Springer's odd degree extension theorem for quadratic forms over semilocal rings
aa r X i v : . [ m a t h . R A ] J a n SPRINGER’S ODD DEGREE EXTENSION THEOREM FORQUADRATIC FORMS OVER SEMILOCAL RINGS
PHILIPPE GILLE AND ERHARD NEHER
Abstract:
A fundamental result of Springer says that a quadratic formover a field of characteristic = 2 is isotropic if it is so after an odd degreeextension. In this paper we generalize Springer’s theorem as follows. Let R be a an arbitrary semilocal ring, let S be a finite R –algebra of odd degree,which is ´etale or generated by one element, and let q be a nonsingular R –quadratic form whose base ring extension q S is isotropic. We show that then q is already isotropic. Keywords:
Quadratic forms, semilocal rings, Springer Odd Degree Exten-sion Theorem.
MSC 2000:
Introduction
A celebrated result of Springer [Sp] says that a quadratic form over afield F of characteristic = 2 that becomes isotropic over an odd degree fieldextension is already isotropic over F . The result was conjectured by Wittand was also proven, but not published by E. Artin. An account of Springer’sresult is given in [Lam, VII, Thm. 2.7] or [Sch, II, Thm. 5.3]. It is provenfor arbitrary fields in [EKM, 18.5].Springer’s Theorem has many important consequences, for example forWitt groups. It is therefore not surprising that it has been generalized byreplacing odd degree extensions by more general field extensions, see forexample, [Ho1, Prop. 5.3], [Ho2, Cor. 4.2] or [Lag, Lem. 2.8], or by replacingthe base field F by more general rings. The main result of this paper goesin the latter direction: Odd Degree Extension Theorem. (Theorem 2.1)
Let R be a semilocalring, M a finite projective R –module, q : M → R a nonsingular quadraticform, and S a finite R –algebra of odd degree, which is ´etale or one-generated.If the base ring extension q S is isotropic, then already q is isotropic. Since the terminology regarding quadratic forms over rings is not stan-dard, see the comparison in 1.2(a), let us explain what we mean with anonsingular quadratic form. Recall that the radical of (
M, q ) with polar
Date : February 1, 2021.P. Gille was supported by the ANR project
Geolie
ANR-15-CE40-0012.E. Neher acknowledges partial support from NSERC through a Discovery grant. form b q is rad( M, q ) = { m ∈ M : q ( m ) = 0 = b q ( m, M ) } . We call q non-singular , if rad(( M, q ) F ) = { } for all R –fields F . An example of such aform is a regular quadratic form, defined by the condition that its polar forminduces an isomorphism between M and its dual space. Nonsingular andregular quadratic forms are the same in case 2 ∈ R × = { u ∈ R : uR = R } ,the units of R , or in case the projective R –module M has constant even rank,but not in general. For example, for u ∈ R × the quadratic form R → R , x ux is always nonsingular, but regular only if 2 ∈ R × . We call a finiteprojective R –algebra S of constant degree d one-generated if it is generatedby one element, equivalently, S ∼ = R [ X ] /f for a monic polynomial of degree d . Special cases of the Odd Degree Extension Theorem have been provenbefore; all of them assumed 2 ∈ R × , so that nonsingular = regular. Specif-ically, it was proven by Panin and Rehmann [PR] under the assumptionthat R is a noetherian local integral domain, which has an infinite residuefield, and that S is ´etale (necessarily one-generated in this case). It wasextended in [PP] by Panin-Pimenov to R a semi-local noetherian integraldomain whose residue fields are all infinite. It was stated in [Scu] for R semilocal, 2 ∈ R × , with a proof that we could not understand.For proving the Odd Degree Extension Theorem one can easily reduce to M of constant rank r ≥
2. The case r = 2 (Lemma 2.2) holds for any finite R –algebra; our proof uses a consequence of Deligne’s trace homomorphismfor the flat cohomology of abelian affine group schemes (Lemma B.3). For r ≥ M ⊗ R ( S/ Jac( R ) S ) to M → R S (Corollary A.6). We obtain this as an application of Demazure’sConjugacy Theorem for reductive R –group schemes. We should point out,that in this part we use the classical version ( R a field, [EKM]) of the OddDegree Extension Theorem. Once the case of one-generated algebras hasbeen settled, the case of ´etale algebras easily follows by applying a recentresult of Bayer-Fluckiger, First and Parimals [BFP] on embeddings of finite´etale R –algebras into one-generated R –algebras. Structure of the paper.
