Download Quadratic Forms Over Semilocal Rings by Ricardo Baeza (auth.) PDF

By Ricardo Baeza (auth.)

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Space for [D(E) ] a n d w(E) (or w(q)) for the (E,q). invariant this of two invariants (E,q). To this in o n l y one, the aim one define the so 42 graded maya Brauer-Wall algebras is a g r a d e d (see group BW(A) [Ba], ch. IV, Azumaya algebra ~(E) called the Clifford mation about exact = so t h a t classes of g r a d e d We have shown it d e f i n e s Azu- t h a t C(E) an e l e m e n t of and Witt (E,q). 15) below), where Ag(A) over A (see algebras the g r o u p of c o n t i n u o u s follow author) this point to h a n d l e the C l i f f o r d Another ~ o C(Spec(A),~ denotes /2~ the g r o u p ch.

Remark. An immediately consequence of this results are the facts. i) Let B = A ( 6 1 ( b ) ) be a q u a d r a t i c B ~ A x A if and o n l y Let D = (a,B] [D] = < 1 , - a > Then it, the n o t a t i o n let D I , D 2 be two q u a t e r n i o n are d e t e r m i n e d and the rank, condition the = n2(~(x)) of q u a d r a t i c ii) with but we m a i n t a i n to § 3. 20) (D,~) 4), w h i c h [D] = < 1 , - a > Proof. n1(x) (of r a n k then full Let A be any r i n g the r i n g A the A z u m a y a by t h e i r space (a,B] to c h e c k to any c o m p l i c a t i o n .

Proof. e. between the n o r m f o r m of q u a d r a t i c algebras. let B be a q u a d r a t i c involution relationships introduce = n2(~(x)) an i s o m o r p h i s m ~ : (B1,n I) (B2,n 2) , as to be shown. L e t us n o w c o n s i d e r a quaternion [B] 6 ~(A) . The r e d u c e d n:D norm map ~A algebra D = of D = B • Be (a,B] over A with (e 2 = a) is g i v e n a 6 A ~, by 29 n(u+ for u , v 6 B, w h e r e ve) n : B ~ A is the n o r m m a p of B. t h a t n is m u l t i p l i c a t i v e , n defines we shall norm on D 6 P ( A ) denote and in the notation [D].

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