × grvn(K).
This map is injective since for every x ∈ K×there is some index i such that y(x) = vi(x).
surjectivity, fix any k, 1 ≤ k ≤ n and any b ∈ K×.
Claim: There is c ∈ K×with vk(c) = vk(b) and vi(c) ≥ vk(b) for i ?= k. This will be proved below.
Now, since the valuations v1,...,vnare incomparable and Vy=
(1.10)
For
n ?
i=1
Viby (1.8), the map Vy→
n ?
i=1
Vi/Mi
is surjective by [EP, Th. 3.2.7(3)]. Therefore, there is d ∈ Vywith d ∈ Mifor i ?= k and d− bc−1∈ Mk.
Let a = cd. Then, vi(a) > vi(c) ≥ vk(b) for i ?= k, and vk(a) = vk(b) with a′= b′in grvk(K). Hence,
y(a) = vk(b) and a′∈ gry(K) maps to (0,...,0,b′,0...,0) ∈ grv1(K) × ... × grvn(K) (b′in the k-th
position). Since these n-tuples span grv1(K) × ... × grvn(K), it follows that the natural map (1.10) is
onto.
Proof of Claim. Assume for simplicity that Γ is the divisible hull of ΓF (= ΓF,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
∆F⊆ ΓF, which is the kernel of the canonical epimorphism ΓF→ ΓF,z. Let ∆ be the divisible hull of
∆Fin Γ, and let Λ = Γ/∆, which is a divisible group with ordering inherited from Γ. Since ΓF∩∆ = ∆F,
the order-preserving inclusion ΓF/∆F֒→ Λ identifies ΓF,zcanonically 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 vi,vj on K with i ?= j there is a valuation vijon K which is the finest
common coarsening of viand vj. (vijis the valuation associated to the valuation ring ViVj.) Let ∆ij⊆ Γ
be the divisible hull of the convex subgroup of ΓFassociated to the restriction of vijto F. Let Γi= ΓK,vi
for 1 ≤ i ≤ n. We say that an n-tuple (β1,...,βn) ∈ Γ1× ... × Γnis compatible if βi− βj∈ ∆ijfor
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 γi= vi(b) ∈ Γi, and note that (γ1,...,γn) is compatible since
vi(b) and vj(b) have the same image vij(b) in Γ?∆ij. For our fixed k and for each i, let ǫi= γk− γi.
|ǫi|−|ǫj| ∈ ∆ij. Because each Γ/Γiis a torsion group, there is a positive integer m such that m|ǫi| ∈ Γi
for each i. Let δi= γi+ m|ǫi| ∈ Γi. Then δk= γk, and for each i we have
δi = γi+ m|ǫi| ≥ γi+ ǫi = γk.
Note that (δ1,...,δn) is compatible, since (γ1,...,γn) and (m|ǫ1|,...,m|ǫn|) are compatible. Since the
valuations v1,...,vnare incomparable, by [R, Th. 1, p. 135] there is c ∈ K×with vi(c) = δifor each i.
This c has the properties of the claim.
So, for all i ?= j, ǫi− ǫj= γj− γi∈ ∆ij. Since 0 ≤
??|ǫi| − |ǫj|??≤ |ǫi− ǫj| and ∆ijis convex, we have
?
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VALUE FUNCTIONS AND ASSOCIATED GRADED RINGS FOR SEMISIMPLE ALGEBRAS9
Let v1, ..., vrbe all the extensions of v to K. For i = 1, ..., r, let ei= e(vi/v) be the ramification
index and fi= f(vi/v) be the residue degree. We say that the fundamental equality holds for K/F if
r ?
Corollary 1.9. 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
?vi(x)?
where v1, ..., vrare all the extensions of v to K.
[K:F] =
i=1
eifi.
y(x) = min
1≤i≤r
for x ∈ K,(1.11)
Proof. Suppose y is an F-gauge on K. By Prop. 1.8, we may find some extensions v1, ..., vnof v to
K such that gry(K)∼=g grv1(K) × ... × grvn(K). Now, [grvi(K):gr(F)] = eifiby [Bl1, Cor. 2], and
[gry(K):gr(F)] = [K:F] since y is an F-norm, so
[K:F] =
n ?
i=1
eifi.
This implies v1, ..., vnis 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 (1.11) defines an F-gauge
on K.
?
The following special case will be used in §1.2:
Corollary 1.10. Let K/F be a finite-degree field extension and let v be a valuation on F. Suppose
char(F) ∤ [K:F]. If v extends uniquely to K, this extension is an F-gauge on K.
Proof. In view of Cor. 1.9, it suffices to show that
[K:F] = [K:F]|ΓK:ΓF|, (1.12)
which may be regarded as a (weak) version of Ostrowski’s theorem. We include a proof for lack of a
convenient reference. The proof is by reduction to the Henselian case.
Because char(F) ∤ [K:F], K lies in the separable closure Fsepof F. Let G = G(Fsep/F), and let
H = G(Fsep/K) ⊆ G. Fix any extension w of v to Fsep, and let Z = {σ ∈ G| w ◦ σ = w}, the
decomposition group of w/v. Let Fhbe the fixed field FZ
p. 121]). Likewise, Z ∩ H is the decomposition group of w/w|K, and for Kh= FZ∩H
Henselization of w|K. Now, G acts transitively on the set E of all extensions of v to Fsep. But because
w|Kis the unique extension of v to K, this E is also the set of all extensions of w|Kto Fsep; so H also
acts transitively on E. Hence, G = HZ. Therefore,
[Kh:Fh] = |Z:(Z ∩ H)| = |HZ:H| = |G:H| = [K:F].
Let δK/F= [K:F]??[K:F]|ΓK:ΓF|?, the defect of K over F. Because the Henselization w|Fh is an
δK/F= δKh/Fh. Let N be the normal closure of Khover Fh. Clearly, δKh/Fh = δN/Fh?δN/Kh. By [EP,
Cor. 5.3.8], if char(F) = 0, then δN/Fh = δN/Kh = 1, so δKh/Fh = 1. But, if char(F) = p ?= 0, then
δN/Fh and δN/Kh are each a power of p. Hence, δKh/Fh = pmfor some integer m. By the Fundamental
Inequality, m ≥ 0. But then δKh/Fh is an integer, so it must divide [Kh:Fh], which is prime to p by
(1.13). Thus, in all cases we have δKh/Fh = 1, hence δK/F= 1, which proves (1.12).
sep; so w|Fh is a Henselization of v (cf. [EP,
sep , w|Kh is a
(1.13)
immediate extension of v and w|Kh is an immediate extension of w|K, equation (1.13) shows that
?
We now turn to the general type of commutative semisimple algebras.
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J.-P. TIGNOL AND A. R. WADSWORTH
Proposition 1.11. Let K1, ..., Kmbe finite-degree field extensions of a field F and A = K1×...×Km.
Let v be a valuation on F, let y be a surmultiplicative v-value function on A, and let yi= y|Kifor i = 1,
..., m. If gry(A) is graded semisimple, then each yihas the form (1.8) and, for a = (a1,...,an) ∈ A,
y(a) = min
1≤i≤m
?yi(ai)?. (1.14)
Moreover, there is a canonical graded isomorphism of graded gr(F)-algebras
gry(A)∼=g gry1(K1) × ... × grym(Km).
There is a v-gauge on A if and only if the fundamental equality holds for each Ki/F. When that
condition holds, there is a unique v-gauge y on A, defined by (1.14) where each yiis the unique v-gauge
on Kias in Cor. 1.9.
(1.15)
Proof. This is immediate from Cor. 1.9 and Prop. 1.6.
?
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
θD: ΓD/ΓF→ Aut?Z(D)?
which for d ∈ D×maps w(d) + ΓF to the automorphism z ?→ dzd−1(see [JW, (1.6)]).
Proposition 1.12. With the notation above, the valuation w is an F-gauge on D if and only if
(1.16)
[D:F] = [D:F]|ΓD:ΓF|. (1.17)
When this condition holds, the gauge w is tame if and only if Z(D) is separable over F and char(F) ∤ |ker(θD)|.
If char(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. [HW2, (1.7)] or [Bl2, p. 4278], we have
[gr(D):gr(F)] = [D:F]|ΓD:ΓF|. (1.18)
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))]. (1.19)
From the definition of θD, it is clear that ΓZ(D)/ΓF⊆ ker(θD), hence there is an induced map
θD: ΓD/ΓZ(D)→ Aut(Z(D)).