Section 1 presents a review of nonsingular qua-dratic forms and proves some results, needed for our proof of the Odd DegreeExtension Theorem in the first part of section 2. In the second part of thatsection we prove some consequences of the theorem, well-known in case ofbase fields. The paper closes with two appendices, A and B. Their resultsare used in our proof of the Odd Degree Extension Theorem, but they areof interest beyond that Theorem.
Basic notation.
Throughout, R is a commutative associative unital ring.We do not assume that 2 ∈ R × , unless this is explicitly stated so. Also,modules, quadratic forms and algebras will be defined over R . We use R - alg to denote the category of commutative associative unital R –algebras. DD DEGREE EXTENSION THEOREM 3
Objects of R - alg are sometimes also referred to as R –rings. If M is an R –module and S ∈ R - alg , we put M S = M ⊗ R S .We say an R –module M is finite projective if it is finitely generatedprojective (= locally free of finite rank). A finite R –algebra is an alge-bra S ∈ R - alg whose underlying R –module is finitely generated, but notnecessarily projective. A finite projective R –algebra is an R –ring S whoseunderlying R –module is finite projective. The norm of such an R –algebra S is denoted N S/R . A finite projective R –algebra S is called finite ´etale if itis separable, see for example [Fo] or [Knu] for other equivalent definitions.We call S ∈ R - alg an ´etale R –algebra of degree d ∈ N if S is a finite ´etalealgebra whose underlying R –module is projective of constant rank d .1. Quadratic forms
This section starts with a review of known facts on bilinear and quadraticforms over arbitrary rings, thereby also establishing our notation and termi-nology, 1.1–1.3. We study quadratic forms over semilocal rings in the secondpart of this section.1.1.
Quadratic and symmetric bilinear forms (terminology, basicfacts). (a) A (symmetric) bilinear module is a pair (
M, b ) consisting of afinite projective R –module and a symmetric R –bilinear map b : M × M → R .We will only consider symmetric bilinear forms, and hence simply speak ofbilinear modules. We use the symbol ( M, b ) ∼ = ( M ′ , b ′ ) to denote isometricbilinear modules. When M is clear from the context or unimportant, wewill sometimes write b for ( M, b ). Given a bilinear module (
M, b ), its adjoint is the R –linear map b ∗ : M → M ∗ = Hom R ( M, R ), m b ( m, · ). We call( M, b ) regular if b ∗ is an isomorphism.(b) A quadratic module is a pair ( M, q ) where M is a finite projective R –module and q : M → R is an R –quadratic form. We use b q to denotethe polar form of q . We call q or ( M, q ) regular if b q is regular. As forbilinear modules, ( M, q ) ∼ = ( M ′ , q ′ ) indicates isometric quadratic modules.We sometimes write q instead of ( M, q ), if M is unimportant or clear fromthe context.(c) ( Base change ) Let S ∈ R - alg , and let ( M, b ) be an R –bilinear module.There exists a unique S –bilinear form b S : M S × M S → S satisfying b S ( s ⊗ m , m ⊗ s ) = s b ( m , m ) s for m i ∈ M and s i ∈ S . Given an R –quadratic module ( N, q ), there exists a unique S –quadratic form q S : N S → S satisfying q S ( s ⊗ n ) = s q ( n ) for s ∈ S and n ∈ N . The polar of q S isthe base change of the polar of q . If ( M, b ) (or (
N, q )) is regular, then so is( M S , b S ) (respectively ( N S , q S )).(d) ( Tensor products ) Let ( M i , b i ), i = 1 ,
2, be bilinear modules. Thereexists a unique symmetric bilinear form b ⊗ b on M ⊗ R M satisfying( b ⊗ b )( m ⊗ m , m ′ ⊗ m ′ ) = b ( m , m ′ ) b ( m , m ′ ) for m i , m ′ i ∈ M i .Given an R –bilinear module ( M, b ) and an R –quadratic form ( N, q ), there
