6

R. HAZRAT AND A. R. WADSWORTH

Remark 3.1. The reduced norm for an Azumaya algebra is defined using a splitting ring, and in general splitting rings can be difficult to find. But for a graded division algebra E we observe that, analogously to the case of ungraded division rings, any maximal graded subfield L of E splits E. For, the centralizer C = C_{E}(L) is a graded subring of E containing L, and for any homogeneous c ∈ C, L[c] is a graded subfield of E containing L. Hence, C = L, showing that L is a maximal commutative subring of E. Thus,

b y L

e m m a 5 . 1 . 1 3 ( 1 )

, p.

1 4 1 o f [ K ]

, as

E i s A

zumaya,

E

⊗

_{T}L

∼

=

E n d L ( E )

∼

=

M n ( L ) . T h

us, we can compute

reduced norms for elements of E by passage to E ⊗_{T }L.

# We have other tools as well for computing Nrd_{E }and Trd_{E}:

Proposition 3.2. Let E be a graded division ring with center T . Let q(T ) be the quotient eld of T , and let q(E) = E ⊗_{T }q(T ), which is the quotient division ring of E. We view E ⊆ q(E). Let n = ind(E) =

ind(q(E)). Then for any a ∈ E, (i) char_{E}(x, a) = char_{q(E)}(x, a), so

Nrd_{E}(a) = Nrd_{q(E) }

(a)

and

Trd_{E}(a) = Trd_{q(E) }

(a).

(3.2)

(ii) If K is any graded sub eld of E containing T and a ∈ K, then

Nrd_{E}(a) = N_{K/T }

(a)^{n/[K :T }^{] }

and

# Trd_{E}(a) =

n [K :T ]

Tr_{K/T }

(a).

(iii) For γ ∈

E , i f a ∈ E γ t h e n N r d E ( a ) ∈ E n γ

a n d T r d ( a ) ∈ E γ . I n p a r t i c u l a r , E ( 1 )

⊆ E 0 .

(iv)

# Set δ = ind(E)

i n d ( E 0

) [ Z ( E 0

) : T 0

]

.

I f a ∈ E 0

, then,

N r d E ( a ) = N Z ( E 0 ) / T 0

Nrd

E 0 ( a )

δ

∈ T 0

and

Trd

E

(a) = δ Tr

Z ( E 0 ) / T 0 T r d

E 0 ( a ) ∈ T 0 .

(3.3)

Proof. (i) The construction of reduced characteristic polynonials described above is clearly compatible with

scalar extension of the ground ring. Hence, char_{E}(x, a) = char_{q(E) }a ⊗ 1 in E ⊗_{T }q(T ) ). The formulas in (3.2) follow immediately.

(x, a) (as we are identifying a ∈ E with

(ii) Let h_{a }= x^{m }+ t_{m }_{1}x^{m 1 }+ . . . + t_{0 }∈ q(T )[x] be the minimal polynomial of a over q(T ). As noted in [HwW_{1}], Prop. 2.2, since the integral domain T is integrally closed and E is integral over T , we have h_{a }∈ T [x]. Let `_{a }= x^{k }+s_{k 1}x^{k 1 }+. . .+s_{0 }∈ T [x] be the characteristic polynomial of the T -linear function

on the free T -module K given by c 7→ ac. By definition, N_{K/T }

(a) = ( 1)^{k}s_{0 }and Tr_{K/T }

(a) =

s_{k 1 }

. Since

q(K) = K ⊗_{T }q(T ), we have [q(K) : q(T )] = [K : T ] = k and `_{a }is also the characteristic polynomial for

the q(T )-linear transformation of q(K) given by q 7→ aq. So, `_{a }= h

k/m a

. Since char_{q(E) }

(x, a) = h

n/m a

(see

# [R], Ex. 1, p. 124), we have char_{q(E) }

( x , a ) = ` n / k a

# . Therefore, using (i),

Nrd_{E}(a) = Nrd_{q(E) }

(a) =

( 1 ) k s 0

n/k

= N K / T ( a )

n/k

.

The formula for Trd_{E}(a) in (ii) follows analogously.

n/m a (iii) From the equalities char_{E}(x, a) = char (x, a) = h noted in proving (i) and (ii), we have q(E) m1 Nrd_{E}(a) = [( 1)^{m}t_{0}]^{n/m }and Trd_{E}(a) = . A s n o t e d i n [ H w W 1 ] , P r o p . 2 . 2 , i f a ∈ E γ , t h e n i t n m t s m1 minimal polynomial h_{a }is γ-homogenizable in T [x] as in (2.3) above. Hence, t_{0 }∈ E_{mγ }and t ∈ E γ .

# Therefore, Nrd_{E}(a) ∈ E_{nγ }

a n d T r d ( a ) ∈ E γ . I f a ∈ E ( 1 )

then a is homogeneous, since it is a unit of E,

and since 1 = Nrd_{E}(a) ∈ E_{n }_{deg(a) }

, necessarily deg(a) = 0.

( i v ) S u p p o s e a ∈ E 0 . T h e n , h a i s 0 - h o m o g e n i z a b l e i n T [ x ] , i . e . , h a ∈ T 0 [ x ] . H e n c e , h a i s t h e m i n i m a l p o l y n o m i a l o f a o v e r t h e fi e l d T 0 . T h e r e f o r e , i f L i s a n y m a x i m a l s u b fi e l d o f E 0 c o n t a i n i n g a , w e h a v e N L / T 0 ( a ) = [ ( 1 ) m t 0 ] [ L : T 0 ] / m . N o w ,

n / m = δ i n d ( E 0 ) [ Z ( E 0 ) : T 0 ] m = δ [ L : T 0 ] / m .