Clearly, |ker(θD)| = |ker(θD)||ΓZ(D):ΓF|. When (1.19) holds, Boulagouaz proved in [Bl2, Cor. 4.4] that
Z(gr(D)) = gr(Z(D)) if and only if Z(D)/Z(D) is separable and char(F) ∤ |ker(θD)|. Therefore, the
following statements are equivalent:
(a) Z(gr(D)) = gr(Z(D)), Z(D)/F is separable and char(F) ∤ |ΓZ(D):ΓF| (i.e., the gauge w is tame);
Page 11
VALUE FUNCTIONS AND ASSOCIATED GRADED RINGS FOR SEMISIMPLE ALGEBRAS 11
(b) Z(D)/F is separable and char(F) ∤ |ker(θD)|.
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] > [Z(D):F]|ΓZ(D):ΓF|,
hence (1.17) does not hold, and w is not a gauge. This proves (i)⇒(iii) (without hypothesis on char(F)).
For the rest of the proof, assume char(F) ∤ [D:F]. Then Z(D)/F is separable and char(F) ∤ |ker(θD)|,
hence (i) ⇐⇒ (ii). Finally, assume (iii). By Cor. 1.10, we have
[Z(D):F] = [Z(D):F]|ΓZ(D):ΓF|.
On the other hand, a noncommutative version of Ostrowski’s theorem [M1, Th. 3] yields
[D:Z(D)] = [D:Z(D)]|ΓD:ΓZ(D)|.
Therefore, (1.17) 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 1.13. 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(F) = 0 or the char(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 [HW2, 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 1.14. As in [HW2], a central division algebra D over a Henselian valued field F is called
tame if the equivalent conditions of Prop. 1.13 hold. Note that by Cor. 3.2 below, w is the unique
F-gauge on D (if any exists).
Example 1.15. 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 F2(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 D = F2(x,y)(√x,√y). The valuation w is an F-gauge on D which is not
tame.
Example 1.16. 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 Q3of 3-adic numbers, hence the
3-adic valuation v on Q does not extend to a valuation on D, by [C, Th. 1] or [M1, proof of Th. 2]. Let
(1,i,j,k) be the quaternion base of D with i2= j2= −1 and k = ij = −ji, and define a Q-norm y
on D by
y(a0+ a1i + a2j + a3k) = min?v(a0),v(a1),v(a2),v(a3)?.
(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) = F3[t,t−1],
where the indeterminate t is the image of 3. With primes denoting images in gr(D), we have i′2= −1′,
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J.-P. TIGNOL AND A. R. WADSWORTH
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)F3⊗F3gr(Q)∼= M2(F3)[t,t−1].
Thus, y is a tame gauge on D with D0∼= M2(F3) and ΓD= ΓQ= Z. One can obtain other gauges on D
by conjugation, by using Prop. 1.17 below.
The residue ring D0in 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 characteristic 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, ΓF = 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 i2= 1 + x, j2= z, and k = ij = −ji. Define an
F-norm y on Q by, for a0,...,a3∈ F,
y(a0+ a1i + a2j + a3k) = min?v(a0),v(a1),v(a2) + (0,1
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
ΓQ= Z ×1
A gauge y on a division ring D is a valuation iff gry(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, as the next proposition shows:
2),v(a3) + (0,1
2)?.
2Z, and Q0∼= k × k, which is clearly not simple.
Proposition 1.17. 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. 3.7 below. (Observe that
Prop. 1.17 is not used in the sequel; thus, the argument is not circular.)
?
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VALUE FUNCTIONS AND ASSOCIATED GRADED RINGS FOR SEMISIMPLE ALGEBRAS13
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 = EndD(M). Supposew: D → Γ ∪ {∞}
is a valuation on D, and let v = w|F. Let α be a D-norm on M.
Lemma 1.18. Let (mi)1≤i≤kbe a splitting base of M for α. For every f ∈ A and nonzero m ∈ M,
α(f(m)) − α(m) ≥
1≤i≤k
min
?α(f(mi)) − α(mi)?.
Proof. Let m =
k ?
i=1
midiwith di∈ D. Then
α(f(m)) = α?k ?
i=1
f(mi)di
?
≥ min
1≤i≤k
?α(f(mi)) + w(di)?. (1.20)
Writing α(f(mi)) + w(di) = α(f(mi)) − α(mi) + α(mi) + w(di), we obtain
min
1≤i≤k1≤i≤k
?α(f(mi)) + w(di)?
≥ min
?α(f(mi)) − α(mi)?
+ min
1≤i≤k
?α(mi) + w(di)?. (1.21)
Since (mi)1≤i≤kis a splitting base of M, the second term on the right side is α(m). The lemma then
follows from (1.20) and (1.21).
?
In view of the lemma, we may define a function yα: A → Γ ∪ {∞} as follows: for f ∈ A,
yα(f) = min
m∈M, m?=0
?α(f(mi))−α(mi)?for any splitting base (mi)1≤i≤kof M.
If f(mj) =
i=1
?α(mi) + w(dij) − α(mj)?.
Now, let E = Endgr(D)(gr(M)). Recall that E is graded as follows: for γ ∈ Γ,
Eγ = {f ∈ E | f(Mδ) ⊆ Mδ+γfor all δ ∈ Γ}.
Proposition 1.19. The map yαof (1.22) is a surmultiplicative F-value function on A, and there is a
canonical gr(F)-algebra isomorphism gr(A)∼=gE. Moreover, yαis 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.
?α(f(m)) − α(m)?. (1.22)
Indeed, the lemma shows that yα(f) = min
1≤i≤k
k ?
midijwith dij∈ D, we have,
yα(f) =min
1≤i,j≤k
(1.23)
Proof. We omit the easy proof that yα 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 = yα(f) ∈ Γ. The definition
of yαsays 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≥δinto M≥δ+γand M>δinto M>δ+γ; so, f induces a well-defined additive
group homomorphism
?fδ: Mδ→ Mδ+γ,
δ∈Γ
by γ. For any d ∈ D×and nonzero m ∈ M, we have
?fα(m)+w(d)(m′· d′) =?fα(m′) · d′∈ Mα(m)+w(d)+γ.
there is a well-defined injective map Aγ→ Eγgiven by f +A>γ?→?f. The direct sum of these maps is a
it is injective on each homogeneous component of gr(A). To prove ρ is onto, let (mi)1≤i≤kbe a splitting
m′?→ f(m) + M>δ+γ
for all m ∈ M with α(m) = δ.
Define?f =?
?fδ: gr(M) → gr(M), an additive group homomorphism which shifts graded components
Thus,?f ∈ Eγ, and the definition of yαassures that?f ?= 0. Since for any g ∈ A>γwe have?
graded (i.e., grade-preserving) gr(F)-algebra homomorphism ρ: gr(A) → E; this map is injective, since
f + g =?f,
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J.-P. TIGNOL AND A. R. WADSWORTH
base of M with respect to α; so, (m′
with 1 ≤ i,j ≤ k and any d ∈ D×, and define g ∈ A by g(mj) = mid and g(mℓ) = 0 for ℓ ?= j, then
yα(g) = w(d) + α(mi) − α(mj); so, ? g(m′
isomorphism, as desired.
It follows from the isomorphism gr(A)∼=gE that gr(A) is a graded simple gr(F)-algebra. Thus, yαis
an F-gauge on A iff yαis an F-norm on A, iff, by Prop. 1.1(ii), [gr(A):gr(F)] = [A:F]. We have
i)1≤i≤kis a homogeneous gr(D)-base of gr(M). If we fix any i, j
j) = (mid) + A>w(d)+α(mi)−α(mj)+α(mj)= m′
i· d′∈ Mw(d)+α(mi)
and ? g(m′
ℓ) = 0 for ℓ ?= j. Since such maps generate E as an additive group, ρ is onto, hence an
[A:F] = k2[D:F]and[gr(A):gr(F)] = [E:gr(F)] = k2[gr(D):gr(F)].
Thus, yαis an F-norm on A iff [gr(D):gr(F)] = [D:F] iff, by (1.18) and Prop. 1.12, w is an F-gauge
on D.
Since Z(A) = Z(D) (up to canonical isomorphism) with yα|Z(A)= w|Z(D)and Z(gr(A))∼=gZ(gr(D))
(a gr(F)-algebra isomorphism), yαis a tame F-gauge on A iff w is a tame F-gauge on D.