P. GILLE AND E. NEHER exists a unique R –quadratic form b ⊗ q : M ⊗ R N → R satisfying( b ⊗ q ) ( m ⊗ n ) = b ( m, m ) q ( n ) , and b b ⊗ q ( m ⊗ n, m ′ ⊗ n ′ ) = b ( m, m ′ ) b q ( n, n ′ )(1.1.1)for all m, m ′ ∈ M and n, n ′ ∈ N , where b q is the polar form of q . It is calledthe tensor product of ( M, b ) and ( N, q ), [Sa, Thm. 1]. The polar form of b ⊗ q is the tensor product of the symmetric bilinear forms b ⊗ b q . If ( M, b )and (
N, q ) are regular, then so is (
M, b ) ⊗ ( N, q ). The tensor product iscompatible with base change: for S ∈ R - alg we have( M, b ) S ⊗ S ( N, q ) S ∼ −→ (cid:0) ( M, b ) ⊗ R ( N, q ) (cid:1) S with respect to m ⊗ s ⊗ n ⊗ s m ⊗ m ⊗ s s .(e) Given a bilinear module ( M, b ), the orthogonal module of a submodule U ⊂ M is U ⊥ = { m ∈ M : b ( m, U ) = 0 } . We call U totally isotropic if U ⊂ U ⊥ . A submodule U of a quadratic module ( M, q ) is totally isotropic if q ( U ) = 0, in which case U is also a totally isotropic submodule of ( M, b q ).If ( M, b ) is a bilinear module and U ⊂ M is a submodule such that ( U, b | U )is regular, then M = U ⊕ U ⊥ .1.2. Nonsingular quadratic forms.
Recall that the radical of a quadraticmodule (
M, q ) is rad(
M, q ) = { m ∈ M : q ( m ) = 0 = b q ( m, M ) } . It is asubmodule of M and satisfies (rad( M, q )) S ⊂ rad(( M, q ) S ), S ∈ R - alg . Wecall ( M, q ) or simply q nonsingular if rad(( M, q ) F ) = { } for all fields F ∈ R - alg . A quadratic space is a quadratic module ( M, q ) with a nonsingular q . We will use the following properties of nonsingular forms (for details seethe references in (a)).(a) ( Comparison of terminology ) Our terminology of a regular bilinearform or regular quadratic form follows [Knu] and [Sch] (except that in thesereferences “regular” and “nonsingular” are used interchangeably), but aregular bilinear form as defined here is called “non singular” in [Ba] and“nondegenerate” in [EKM]. A nonsingular quadratic form is called “non-degenerate” in [Co1], “nondegenerate” in [EKM], “semiregular” in [Knu] incase of odd rank, cf. (e), and “separable” in [Pe].(b) A regular quadratic form is nonsingular, since rad((
M, q ) S ) = 0 for aregular form and any S ∈ R - alg by (c).(c) If 2 ∈ R × , a nonsingular form is regular. Indeed, if 2 ∈ R × , thenrad( q S ) = { m ∈ M S : b q S ( m, M S ) = 0 } , S ∈ R - alg , and so the adjoint of b q is an isomorphism by Nakayama.(d) If q is nonsingular, then so is q S for any S ∈ R - alg .(e) Let ( M, q ) be a quadratic module with M of constant even rank. Then q is nonsingular if and only if q is regular, cf. (h) below.(f) A quadratic R –space ( M, q ) with M faithfully projective is primitive inthe sense that q ( M ) generates R as ideal. If R is semilocal, then q ( m ) ∈ R × for some m ∈ M (which is unimodular in the sense of 1.8). DD DEGREE EXTENSION THEOREM 5 (g) Let (
M, q ) = ( M , q ) ⊥ ( M , q ) be a direct sum of quadratic modules.If q is nonsingular, then so are q and q . Conversely, if q is regular, then q is nonsingular if and only if q is nonsingular. If q and q are nonsingular,then q need not be nonsingular, see the example in 1.7.(h) (1 –dimensional forms ) Let u ∈ R × . We define h u i b : R × R → R ,( r , r ) ur r and h u i q : R → R , r ur . Then h u i b is a regular bilinearform. The polar of the quadratic form h u i q is 2 h u i , whence h u i q is regular ifand only if 2 ∈ R × , but h u i q is always nonsingular. We abbreviate h u i = h u i b if the meaning of h u i is clear from the context.If ( M, b ) and (
