- Email: [email protected]
Quadratic forms over polynomial rings over global fields
JOURNAL
OF NUMBER
THEORY
Quadratic
17, 113-115 (1983)
Forms over Polynomial over Global Fields
Rings
RAMAN PARIMALA School
of Mathematics,
Tata Institute of Fundamental Bombay 400 005, India
Communicated
Research,
by M. Kneser
Received October 26. 1981
K
It is proved that a quadratic space over the polynomial extension of a global field is extended from K if it is extended from K,. for every completion K,. of K.
1 The aim of this note is to prove the following: THEOREM. Let K be a global field of characteristic # 2. A quadratic space q over K[X,,..., X,] is extended from K if and only if q OK K, is extended from K, for every completion K, of K. This theorem was proved for quadratic spaces of rank <4 in [3]. We use the same technique of Amitsur cohomology as in [3].
2. PROOF OF THE THEOREM
Let q be a quadratic space over K[X,,...,X,] which is not extended from K. We show that for at least one completion K,, of K, q OK K,. is not extended from K,. Since any isotropic quadratic space over K[X, ,..., X,] is extended from K [2], it follows that 4 is anisotropic, - denoting “reduction modulo the variables.” By the Hasse-Minkowski theorem, qOK K, is anisotropic for some completion K, of K. Then, q OK K, is not extended from K,. This follows from: LEMMA. Let q be a quadratic space over K[X, ,..., X,]. Let L/K be any field extension such that QOK L is anisotropic. If q OK L is extendedfrom L, then q is extended from K. 113 0022-3 14x/83 $3.00 Copyright 0 1983 by Academic Press, Inc. All rights of reproduction in any form reserved.
114
R. PARIMALA
Let 4: L ,& q 1 L IX, ,..., X,] x8 S be an isometry Proof of the Lemma. over L. The Amitsur complex
flat extension L/K, gives by functoriality, maps d,@ d, 9: O(L &L &q)+ O(L KOLIX,,..., X,]&I~) and ~,=&~)~(d,~)~'~ = d, a,,. Hence czq defines O(L Kc4 L [X, ,***, X,] & 4) satisfies (d,a,)(d,a,)
of the faithfully
an element [a,] of the Amitsur
H,(L[X,
,..., X,]/K[X,
cohomology
set
.... . x, 1, o(q)) = (a E O(L &3 L [Xl .-*.3x,1 Kc3 4) f (d,a)(d,a)
= d, a))/-.
where - is the equivalence relation defined by a - ,8 if and only if there exists y E O(L[X,,...,X,] & S) with a = (d, y) ,f3(d, 7))‘. The assignment [q] w [a,] defines a bijection between the set Q of isometry classes of quadratic spaces over K[X,,..., X,] which become isometric to qK@ L [X, ,..., X,,] over L[X, ,..., X,,], and the set H,(L[X, ,..., X,]/ K[X, ,..., X,], O(q)) [ 1, 10.20, Corollary 41. We show that the inclusion L + L [X, )...) X,,] induces a bijection
i:H'(L/K,
O(q))z
H'(L[X,,...,X,]/K[X,,...,X,],
O(q)).
If Q denotes the set of isometry classes of quadratic spaces over K which become isometric to qK@ L over L, then we have an inclusion Q 4 Q and the commutative diagram
0~ H'(L/K,O(q)) r &H,(L;X
(*I
ri , ,..., X,1/W,
,...>Xi,], O(q))
and if i is a bijection, then Q 4 Q is also a bijection, thus proving that q is extended from K. We now prove that the inclusion i is indeed surjective. We do this by induction on n. If n = 1, since any quadratic space over K[X,] is extended from K, i is a bijection in view of (*). Let n > 1 and let a E w Kc3 L [Xl 3***,x,1 Kc3 4) with d,a . d,a = d, a. By induction, over the ring L(X,) KcX,@~(~,)[~,~...~X,l which contains (L &L)[X1,...,X,], a is equivalept to 0 E O&(X,) K(X,,@ L(X,) ,@ 4). There exists y E X,] ,@ q) such that a = (d, y) j?(d, y)-‘. Since q is O(Jw,)[X,,..., anisotropic over L, 4 is anisotropic over L(X,) and in view of [3,
115
QUADRATICFORMS
Lemma 2.21 we have O&(X,) ,& 4) = O(L(X,)(X,,...,X,] &3 4). Thus, both /I and y are independent of X2,..., X, and hence a is independent of X, ,..., X,, i.e., a E O(L ,@ L[X,] & q) with (d,a)(d,a) = (~?,a). Since the result is true for n = 1, there exist /3’ E O(L &!J L &I q), y’ E O(L [X,] & 4) such that a = (d,y’)P’(d, y’)-‘. For the same reason as earlier, O(L[X,] ,&2 tj) = O(L K@4) so that a is independent of X, and hence the map i is surjective, thus completing the proof.
REFERENCES 1. M. ARTIN, “Commutative 1966. 2. M. OJANGUREN, Formes
rings,”
Mimeographed
Quadratiques
Lecture
sur les algibres
Notes,
MIT,
de polyndmes,
Cambridge.
Mass..