?
We now compare the “gauge ring” VA= {f ∈ A | yα(f) ≥ 0} with the valuation rings VDand VF.
Lemma 1.20. The following conditions are equivalent:
(i) VAis integral over VF;
(ii) VDis integral over VF;
(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 (mi)1≤i≤k be a splitting base of M with respect to α, and let f ∈ A.
f(mj) =
i=1
the orthogonal idempotents of A for the splitting base (mi)1≤i≤kof M, i.e., ei(mj) = δijmi(Kronecker
delta). We have yα(ei) = 0, so each ei∈ VA. For f with matrix (dij) as above, eifeihas matrix with
ii-entry diiand all other entries 0. Hence, eiVAei∼= VD, a VF-algebra isomorphism. If VAis integral
over VF, then VD∼= e1VAe1must also be integral over VF. Conversely, suppose VDis integral over VF.
Then, each eiVAeiis integral over VF, since it is isomorphic to VD. For j ?= i, each element of eiVAej
has square 0, so is integral over VF. Since
i=1
of each summand integral over VF. Because A is a p.i.-algebra over F, the theorem [AS, Th. 2.3] of
Amitsur and Small shows that VAis integral over VF.
(ii) ⇐⇒ (iii) Let Z = Z(D), the center of D, and let VZ be the valuation ring of w|Z. Then VDis
always integral over VZ, by [W1, Cor.], and VZis integral over VFiff w|Zis the unique extension of the
valuation v to Z, see [B, Ch. VI, §8.3, Remark] or [EP, Cor. 3.1.4].
Remark 1.21. The value function yαon EndD(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 carr´ ee. In [BT, 2.13,
Cor.] (see also the Appendix of [BT]), Bruhat and Tits establish a bijection between the set of normes
carr´ ees on EndD(M) and the building of GL(M), when the rank of the ordered group Γ is 1.
If
k ?
midij, eq. (1.23) shows that f ∈ VAiff w(dij) ≥ α(mi)−α(mj) for all i, j. Let e1,...,ekbe
k ?
ei= 1, we have VA=
?
1≤i,j≤k
eiVAej, with the elements
?
The following result is in [BT, 1.13]:
Proposition 1.22. 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) yα= yβ.
Page 15
VALUE FUNCTIONS AND ASSOCIATED GRADED RINGS FOR SEMISIMPLE ALGEBRAS15
Proof. (i) ⇒ (ii) is clear from the definition (1.22). 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 yα, we have
yα(f) ≤ α(n) − α(m)
On the other hand, we may choose a splitting base (mi)1≤i≤kof M with m1= m (see [RTW, Prop. 2.5])
and define g ∈ A(m,n) by g(m1) = n, g(mi) = 0 for i > 1; then yα(g) = α(n) − α(m). Therefore,
α(n) − α(m) =
g∈A(m,n)
for f ∈ A(m,n).
max
?yα(g)?. (1.24)
If (ii) holds, it follows from (1.24) 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 ⊗KQ is given by (P ⊗KQ)γ=?
Proposition 1.23. 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 ⊗FN such
that there is a graded isomorphism of gr(F)-vector spaces
grt(M ⊗FN)∼=ggrα(M) ⊗gr(F)grβ(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.
δ∈Γ
Pδ⊗K0Qγ−δ.
The unique value function t on M ⊗FN 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. 1.23.
Lemma 1.24. 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 χt,u: grt(P) → gru(P) which is injective iff t = u.
Proof. Clear.
?
Proof of Prop. 1.23. Let (mi)1≤i≤kbe a splitting base of M with respect to α. Define t: M ⊗FN →
Γ ∪ {∞} by
t?k ?
Clearly, t is an F-value function on M ⊗FN, and
(M ⊗FN)≥γ=
i=1
Also, since (m′
i)1≤i≤kis a homogeneous gr(F)-base of gr(M), we have
?gr(M) ⊗gr(F)gr(N)?
For γ ∈ Γ, let πγ: N≥γ→ Nγ be the canonical map. Define a surjective map ψγ: (M ⊗FN)≥γ→
?gr(M) ⊗gr(F)gr(N)?
ψγ
i=1i=1
i=1
mi⊗ ni
?
= min
1≤i≤k
?α(mi) + β(ni)?
for any ni∈ N. (1.25)
k ?
mi⊗ N≥γ−α(mi)
for γ ∈ Γ.
γ=
k ?
i=1
m′
i⊗ Nγ−α(mi).
γas follows:
?k ?
mi⊗ ni
?
=
k ?
m′
i⊗ πγ−α(mi)(ni)for ni∈ N≥γ−α(mi).
Page 16
16
J.-P. TIGNOL AND A. R. WADSWORTH
This ψγis clearly an additive group homomorphism. Moreover, ψγ
iff?mi⊗ni∈ (M⊗FN)>γ. Thus, ψγinduces a group isomorphism ϕγ: (M ⊗FN)γ→?gr(M) ⊗gr(F)gr(N)?
To see that ϕ?(m ⊗ n)′?
m ⊗ n =
i=1
?α(mi) + β(n) + v(ri)?
and
?k ?
Note that
?
0
??mi⊗ni
?= 0 iff each ni∈ N>γ−α(mi),
γ
and ϕ = ⊕
γ∈Γϕγ is a graded gr(F)-vector space isomorphism grt(M ⊗FN) → grα(M) ⊗grv(F)grβ(N).
= m′⊗ n′for m ∈ M and n ∈ N, let m =
k ?
i=1
miri with ri ∈ F. Then
k ?
mi⊗ rin and α(m) = min{α(mi) + v(ri)}, so
t(m ⊗ n) =min
1≤i≤k
= α(m) + β(n)
ϕ((m ⊗ n)′) = ψα(m)+β(n)
i=1
mi⊗ rin
?
=
k ?
i=1
m′
i⊗ πα(m)−α(mi)+β(n)(rin).
πα(m)−α(mi)+β(n)(rin) =
r′
in′
if α(mi) + v(ri) = α(m),
if α(mi) + v(ri) > α(m).
Changing the indexing if necessary, we may assume α(mi) + v(ri) = α(m) for i = 1, ..., ℓ and
ℓ ?
ϕ((m ⊗ n)′) =
i=1
α(mi) + v(ri) > α(m) for i > ℓ. Then m′=
i=1
m′
ir′
iand
ℓ ?
m′
i⊗ r′
in′= m′⊗ n′,
as desired.
Now, suppose u: M ⊗FN → 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 gru(M ⊗ N) is the same
as the degree of m′⊗ n′in grα(M) ⊗ grβ(N). Therefore, for ni∈ N we have
u?k ?
We have graded gr(F)-vector space isomorphisms ϕt: grt(M ⊗F N) → grα(M) ⊗gr(F)grβ(N) and
ϕu: gru(M ⊗FN) → grα(M) ⊗gr(F)grβ(N). Because of the inequality in (1.26), Lemma 1.24 yields
a canonical gr(F)-vector space homomorphism χt,u: grt(M ⊗FN) → gru(M ⊗FN). Our hypotheses on
ϕtand ϕuimply that ϕu◦χt,uand ϕtagree on (m⊗n)′for all m ∈ M and n ∈ N. Since such (m⊗n)′
form a generating set for grt(M ⊗FN), we have ϕu◦ χt,u= ϕt. Then, χt,uis an isomorphism, since ϕt
and ϕuare each isomorphisms. Lemma 1.24 then shows that u = t, proving the desired uniqueness of t.
If β is an F-norm on N, say with splitting base (ni)1≤i≤ℓ, then it follows easily from (1.25) that
(mi⊗ nj)1≤i≤k
1≤j≤ℓ
i=1
mi⊗ ni
?
≥ min
1≤i≤k
?u(mi⊗ ni)?
=min
1≤i≤k
?α(mi) + β(ni)?
= t?k ?
i=1
mi⊗ ni
?.(1.26)
is a splitting base for t on M ⊗FN, so t is an F-norm for M ⊗FN.
?
Remark 1.25. A basis-free description of α ⊗ β is stated in [BT, p. 269]: for s ∈ M ⊗ N,
α ⊗ β(s) = sup?
To see this equality, for s ∈ M ⊗FN, let u(s) = sup?
representation for s as
j=1
min
1≤j≤ℓ
?α(pj) + β(qj)? ??s =
min
1≤j≤ℓ
ℓ ?
j=1
pj⊗ qj
?.