N, q ) are bilinear and quadratic modules respectively, then h i b ⊗ b ∼ = b, h i b ⊗ q ∼ = q under the standard isomorphism R ⊗ R M ∼ −→ M . However, b ⊗ h i q = b q ,where b q : M → R , m b ( m, m ), is the quadratic form associated with b .Its polar is b q b = 2 b .(i) ( Reduction to constant rank ) Let R = R × · · ·× R n be a direct productof rings. A quadratic module ( M, q ) over R uniquely decomposes as anorthogonal sum(1.2.1) ( M, q ) = ( M , q ) ⊥ · · · ⊥ ( M n , q n )where each ( M i , q i ) is a quadratic module over R i . Conversely, given qua-dratic R i –modules ( M i , q i ), the formula (1.2.1) defines a quadratic R –module.The quadratic module ( M, q ) is regular (nonsingular) if and only if all qua-dratic R i –modules ( M i , q i ) are regular (nonsingular respectively).For arbitrary R , a finite projective R –module M gives rise to a decompo-sition R = R × · · · × R n and hence to M = M × · · · × M n such that M i isa projective R i –module of constant rank i . The discussion above then de-scribes the reduction of quadratic modules to quadratic modules of constantrank.(j) ( Witt cancellation ) Let R be a semilocal ring, let q , q , q ′ and q ′ bequadratic forms where q is regular and q is nonsingular and of positiverank. If q ∼ = q ′ and q ⊥ q ∼ = q ′ ⊥ q ′ , then q ∼ = q ′ . Indeed, since the non-singular form q is primitive by (f), this follows from [Kne, K¨urzungssatz].Witt cancellation with all forms q i and q ′ i being regular, is proven in [Ba,III, (4.3)].1.3. Metabolic and hyperbolic spaces.
Let (
U, b ) be a bilinear module.The metabolic space associated with (
U, b ) is the bilinear module M ( U, b ) =( U ⊕ U ∗ , b M ( U,b ) ) whose bilinear form b M ( U,b ) is defined by b M ( U,b ) ( u + ϕ, v + ψ ) = b ( u, v ) + ϕ ( v ) + ψ ( u ) . It is a regular bilinear form, justifying the terminology “space”. By defini-tion, the hyperbolic bilinear space is H b ( U ) = M ( U,
0) where 0 is the nullform.
P. GILLE AND E. NEHER
Given a finite projective R –module U , the associated hyperbolic space H ( U ) is the quadratic module ( U ∗ ⊕ U, hyp) with hyp( ϕ ⊕ u ) = ϕ ( u ). Thepolar form of the hyperbolic quadratic form hyp is the bilinear form b H b ( U, ,in particular, hyp is nonsingular. We call H ( R ) the hyperbolic plane .We say a bilinear module ( M, b ) metabolic if there exists a bilinear module( U, b U ) such that ( M, b ) ∼ = M ( U, b U ). The same terminology will be appliedto hyperbolic quadratic spaces. The following facts are for example provenin [Ba, I, § § M, b ) be a regular bilinear module and let U ⊂ M be a totallyisotropic complemented submodule. Then there exists a submodule V ⊂ M such that U ∩ V = 0, ( U ⊕ V, b | U ⊕ V ) is metabolic and hence M = ( U ⊕ V ) ⊥ ( U ⊕ V ) ⊥ . If b ( m , m ) = b ( m , m ) + b ( m , m ) for some bilinear form b : M × M → R , one can choose V such that ( U ⊕ V, b U ⊕ V ) is a hyperbolicbilinear space(b) ( Characterization of metabolic spaces ) A regular bilinear module (
M, b )is metabolic if and only if one of the following conditions holds:(i) M contains a totally isotropic complemented submodule V with V = V ⊥ , a so-called Lagrangian ,(ii) M contains a totally isotropic and complemented submodule U sat-isfying rank p M = 2 rank p U for all p ∈ Spec( R ).In this case ( M, b ) ∼ = M ( U ) for U ∼ = V ∗ . In particular, M ( U ) is free when-ever V is free, and any decomposition V = V ⊕ · · · ⊕ V n gives rise to andecomposition M ( U ) = M ( U ) ⊥ · · · ⊥ M ( U n ) with U i ∼ = V ∗ i .To see that M is metabolic in case (ii), apply (a) to get M = ( U ⊕ V ) ⊥ ( U ⊕ V ) ⊥ where U ⊕ V is metabolic and rank p = 2 rank p U = rank p M ,whence ( U ⊕ V ) ⊥ = 0. The last claim follows from V ∗ = V ∗ ⊕ · · · ⊕ V ∗ n .In the remainder of this section we consider quadratic spaces ( M, q ) over R in two settings: (a) q is regular, and (b) R is semilocal. The results in case(a) are proven in [Ba, § § Lemma.