C. R. Acad. Sci. Ser.
A 287 (1978). 695-698. 3. R. PARIMALA AND R. SRIDHARAN, A local global polynomial rings, J. Algebra 74 (1982) 264-269.
principle
for
quadratic
forms
over
OF NUMBER
THEORY
Quadratic
17, 113-115 (1983)
Forms over Polynomial over Global Fields
Rings
RAMAN PARIMALA School
of Mathematics,
Tata Institute of Fundamental Bombay 400 005, India
Communicated
Research,
by M. Kneser
Received October 26. 1981
K
It is proved that a quadratic space over the polynomial extension of a global field is extended from K if it is extended from K,. for every completion K,. of K.
1 The aim of this note is to prove the following: THEOREM. Let K be a global field of characteristic # 2. A quadratic space q over K[X,,..., X,] is extended from K if and only if q OK K, is extended from K, for every completion K, of K. This theorem was proved for quadratic spaces of rank <4 in [3]. We use the same technique of Amitsur cohomology as in [3].
2. PROOF OF THE THEOREM
Let q be a quadratic space over K[X,,...,X,] which is not extended from K. We show that for at least one completion K,, of K, q OK K,. is not extended from K,. Since any isotropic quadratic space over K[X, ,..., X,] is extended from K [2], it follows that 4 is anisotropic, - denoting “reduction modulo the variables.” By the Hasse-Minkowski theorem, qOK K, is anisotropic for some completion K, of K. Then, q OK K, is not extended from K,. This follows from: LEMMA. Let q be a quadratic space over K[X, ,..., X,]. Let L/K be any field extension such that QOK L is anisotropic. If q OK L is extendedfrom L, then q is extended from K. 113 0022-3 14x/83 $3.00 Copyright 0 1983 by Academic Press, Inc. All rights of reproduction in any form reserved.
114
R. PARIMALA
Let 4: L ,& q 1 L IX, ,..., X,] x8 S be an isometry Proof of the Lemma. over L. The Amitsur complex
flat extension L/K, gives by functoriality, maps d,@ d, 9: O(L &L &q)+ O(L KOLIX,,..., X,]&I~) and ~,=&~)~(d,~)~'~ = d, a,,. Hence czq defines O(L Kc4 L [X, ,***, X,] & 4) satisfies (d,a,)(d,a,)
of the faithfully
an element [a,] of the Amitsur
H,(L[X,
,..., X,]/K[X,
cohomology
set
.... . x, 1, o(q)) = (a E O(L &3 L [Xl .-*.3x,1 Kc3 4) f (d,a)(d,a)
= d, a))/-.
where - is the equivalence relation defined by a - ,8 if and only if there exists y E O(L[X,,...,X,] & S) with a = (d, y) ,f3(d, 7))‘. The assignment [q] w [a,] defines a bijection between the set Q of isometry classes of quadratic spaces over K[X,,..., X,] which become isometric to qK@ L [X, ,..., X,,] over L[X, ,..., X,,], and the set H,(L[X, ,..., X,]/ K[X, ,..., X,], O(q)) [ 1, 10.20, Corollary 41. We show that the inclusion L + L [X, )...) X,,] induces a bijection
i:H'(L/K,
O(q))z
H'(L[X,,...,X,]/K[X,,...,X,],
O(q)).
If Q denotes the set of isometry classes of quadratic spaces over K which become isometric to qK@ L over L, then we have an inclusion Q 4 Q and the commutative diagram
0~ H'(L/K,O(q)) r &H,(L;X
(*I
ri , ,..., X,1/W,
,...>Xi,], O(q))
and if i is a bijection, then Q 4 Q is also a bijection, thus proving that q is extended from K. We now prove that the inclusion i is indeed surjective. We do this by induction on n. If n = 1, since any quadratic space over K[X,] is extended from K, i is a bijection in view of (*). Let n > 1 and let a E w Kc3 L [Xl 3***,x,1 Kc3 4) with d,a . d,a = d, a. By induction, over the ring L(X,) KcX,@~(~,)[~,~...~X,l which contains (L &L)[X1,...,X,], a is equivalept to 0 E O&(X,) K(X,,@ L(X,) ,@ 4). There exists y E X,] ,@ q) such that a = (d, y) j?(d, y)-‘. Since q is O(Jw,)[X,,..., anisotropic over L, 4 is anisotropic over L(X,) and in view of [3,
115
QUADRATICFORMS
Lemma 2.21 we have O&(X,) ,& 4) = O(L(X,)(X,,...,X,] &3 4). Thus, both /I and y are independent of X2,..., X, and hence a is independent of X, ,..., X,, i.e., a E O(L ,@ L[X,] & q) with (d,a)(d,a) = (~?,a). Since the result is true for n = 1, there exist /3’ E O(L &!J L &I q), y’ E O(L [X,] & 4) such that a = (d,y’)P’(d, y’)-‘. For the same reason as earlier, O(L[X,] ,&2 tj) = O(L K@4) so that a is independent of X, and hence the map i is surjective, thus completing the proof.
REFERENCES 1. M. ARTIN, “Commutative 1966. 2. M. OJANGUREN, Formes
rings,”
Mimeographed
Quadratiques
Lecture
sur les algibres
Notes,
MIT,
de polyndmes,
Cambridge.
Mass..
C. R. Acad. Sci. Ser.
A 287 (1978). 695-698. 3. R. PARIMALA AND R. SRIDHARAN, A local global polynomial rings, J. Algebra 74 (1982) 264-269.
principle
for
quadratic
forms
over