?α(pj)+β(qj)? ??s =
mirijwith rij∈ F. Then, s =
ℓ ?
j=1
pj⊗qj
k ?
?. Take any
mi⊗?
ℓ ?
pj⊗qj. Write each pj=
k ?
i=1i=1
ℓ ?
j=1
rijqj
?,
Page 17
VALUE FUNCTIONS AND ASSOCIATED GRADED RINGS FOR SEMISIMPLE ALGEBRAS 17
and
min
1≤j≤ℓ
?α(pj) + β(qj)?
= min
1≤j≤ℓ
?α(
?α(mi) + v(rij) + β(qj)?
?α(mi) + β?
k ?
i=1
mirij) + β(qj)?
=min
1≤j≤ℓ
??min
?α(mi) + min
= α ⊗ β? k ?
1≤i≤kα(mi) + v(rij)?
1≤j≤ℓ
mi⊗?
+ β(qj)?
= min
i,j
= min
1≤i≤k
?v(rij) + β(qj)?
rijqj
≤ min
1≤i≤k
ℓ ?
j=1
rijqj
??
i=1
ℓ ?
j=1
??
= 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=1j=1
and u(s) = α ⊗ β(s).
Corollary 1.26. 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 ⊗FL and the canonical isomorphism
gr(A ⊗FL)∼=ggr(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.
k ?
mi⊗?
ℓ ?
rijqj
?, we see that α⊗β(s) is one of those quantities. Hence, the sup exists,
Proof. Let z = y ⊗ w. Prop. 1.23 shows that z is a well-defined F-value function on A ⊗FL 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 (ai)1≤i≤kis an F-splitting base of A with respect to y, then formula (1.25) shows that
(ai⊗ 1)1≤i≤kis an L-splitting base of A ⊗FL with respect to z. Since
z?(ai⊗ 1)(aj⊗ 1)?
Lemma 1.2 shows that z is surmultiplicative. The value-function compatible F-algebra homomorphism
A → A ⊗FL induces a graded gr(F)-algebra homomorphism gr(A) → gr(A ⊗FL), and hence a graded
gr(L)-algebra homomorphism ϕ: gr(A)⊗gr(F)gr(L) → gr(A⊗FL). This ϕ is bijective, since it coincides
with the gr(F)-vector space isomorphism of Prop. 1.23. Since z satisfies all the other conditions for an
L-gauge, z is an L-gauge iff gr(A⊗FL) 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)∼=ggr(A)⊗Z(gr(A))
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 [HW2, 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 ⊗KM 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 ⊗FL)?= gr(Z(A) ⊗FL)∼=ggr?Z(A)?⊗gr(F)gr(L).
On the other hand,
Z?gr(A ⊗FL)?∼=gZ?gr(A) ⊗gr(F)gr(L)?= Z?gr(A)?⊗gr(F)gr(L),
hence
[gr?Z(A ⊗ L)?:gr(L)] = [gr?Z(A)?:gr(F)]
same dimension. Similarly, gr?Z(A)?= Z?gr(A)?iff [gr?Z(A)?:gr(F)] = [Z?gr(A)?:gr(F)]. Therefore,
= z(aiaj⊗ 1) = y(aiaj) ≥ y(ai) + y(aj) = z(ai⊗ 1) + z(aj⊗ 1),
?Z(gr(A))⊗gr(F)gr(L)?.
(1.27)
and[Z?gr(A ⊗ L)?:gr(L)] = [Z?gr(A)?:gr(F)].(1.28)
Since gr?Z(A ⊗ L)?⊆ Z?gr(A ⊗ L)?, these finite-dimensional gr(L)-algebras coincide iff they have the
Page 18
18
J.-P. TIGNOL AND A. R. WADSWORTH
(1.28) shows that
gr?Z(A ⊗ L)?= Z?gr(A ⊗ L)?
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.
iffgr?Z(A)?= Z?gr(A)?.
Moreover, since gr(L) is a free gr(F)-module, it follows from (1.27) that Z?gr(A⊗L)?is separable over
gr(L) iff Z?gr(A)?
Remark 1.27. In the context of Cor. 1.26, 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 [HW1,
Th. 3.11, Def. 3.4] that if [gr(L):gr(F)] < ∞, then gr(L) is separable over gr(F) iff L is separable over F
and char(F) ∤
?
??ΓL:ΓF
??.
Corollary 1.28. 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⊗FB, and
the canonical isomorphism
gr(A ⊗FB)∼=ggr(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 (ai)1≤i≤k, and z is an F-norm on B, say with
splitting base (bj)1≤j≤ℓ, we saw in the proof of Prop. 1.23 that (ai⊗ bj)1≤i≤k
F-norm y ⊗ z on A ⊗FB. For any ai, ap, bj, bq, we have, by Prop. 1.23,
(y ⊗ z)?(ai⊗ bj) · (ap⊗ bq)?
≥ y(ai) + y(ap) + z(bj) + z(bq) = (y ⊗ z)(ai⊗ bj) + (y ⊗ z)(ap⊗ bq).
Therefore, Lemma 1.2 shows that y⊗z is surmultiplicative. We have gr(A⊗FB)∼=ggr(A)⊗gr(F)gr(B)
by Prop. 1.23. Thus, y ⊗ z is an F-gauge on A ⊗FB 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 [HW1, 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 [HW2, Prop. 1.1].
1≤j≤ℓ
is a splitting base for the
= (y ⊗ z)(aiap⊗ bjbq) = y(aiap) + z(bjbq)
?
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∗= HomD(M,D) and check that the tensor product
α⊗α∗corresponds to the gauge yαunder the canonical isomorphism M ⊗DM∗ ∼= EndD(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., [HW2, 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∼=gEndE(N). We identify B with EndE(N). The grading on B is given as follows: for any ǫ ∈ Γ,
Bǫ = {f ∈ B | f(Nδ) ⊆ Nδ+ǫfor all δ ∈ Γ}.
Page 19
VALUE FUNCTIONS AND ASSOCIATED GRADED RINGS FOR SEMISIMPLE ALGEBRAS19
Since E is a graded division ring, E0is a division ring, and for each γ ∈ ΓE, Eγis a 1-dimensional left
and right E0-vector space. The grade set of N, ΓN= {γ ∈ Γ| Nγ?= {0}}, need not be a group, but
it is clearly a union ΓN= Γ1∪ ... ∪ Γkwhere each Γiis a (non-empty) coset of the group ΓE. Then,
N has a canonical direct sum decomposition into graded E-vector subspaces
N =
k ?
i=1
NΓi
whereNΓi=
?
γ∈Γi
Nγ.(2.1)
For each coset Γi, choose and fix a representative γi∈ Γi.
Proposition 2.1. The grade set of B is
ΓB =
k ?
i,j=1
(γi− γj) + ΓE
(2.2)
and there is a canonical isomorphism of K0-algebras
B0∼=
k ?
i=1
EndE0(Nγi). (2.3)
Moreover,
dimE(N) =
k ?
i=1
dimE0(Nγi). (2.4)
Proof. Eq. (2.2) readily follows from the description of the grading on B, and (2.4) from (2.1), since
NΓi= Nγi⊗E0E. To prove (2.3), observe that every element in B0maps each NΓito itself, so
k ?
Since NΓi= Nγi⊗E0E, restriction to Nγidefines a canonical isomorphism of K0-algebras EndE(NΓi)0∼=
EndE0(Nγi). The proof is thus complete.
B0 =
i=1
EndE(NΓi)0.
?
This proposition shows that B0 is in general semisimple but not simple; however all its simple
components are equivalent to E0in the Brauer group Br(Z(E0)). Also, the grade set ΓBis in general
not a group. We next show how ΓEcan be detected within ΓB.
Let HB be the multiplicative group of homogeneous units of B and let ∆B⊆ ΓB be the image of
HBunder the grade homomorphism mapping each nonempty HB∩ Bγto γ. The group action of HB
by conjugation on B preserves the grading, so sends B0, hence also Z(B0), to itself. If b, c ∈ HB∩ Bγ,
then b−1c ∈ B×
well-defined homomorphism
θB: ∆B/ΓK → Aut(Z(B0)/K0),
which maps γ +ΓKto z ?→ bzb−1for z ∈ Z(B0) and b ∈ HB∩ Bγ. Of course, if B = E then ∆B= ΓE.
The homomorphism θBthen coincides with the homomorphism θEdefined in [HW2, (2.2)].