Let R be a semilocal ring, let M be a finite projective R –module and let U ⊂ M be a complemented submodule. For a maximalideal m ⊳ R we put κ ( m ) = R/ m , M ( m ) = M ⊗ R κ ( m ) = M/ m M , andanalogously for U ( m ) . We further assume that r ∈ N + and that for every m ∈ Specmax( R ) there exists an r –dimensional subspace W [ m ] ⊂ M ( m ) with U ( m ) ∩ W [ m ] = { } .Then there exists a free submodule W ⊂ M of rank r which satisfies W ( m ) = W [ m ] , U ∩ W = { } and which has the property that U ⊕ W iscomplemented in M . DD DEGREE EXTENSION THEOREM 7
Proposition.
Let ( M, q ) be a quadratic space over R , let U ⊂ M be acomplemented totally isotropic submodule, and assume one of the following. (a) q is regular, or (b) R is semilocal.Then there exists a totally isotropic submodule V ⊂ M such that U ∩ V = { } , ( U ⊕ V, q U ⊕ V ) ∼ = H ( U ) and hence M = ( U ⊕ V ) ⊕ ( U ⊕ V ) ⊥ .Proof. (I) ( Intermediate step ) Suppose there exists a submodule W ⊂ M satisfying U ∩ W = { } , and for which the canonical map β : U ∼ −→ W ∗ , u ( w b q ( u, w ))(1.5.1)is an isomorphism of R –modules. By [Ba, I, (1.7)], we can choose a notnecessarily symmetric bilinear form b satisfying b ( m, m ) = q ( m ) for all m ∈ M . By (1.5.1), for every w ∈ W there exists a unique u w ∈ U such that b q ( u w , w ′ ) = b ( w, w ′ ) holds for all w ′ ∈ W . Because of uniqueness of the u w ,the map U → W , w u w , is R –linear. Then V = { w − u w : w ∈ W } ∼ = W is a totally isotropic submodule: since q ( u w ) = 0 we have q ( w − u w ) = q ( w ) − b q ( w, u w ) = b ( w, w ) − b q ( u w , w ) = 0 . Moreover, U ∩ V = 0 and the canonical map U ∼ −→ V ∗ , u ( v b q ( u, v ))is an isomorphism. Hence ( U ⊕ V, q | U ⊕ V ) ∼ = H ( U ). Since H ( U ) is a regularquadratic module, 1.1(e) applies and yields M = ( U ⊕ V ) ⊕ ( U ⊕ V ) ⊥ .In the remainder of the proof we will establish the existence of a submod-ule W satisfying (1.5.1). We point out that (I) applies to the two cases (a)and (b).(II) ( Case (a) in general and case (b) with R a field ) Let U ⊥ = { m ∈ M : b q ( m, U ) = 0 } . We have a well-defined pairing α : U × M/ ( U ⊥ ) → R, ( u, ¯ m ) b q ( u, m )which is regular: if α ( u, ¯ m ) = b q ( u, m ) = 0 for all m ∈ M , then u ∈ rad( b q ).Hence u = 0 in case (a), while in case (b) we get u ∈ rad( q ) because q ( u ) = 0, so that again u = 0 follows, using that rad( q ) = 0 if R is a field.Also, if α ( U, ¯ m ) = b q ( U, m ) = 0, then m ∈ U ⊥ , and therefore ¯ m = 0 fol-lows. We now get that α induces an isomorphism α ∗ : U ∼ −→ ( M/U ⊥ ) ∗ . Weclaim that there exists a submodule W ⊂ M such that M = U ⊥ ⊕ W ,and therefore can : W ∼ −→ M/U ⊥ under the canonical map. The exis-tence of W follows in case (a) from [Ba, I, (3.2)(a)], saying that U ⊥ iscomplemented in M . It is obvious in case (b) with R a field. Denot-ing by can ∗ : ( M/U ⊥ ) ∗ ∼ −→ W ∗ the dual of the isomorphism can, we have (cid:0) (can ∗ ◦ α ∗ )( u ) (cid:1) ( w ) = α ∗ ( u ) (cid:0) can( w ) (cid:1) = α ∗ ( u )( ¯ w ) = b q ( u, w ) = ( β ( u ) (cid:1) ( w ),i.e., W satisfies (1.5.1). Now (I) finishes the proof in case (a), and in case(b) with R a field. Before we can deal with a semilocal R in case (b) wemake a further reduction. P. GILLE AND E. NEHER (III) (