0centralizes Z(B0), hence b and c have the same action on Z(B0). Therefore, there is a
(2.5)
Proposition 2.2. Let e be any primitive idempotent of Z(B0). Then
ΓE = {γ ∈ ∆B| θB(γ + ΓK)(e) = e}.
Moreover, the following diagram is commutative:
ΓE/ΓK
?
i
− − − − →∆B/ΓK
?θB
θE
Aut(Z(E0))
d
− − − − → Aut(Z(B0))
(2.6)
Page 20
20
J.-P. TIGNOL AND A. R. WADSWORTH
where i is induced by the inclusion ΓE⊆ ∆Band d is the diagonal map, letting an automorphism of
Z(E0) act on each component of Z(B0)∼= Z(E0) × ... × Z(E0).
Proof. From the description of B0in Prop. 2.1, it follows that the primitive idempotents of Z(B0) are
the maps e1, ..., eksuch that ei|NΓi= id and ei|NΓj= 0 for i ?= j. Suppose γ ∈ ∆B is such that
θB(γ + ΓK) fixes some ei, and let h ∈ HB∩ Bγ; then hei= eih, hence h(NΓi) = NΓiand therefore
γ ∈ ΓE.
Conversely, suppose γ ∈ ΓE. Take any nonzero c ∈ Eγ and any homogeneous E-vector space base
of N built from bases of the NΓi; let f ∈ B be defined by mapping each base vector n to nc. Then
fei = eif for all i and f ∈ HB∩ Bγ, so θB(γ + ΓK) fixes each e1, ..., ek. Moreover, θB(γ + ΓK)
induces on each component eiZ(B0)∼= Z(E0) of Z(B0) the automorphism of conjugation by c, which is
θE(γ + ΓK). Therefore, diagram (2.6) commutes.
?
Corollary 2.3. If B0is simple, then ΓB= ∆B= ΓE and θB= θE. Moreover, B and B0have the
same matrix size, and [B:K] = [B0:K0]|ΓB:ΓK|.
Proof. If B0is simple, Prop. 2.1 yields ΓB= ΓE. It also yields B0∼= EndE0(Nγ1) where dimE0(Nγ1) =
dimE(N), hence B and B0 have the same matrix size.
Prop. 2.2. Since B∼=gEndE(N), we have
[B:K] = (dimE(N))2[E:K] = (dimE0(Nγ1))2[E0:K0]|ΓE:ΓK| = [B0:K0]|ΓB:ΓK|.
The equalities for ΓB and θB follow from
?
Example 2.4. With the notation of Prop. 2.1, if the dimensions dimE0(Nγ1), ..., dimE0(Nγk) are all
different then every invertible homogeneous element in B has grade in ΓE; therefore, ∆B = ΓE and
θB= θE.
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 [M2] 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 Vy= A≥0, the “valuation ring” of y, and let A0= A≥0/A>0, the
degree 0 part of gry(A). Then, y is a Morandi value function if
(i) A0is a simple ring;
(ii) y(st(y)) = ΓA.
When this occurs, it is known that Vy is a Dubrovin valuation ring integral over its center, which
is Vv, and y is completely determined by Vy. Conversely, to every Dubrovin valuation ring B of A with
B integral over its center, there is a canonically associated Morandi value function yBwith VyB= B.
(For the theory of Dubrovin valuation rings, see [MMU], [W2], 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 = Vv, and such a B is unique up to conjugacy,
so unique up to isomorphism. See [W2, Th. F] for characterizations of when B is integral over Vv.) For
a Morandi value function y on A, the defect δ(y) (an integer) is defined by
δ(y) = [A:F]?[A0:F0]|ΓA:ΓF|.
If (Fh,vh) is a Henselization of (F,v), and Dhis the division algebra Brauer-equivalent to A ⊗FFh,
it is known (see. [W2, Th. C]) that δ(y) coincides with the defect of the valuation on Dhextending vh
on Fh. Hence, δ(y) = 1 if char(F) = 0 and δ(y) is a power of char(F) otherwise. The requirement
Page 21
VALUE FUNCTIONS AND ASSOCIATED GRADED RINGS FOR SEMISIMPLE ALGEBRAS 21
of integrality of Vy over Vv 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 A0simple.
Proof. ⇒ Suppose y is a Morandi value function on A with δ(y) = 1. For any γ ∈ ΓA = y(st(y)),
there is uγ ∈ st(y) with y(uγ) = γ. Then, u′γis a homogeneous unit of gr(A), by Lemma 1.3. Take
any nonzero homogeneous ideal I of gr(A). Then, there is a γ ∈ ΓAwith I ∩ Aγ ?= {0}. So, for the
ideal I ∩ A0of A0we have I ∩ A0⊇ u′−1
I ∩A0= A0, 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
γ (I ∩ Aγ) ?= {0}. Because A0is a simple ring, we must have
[A:F] = [A0:F0]|ΓA:ΓF| = [gr(A):gr(F)].
Hence, y is a gauge on A. By hypothesis, A0is simple.
⇐ Suppose y is a gauge on A with A0 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 A0= C0× D0with C0and D0nontrivial. This cannot occur as A0is simple.
Hence, gr(A) is graded simple. (This also follows from the simplicity of A. See Cor. 3.7 below.) Since
A0is simple, Cor. 2.3 applies with B = gr(A). The equality Γgr(A)= ∆gr(A)from Cor. 2.3 shows that
for each γ ∈ ΓA= Γgr(A), there is a homogeneous unit u in Aγ. Pick any a ∈ A with y(a) = γ and
a′= u. Then, a ∈ st(y) by Lemma 1.3. Hence, y(st(y)) = ΓA, 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]?[A0:F0]|ΓA:ΓF| = [A:F]?[gr(A):gr(F)] = 1.
?
3. Gauges over Henselian fields
The next theorem provides the key link between a gauge on a simple algebra over a Henselian field
and the valuation on the associated division algebra. 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. Let A be a simple F-algebra with an
F-gauge y. Let D be the division algebra Brauer-equivalent to A, and let w be the valuation on D
extending v on F. Then, gr(A) is simple and w is defectless. Furthermore, there is a finite-dimensional
right D-vector space M with a w-norm α such that
A∼= EndD(M),
and y on A corresponds to the F-gauge yα on EndD(M) induced by α as in §1.3.
ms(A) = ms(gr(A)), and the gauge y is tame if and only if w is a tame F-gauge.
gr(A)∼=g Endgr(D)(gr(M)),
In particular,
Proof. We first show that the gauge ring VA= {a ∈ A| y(a) ≥ 0} is integral over VF. Let C = EndF(A),
and let z = zy: C → Γ ∪ {∞} be the gauge on C arising from the norm y on A, as in §1.3. Take any
a ∈ VA. 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 λa∈ C given by b ?→ ab, we have z(λa) ≥ 0. That is,
λa∈ VC. Lemma 1.20 shows that VCis integral over VF, hence λais integral over VF, and so therefore
is a. This shows that VAis integral over VF.
Page 22
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J.-P. TIGNOL AND A. R. WADSWORTH
Because VF is Henselian and VAis integral over VF, we can lift idempotents from any homomorphic
image of VA to VA, by [MMU, Th. A.18, p. 180]. (This is also deducible from results in Nagata’s
book [N]: if J is any ideal of VA, take any b ∈ VA with b2− b ∈ J, and let T = VF[b], which is a
commutative ring finitely generated as a VF-module, as b is integral over VF. By [N, Th. 43.15, p. 185],
T is a direct product T = S1× ... × Skwhere each Siis a local ring. Let J ∩ T = K1× ... × Kk, so
T/(J ∩T) ≃ S1/K1×...×Sk/Kk. Since each Si/Kiis local, it has no nontrivial idempotent. Therefore,
by working componentwise, we see that there is an idempotent e of T with e ≡ b mod J ∩ T.) We
use this idempotent-lifting property first to show that gr(A) is graded simple. Since gr(A) is graded
semisimple, if it were not graded simple, it would have a nontrivial central homogeneous idempotent ? c.
is an idempotent c ∈ VAwith c′= ? c in gr(A)0. So, y(c) = 0. For the complementary idempotent 1 − c,
Since ? c2= ? c, necessarily ? c ∈ gr(A)0= VA
we have (1 − c)′= 1′− c′, since y(c) = y(1) and c′?= 1′. Now, by two applications of Lemma 1.7,
gr?cA(1 − c)?
as ? c is a central idempotent of gr(A). Hence, cA(1−c) = {0}, and likewise (1−c)Ac = {0}. Therefore,
graded simple.
Now, let ? e be any primitive homogeneous idempotent of the graded simple ring gr(A). Just as for
Now, y|eAe is a surmultiplicative F-value function (when we identify F with eF). By Lemma 1.7,
gr(eAe) = e′gr(A)e′, which is a graded division ring as e′is primitive. Because gr(eAe) has no zero
divisors, (1.5) shows that
y(bd) = y(b) + y(d)
Therefore, the finite-dimensional F-algebra eAe has no zero divisors, and is thus a division ring. Hence,
e is a primitive idempotent of A, and eAe∼= D. We identify eAe with D by any such F-isomorphism.
Formula (3.1) further shows that y|Dis a valuation restricting to v on F. Hence, y|D= w, as v has a
unique extension to D. Since the gauge y is an F-norm on A, its restriction to D is also an F-norm by
Prop. 1.1(iii). So, D is defectless over F.
?MA. Therefore, the idempotent lifting result says that there
= c′gr?A(1 − c)?
= ? cgr(A)(1′− ? c) = {0},
c ∈ Z(A), so A∼= cAc ×(1−c)A(1 −c), contradicting the graded simplicity of A. Thus, gr(A) must be
? c above, we have ? e ∈ gr(A)0and there is an idempotent e ∈ VAwith e′= ? e in gr(A). So, y(e) = 0.
for all b,d ∈ eAe.(3.1)
Set M = Ae, which is a right D = eAe-vector space.
Prop. 1.1(iii). By Lemma 1.7, gr(M) = gr(A)? e, which is a (unital) right ? egr(A)? e = gr(D)-module.
α(md) = y(md) = y(m) + y(d) = α(m) + w(d)
Let α = y|M, which is an F-norm by
Hence, as m′d′?= 0 in gr(M) whenever m′?= 0 and d′?= 0, formula (1.5) yields,
for all m ∈ M, d ∈ D.
This shows that α is actually a D-value function on M. Furthermore,
dimgr(D)(gr(M)) = dimgr(F)(gr(M))?[gr(D):gr(F)] = dimF(M)?[D:F] = dimD(M),
which shows that α is a D-norm on M.
By Wedderburn’s Theorem, the map
β: A → EndD(M) given byβ(a)(b) = ab
is an F-algebra isomorphism. We prove that β induces an isomorphism of graded rings. Let yα: EndD(M) →
Γ ∪ {∞} be the F-gauge on EndD(M) induced by the D-norm α on M, as in §1.3.
f ∈ EndD(M), we have yα(f) =
m∈M, m?=0
yα(β(a)) = y(a).
?y(am) − y(m)?
Suppose yα(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.
That is, for
min
?α(f(m)) − α(m)?. We claim that for every nonzero a ∈ A,
(3.2)
For, we have yα(β(a)) =min
m∈M,m?=0
≥ y(a), since y(am) ≥ y(a) + y(m) for all m.
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VALUE FUNCTIONS AND ASSOCIATED GRADED RINGS FOR SEMISIMPLE ALGEBRAS23
But, gr(M) = gr(A)e′. Thus, 0 = a′gr(M) = a′gr(A)e′. Because gr(A) is graded simple, we have
gr(A)e′gr(A) = gr(A), and hence 0 = a′gr(A)e′gr(A) = a′gr(A), which shows that a′= 0.
contradicts a ?= 0, proving (3.2). From (3.2) and Prop. 1.19 we have
gry(A)∼=g gryα(EndD(M))∼=g Endgrw(D)(grα(M)),
This
as desired. Note that ms(A) = dimD(M) = dimgr(D)(gr(M)) = ms(gr(A)). Moreover, Prop. 1.19 shows
that the gauge yαis tame if and only if w is a tame F-gauge. This completes the proof of Th. 3.1.
?
Corollary 3.2. Let F be a field with a Henselian valuation v, let D be a (finite-dimensional) division
algebra over F, and let w be the valuation on D extending v. Then, w is the only possible v-gauge
on D; w is a v-gauge if and only if D is defectless over F.
Proof. Let y be a v-gauge on D. Th. 3.1 shows that gry(D)∼=ggrw(D), which is a graded division ring.
Therefore, as we saw in the proof of Th. 3.1, y is a valuation on D, extending v on F. But w is the
unique extension of v to D, so y = w. By Prop. 1.12, w is a v-gauge if and only if D is defectless over
F.
?
Because of the compatibility of gauges with direct products, we can easily extend Th. 3.1 to semisimple
algebras over a Henselian field.
Theorem 3.3. Let F be a field with a Henselian valuation v, and let A be a semisimple F-algebra with
an F-gauge y. Let A1, ..., Anbe the simple components of A,
A = A1× ... × An.
For i = 1, ..., n, the restriction y|Aiis a gauge on Ai, the graded algebra gry|Ai(Ai) is graded simple
with ms(Ai) = ms(gr(Ai)), and gr(A1), ..., gr(An) are the graded simple components of gr(A),
gr(A) = gr(A1) × ... × gr(An).
For a = (a1,...,an) ∈ A1× ... × An,
y(a) = min
1≤i≤n
?y|Ai(ai)?.
Moreover, the gauge y is tame if and only if each y|Aiis tame.
Proof. This is immediate from Th. 3.1 and Prop. 1.6.
?
If the valuation v is not Henselian, we may still apply Th. 3.3 after scalar extension to a Henselization
(Fh,vh) of (F,v). If y is a v-gauge on the semisimple F-algebra A, then Cor. 1.26 shows that y ⊗ vh
is a vh-gauge on A ⊗FFhwith gr(A ⊗FFh) = gr(A), since gr(Fh) = gr(F). Let B1, ..., Bnbe the
simple components of A ⊗FFh,
A ⊗FFh= B1× ... × Bn.
For i = 1, ..., n, let Dibe the division algebra Brauer-equivalent to Bi.
Proposition 3.4. 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 ⊗ vhis a tame vh-gauge;
(iii) for i = 1, ..., n, the unique valuation wion Diextending vhis a tame vh-gauge.
Page 24
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J.-P. TIGNOL AND A. R. WADSWORTH
Proof. The equality ℓ′= n follows from Th. 3.3 and the equality ℓ ≤ n is clear. The equivalence
(i) ⇐⇒ (ii) readily follows from Cor. 1.26. For i = 1, ..., n, let yibe the restriction of y ⊗ vhto Bi.
Th. 3.3 shows that y ⊗vhis a tame vh-gauge if and only if each yiis a tame vh-gauge. By Th. 3.1, this
condition is equivalent to (iii).
?
Corollary 3.5. 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 (Fh,vh) be a Henselization of (F,v), and let K = Z(A). Since Fhis separable over F, we can
write K⊗FFh= L1×...×Ln, where each Liis a field. Then, A⊗FFh ∼= A⊗K(K⊗FFh)∼= B1×...×Bn,
where each Bi= A⊗KLi, a central simple Li-algebra. With respect to the Fh-gauge y⊗vhon A⊗FFh,
Th. 3.3 shows that
gr(A ⊗FFh)∼=g gr(B1) × ... × gr(Bn)
with each gr(Bi) 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. 1.8 and Cor. 1.9, the number of simple summands of gr(K) equals the number of
extensions of v to K. We have y|K⊗ vhis a gauge on K ⊗FFh, and
gr(K ⊗FFh)∼=g gr(L1) × ... × gr(Ln)
by Prop. 1.11. Furthermore, each gr(Li) is simple by Cor. 1.9 and Prop. 1.8 because the Henselian valu-
ation vhhas a unique extension from Fhto Li. Since gr(A)∼=ggr(A⊗FFh) and gr(K)∼=ggr(K ⊗FFh)
by Cor. 1.26, equations (3.3) and (3.4) show that
(3.3)
(3.4)
n =number of simple components of gr(A)
= number of simple components of gr(K)
= number of extensions of v to K .
?
Corollary 3.6. If char(F) = 0, then every F-gauge on a semisimple F-algebra is tame.
Proof. Condition (iii) of Prop. 3.4 holds if char(F) = 0, by Prop. 1.13.
?
In the rest of this section, we consider the case where the semisimple F-algebra A is central simple.
Recall from [HW2] 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 3.7. Let (F,v) be a valued field and let (Fh,vh) be a Henselization of (F,v). Let A be a
central simple F-algebra, let D be the division algebra Brauer-equivalent to A⊗FFh, and let w be the
valuation on D extending vh. Let y be any v-gauge on A. Then, the grv(F)-algebra gry(A) is graded
simple and Brauer-equivalent to grw(D). Moreover, if y is tame then D is tame (see Def. 1.14) and
gry(A) is central simple over grv(F).
Proof. Let Ah= A⊗FFhand let yh= y⊗vh. Th. 3.3 shows that gryh(Ah) is graded simple, and Th. 3.1
yields gryh(Ah)∼=gEndgr(D)
is Brauer-equivalent to gr(D). These properties carry over to gry(A) because gry(A) = gryh(Ah) by
Cor. 1.26, since gr(Fh) = gr(F). When y is tame, it follows by definition that Z(gr(A)) = gr(F), and
from Prop. 3.4 that D is tame.
?gr(M)?for some finite-dimensional right D-vector space M, hence gryh(Ah)
?
Page 25
VALUE FUNCTIONS AND ASSOCIATED GRADED RINGS FOR SEMISIMPLE ALGEBRAS 25
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. 3.7 shows that if y is a tame v-gauge on a central simple F-algebra A, then the Brauer class
[gry(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 ⊗FFh. Therefore, there is a well-defined map
Ψ: TBr(F) → GBr(gr(F)),
By Cor. 1.28, TBr(F) is a subgroup of Br(F) and the map Ψ is a group homomorphism.
[A] ?→ [gry(A)] for any v-gauge y on A.
Theorem 3.8. The kernel of Ψ consists of the elements in TBr(F) which are split by any Henselization
Fhof F with respect to v. Moreover, for any valued field (L,w) extending (F,v), there is a commutative
diagram
TBr(F)− − − − → GBr(gr(F))
?
If v is Henselian, then Ψ is an index-preserving group isomorphism.
ΨF
?
TBr(L)
ΨL
− − − − → GBr(gr(L)).
(3.5)
Proof. Cor. 1.26 shows that the scalar extension map −⊗FL sends TBr(F) to TBr(L), and that diagram
(3.5) is commutative. When v is Henselian, Th. 3.1 shows that ms(gry(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 [HW2, 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 = Fhin commutative diagram (3.5); since ΨFh is bijective and gr(Fh) = gr(F), the
kernel of ΨF is the kernel of the scalar extension map TBr(F) → TBr(Fh).
This theorem generalizes [HW2, Th. 5.3], which showed that ΨF is a group isomorphism when v is
Henselian. The proof given here is vastly simpler than the one in [HW2].
?
4. Applications
The utility of Theorems 3.3 and 3.1 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 Endgr(D)(N) for some graded right gr(D)-
vector space N. Hence, D is Brauer equivalent to any simple component of A0, and ΓDand θDare
determinable from gr(A), as described in Prop. 2.2.
Page 26
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J.-P. TIGNOL AND A. R. WADSWORTH
Proof. Let (Fh,vh) be a Henselization of (F,v).
Ah ∼= Mn(Dh), where A∼= Mn(D). As noted in the proof of Cor. 3.7, y ⊗ vhis an Fh-gauge on
Ahwith gr(Ah)∼=g gr(A). Since Ahis simple, Th. 3.3 shows that gr(Ah) (so also gr(A)) is graded
simple, and that ms(gr(A)) = ms(Ah). Of course, ms(Ah) ≥ ms(A). Thus, ms(gr(A)) = ms(A) iff
ms(Ah) = ms(A), iff Dhis a division ring, iff v extends to a valuation on D, by Morandi’s theorem [M1,
Th. 2].
Let?D be the division algebra Brauer-equivalent to Ah. By Th. 3.1, we have Ah ∼= Ende D(N) for some
uniqueness part of the graded Wedderburn theorem. Since Th. 3.1 also says that the valuation on?D
Hence, [E:gr(F)] = [D:F] iff [?D:Fh] = [Dh:Fh] iff Dhis a division ring, iff (as above) v extends
ι: gr(D) ֒→ gr(Dh); but ι is actually an isomorphism, since D∼= Dhand ΓD= ΓDh by [M1, Th. 2], hence
D0∼= Dh
Hence the Brauer class of D = D0 coincides with that of any simple component of A0, and θD is
determinable from gr(A) as described in Prop. 2.2.
Let Ah= A ⊗F Fhand Dh= D ⊗F Fh.So,
right?D-vector space N with a?D-norm, and gr(Ah)∼=gEndgr(e D)(gr(M)). Hence, E∼=ggr(?D), by the
extending vhis an Fh-gauge, we have [E:gr(F)] = [gr(?D):gr(Fh)] = [?D:Fh]. But, [D:F] = [Dh:Fh].
to D.When this occurs, since the valuation on Dhextends the one on D, we have an inclusion
0and Γgr(D)= Γgr(Dh). Thus, E∼=ggr(? D)∼=ggr(Dh)∼=ggr(D), so that gr(A)∼=gEndgr(D)(N).
?
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 Z2(G,K×), and assume for convenience
that f is normalized. Explicitly, write
A =
σ∈G
where
?
Kxσ,
xid = 1,xσcx−1
σ
= σ(c) for all c ∈ Kandxσxτ = f(σ,τ)xστ 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 K is separable over F and that char(F) ∤ |ΓK:ΓF|.
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′∈ Z2(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 [HW2, Lemma 3.1].
Toward defining a v-gauge on A, we set for σ ∈ G
y(xσ) =
|G|
ρ∈G
1
?
w(f(σ,ρ)). (4.1)
We extend y to a value function on A by letting
y? ?
σ∈G
cσxσ
?
= min
σ∈G
?w(cσ) + y(xσ)?.(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 (ai)1≤i≤nbe any splitting base of w on K as an F-norm. Then, it follows from the definition
in (4.2) that (aixσ| 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
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VALUE FUNCTIONS AND ASSOCIATED GRADED RINGS FOR SEMISIMPLE ALGEBRAS27
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(xστ) = |G|y(xτ) + |G|y(xσ).
Since Γ is torsion-free this yields
y(xσ) + y(xτ) = w(f(σ,τ)) + y(xστ),(4.3)
for all σ, τ ∈ G. Therefore, for any i, j, σ, τ we have
y?(aixσ)(ajxτ)?
By Lemma 1.2 it follows that y is surmultiplicative.
Now consider gr(A). Since (a′
i ≤ n, σ ∈ G?
σ∈G
in gr(A),
x′
Moreover, formula (4.3) shows that x′σx′τ= f(σ,τ)′x′στfor all σ, τ ∈ 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.
= y?aiσ(aj)f(σ,τ)xστ
= w(ai) + w(aj) + y(xσ) + y(xτ) = y(aixσ) + y(ajxτ).
?
= w(ai) + w(aj) + w(f(σ,τ)) + y(xστ)
i)1≤i≤n is a homogeneous gr(F)-base for gr(K) and
is a homogeneous gr(F)-base of gr(A), and since (aixσ)′= a′
gr(K)x′σ. For any c ∈ K×and σ ∈ G, we have y(xσc) = y(σ(c)xσ) = w(c) + y(xσ); hence,
?(aixσ)′| 1 ≤
ix′σby (4.2), we have
gr(A) =
?
σc′= (xσc)′= (σ(c)xσ)′= σ(c)′x′
σ= σ′(c′)x′
σ.
Therefore, gr(A)∼=g
?
We can describe A0to some extent for this A. The value function y yields a map
λ: G → Γ/ΓK
given byσ ?→ y(xσ) + ΓK,(4.4)
and (4.3) shows that λ is a group homomorphism. Let H = ker(λ). Write A =
?
σ∈G
Kzσ, where
zσ = dσxσ, with the dσ∈ K×chosen so that w(dσ) = −y(xσ) if σ ∈ H .
Thus, dσdτ= g(σ,τ)dστ, where g(σ,τ) = dσσ(dτ)d−1
have y(zρ) = 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(zσ) / ∈ ΓKthe summand gr(K)z′σmakes no contribution to A0. Thus,
A0 =
ρ∈H
(4.5)
στf(σ,τ), so g is a 2-cocycle cohomologous to f. We
?
K0z′ρ.
This A0is 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 K0. Recall that K0= K. Because the extension of v to K is
indecomposed, each σ ∈ G induces an automorphism ? σ of K which coincides with the restriction of σ′
Proposition 4.3. In the situation just described where A = (K/F,G,f) =
to K0.
?
σ∈G
Kxσand H = ker(λ)
for λ as in (4.4), suppose that each ρ ∈ H induces a different automorphism of K. Then, Z(A0) = KH
the subfield of K0fixed by H, and A0=
ρ∈H
the graded division algebra Brauer-equivalent to gr(A). Then, ΓE= Γgr(A)= ΓK+?y(xσ) | σ ∈ G?and
?θ: Γgr(A)
G− − − − → Γgr(A)
?
0,
?
K0z′
ρis a crossed product algebra over Z(A0). Let E be
E0is the division algebra Brauer-equivalent to A0. The map θgr(A): ΓE/ΓF → G(Z(A0)/F0) induces
?ΓK→ G(KH
0
?F0), and we have a commutative diagram,
λ
?ΓK
?F0).
?eθ
G(K/F) − − − − → G(KH
0
(4.6)
Page 28
28
J.-P. TIGNOL AND A. R. WADSWORTH
Proof. Since each ρ ∈ H induces a different automorphism of K0= K, it is clear that A0=
a crossed product algebra over its center KH
is simple, we have ΓE= Γgr(A)and θE= θgr(A), by Cor. 2.3. Furthermore, since A0has only one simple
component, for any γ ∈ ΓE, θE(γ + ΓF) is the automorphism of Z(A0) induced by conjugation by any
a ∈ Aγ∩A×. If γ ∈ ΓK, then a can be chosen in gr(K), and the conjugation is trivial, as Z(A0) ⊆ K0.
Hence, θgr(A)induces?θ. For each γ ∈ Γgr(A), there is σ ∈ G with λ(zσ) ≡ γ (mod ΓK). Then?θ(γ+ΓK) is
where the left map is σ ?→ ? σ and the bottom map is restriction of the automorphism from K = K0to
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, K is Galois over F and G(K/F)∼= G(K/F). Let G = G(K/F). The short exact
sequence of trivial G-modules
0 → ΓF→ Γ → Γ/ΓF→ 0
yields a connecting homomorphism ∂: H1(G,Γ/ΓF) → H2(G,ΓF). In fact, ∂ is an isomorphism, since
for the divisible torsion-free group Γ we have H1(G,Γ) = H2(G,Γ) = {0}. Thus, we have a succession
of maps
H2(G,K×) → H2(G,ΓF)
where the left map is induced by the G-module homomorphism w: K×→ ΓK= ΓF.
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 ∈ Z2(G,K×). Let A = (K/F,G,f) =
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 zσas defined in (4.5). So, for the
graded simple ring gry(A), we have A0=
ρ∈H
KH. Furthermore,
?
ρ∈H
K0z′ρis
0= KH. Clearly, Γgr(A)= ΓK+?y(xσ) | σ ∈ G?. Since A0
given by conjugation by z′
σon KH
0, which coincides with ? σ|KH. Therefore, diagram (4.6) is commutative,
KH= KH
0.
?
∂−1
− − → H1(G,Γ/ΓF) = Hom(G,Γ/ΓF),(4.7)
?
σ∈G
Kxσ,
?
K0z′ρ, as above, and A0is a crossed product algebra over
(i) The map λ of (4.4) is the image of [f] ∈ H2(G,K×) under the maps of (4.7).
(ii) v extends to a valuation on D iff ms(A0) = ms(A).
(iii) Suppose v extends to D. Then, ΓD/ΓF= im(λ); Z(D) = KH; D is the division algebra Brauer-
equivalent to A0; and θD: ΓD/ΓF→ G(Z(D)/F) is the isomorphism which is the inverse to the
composite map G(Z(D)/F)
Proof. Note that Prop. 4.3 applies here, since the map G(K/F) → G(K/F) is injective. Hence, A0is a
crossed product with center KH.
∼=
− → G/H
∼=
− → ΓD/ΓF induced by λ.
(i) follows from (4.3) above and the definition of the connecting homomorphism in group cohomology.
For (ii), we have ms(gr(A)) = ms(A0), as A0is 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)∼=gEndE(N) for some graded
right E-vector space N. Then, using Cor. 2.3 and Prop. 4.3, we have ΓD/ΓF= ΓE/ΓF= Γgr(A)/ΓF= im(λ).
Also, D = E0, which is the division algebra Brauer-equivalent to A0, so Z(D)∼= Z(A0) = KH
Finally, again using Cor. 2.3, we have θD= θE= θgr(A). In commutative diagram (4.6),?θ = θgr(A), as
?θ is an isomorphism, and the diagram shows that?θ is the inverse of the isomorphism induced by λ.
0= KH.
ΓK= ΓF. The diagram shows that the surjective map?θ, is also injective, as ker(λ) = H = ker(?θ◦λ). So,
?
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VALUE FUNCTIONS AND ASSOCIATED GRADED RINGS FOR SEMISIMPLE ALGEBRAS29
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 in∈ F×,
jn∈ 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 µs⊆ 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 xs∈ F×for all x ∈ X. Therefore, there is a commutative diagram
with exact rows:
1 − − − − → F×− − − − → X
s
π
− − − − →Z
?ρ
− − − − → 1
?
?s
1 − − − − → F×s− − − − → F×− − − − → F×/F×s− − − − → 1.
Let K = ker(ρ) and L = im(ρ), and let L = F({s√c | cF×s∈ L}) be the s-Kummer extension field of F
associated with L. Let also G = Hom(Z,µs), 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]:
(4.8)
Proposition 4.5. The F-algebra Z is G-Galois, and contains r primitive idempotents e1, ..., er, which
form an F-base of K and are conjugate under the G-action. The isotropy subgroup of any eiis H, and
eiZ∼= L is a Galois extension of F with Galois group isomorphic to H. In particular, Z∼= Lr, a direct
product of r copies of L.
Proof. For each z ∈ Z, choose xz∈ X such that π(xz) = z. By definition of an armature, (xz)z∈Z is
an F-base of Z. LetsX = {x ∈ X | xs= 1}. Applying the snake lemma to (4.8), we get the exact
sequence
1 → µs→sX
Since µsis a cyclic group of order s in the finite abelian groupsX of exponent s, this exact sequence
splits. Therefore, we may assume that the elements xzsatisfy
π− → K → 1.
xkxk′ = xkk′
for k, k′∈ K. (4.9)
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J.-P. TIGNOL AND A. R. WADSWORTH
In particular, x1= 1. In the base (xz)z∈Z, the matrix of multiplication by xz 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 TZ/F: Z → F satisfies
TZ/F(x1) = |Z| ?= 0
It is then straightforward to check that the bilinear trace form on Z is not degenerate, hence Z is ´ etale.
An action of G on Z is defined by
andTZ/F(xz) = 0for z ?= 1.
χ ∗ xz = χ(z)xz
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 =1
r
z1, ..., zm∈ Z be representatives of the cosets modulo K. Since xzixk∈ xzikF×, the products xzixk
for k ∈ K and i = 1, ..., m form a base of Z. For i = 1, ..., m the product exziis in the F-span of
(xzixk)k∈K, hence exz1, ..., exzmare linearly independent. These elements span eZ since exzixk= exzi
for k ∈ K, hence they form a base of eZ. Let
?
k∈Kxk. In view of (4.9), we have exk= e for all k ∈ K, hence e2= e. Let
eX = {ex | x ∈ X} =
m ?
i=1
exziF×⊆ (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 = IndG
H(eZ).
For an armature A of an arbitrary finite-dimensional F-algebra A, there is an associated armature
pairing
βA: A × A → µ(F) given by(aF×,bF×) ?→ aba−1b−1,
where µ(F) denotes the group of roots of unity in F. It is shown in [TW, §2] that βA is a well-
defined symplectic bimultiplicative pairing, and if βA 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 | βA(a,b) = 1 for all b ∈ B},
which is a subgroup of A.
|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 βAis nondegenerate. Let s = exp(A). The nondegeneracy of βAimplies that
µs⊆ F. We denote by π: A×→ A×/F×the canonical map. Let v: F → Γ∪{∞} be a valuation on F.
Assume that char(F) ∤ s. Hence, the group µs= µs(F) of s-th roots of unity in F maps bijectively
to µs(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) =
Note that when βA is nondegenerate, i.e., A ∩ A⊥= {1}, we have
1
sv(xs), and
? w: A → Γ/ΓF, the map induced by w.
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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.
Page 34
34
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