Reduction to constant rank ) Since U is complemented, it is finitelygenerated projective. By 1.1(i), we can therefore decompose R = R × · · · × R n and correspondingly( M, q ) = ( M , q ) ⊥ · · · ⊥ ( M n , q n ) , U = U × · · · × U n such that each U i ⊂ M i is a complemented submodule of constant rankand a totaly isotropic submodule of the R i –quadratic space ( M i , q i ). If V i ⊂ M i , 1 ≤ i ≤ n , are submodules as in the claim of the lemma, then= V × · · · × V n satisfies the conditions for ( M, q ). Without loss of generalitywe can therefore assume that U has constant rank, say rank r . It is thenfree of rank r .(IV) ( Case (b) in general ) By (III) we can assume that U is free of rank r . Using the notation of Lemma 1.4, we know that q κ ( m ) is nonsingularby 1.2(d). Thus, by (II), the lemma holds for κ ( m ). We can thereforechoose an r –dimensional subspace W [ m ] such that U ( m ) ∩ W [ m ] = { } and U ( m ) ∼ −→ W [ m ] ∗ via b κ ( m ) . By Lemma 1.4, the W [ m ] lift to a submodule W satisfying U ∩ W = { } and W ( m ) = W [ m ]. Moreover, the map U −→ W ∗ ,induced by b q , is an isomorphism by Nakayama, since it is an isomorphismafter passing to each κ ( m ). Again (I) finishes the proof. (cid:3) Corollary ( Characterization of hyperbolic spaces).
Let ( M, q ) bea quadratic R –space and assume that q is regular or that R is semilocal.Then the following are equivalent: (i) ( M, q ) is hyperbolic; (ii) M admits a direct summand U satisfying q ( U ) = 0 and p U =rank p M for all p ∈ Spec( R ) ; (iii) M admits a direct summand U satisfying q ( U ) = 0 and U = U ⊥ .In this case ( M, q ) ∼ = H ( U ) .Proof. (i) = ⇒ (ii) being obvious because rank p U = rank p U ∗ , let us assume(ii) and prove (iii). By Proposition 1.5, there exists a submodule V ⊂ M such that ( U ⊕ V, q | U ⊕ V ) ∼ = H ( U ) is hyperbolic and M = ( U ⊕ V ) ⊕ ( U ⊕ V ) ⊥ .Since V ∼ = U ∗ as R –modules, rank p ( U ⊕ V ) = rank p M , whence U ⊕ V = M and U ⊥ = U ⊕ ( U ⊥ ∩ V ) with U ⊥ ∩ V ∼ = { ϕ ∈ U ∗ : ϕ ( U ) = 0 } = { } . Theproof of (iii) = ⇒ (i) follows the same pattern. (cid:3) Corollary.
Let ( M, q ) and ( M ′ , q ′ ) be regular quadratic modules. (a) [Ba, I, (4.7.i)] If ( M, q ) and ( M ′ , q ′ ) are isometric, then the quadraticmodule ( M, q ) ⊥ ( M ′ , − q ′ ) is hyperbolic: ( M, q ) ⊥ ( M ′ , − q ′ ) ∼ = H ( M ) . (b) Conversely, if R is semilocal and if ( M, q ) ⊥ ( M ′ , − q ′ ) ∼ = H ( M ) , then ( M, q ) ∼ = ( M ′ , q ′ ) .Proof. (a) Let f : ( M, q ) → ( M ′ , q ′ ) be an isometry. The quadratic form q ⊥ ( − q ′ ) is regular. The diagonal submodule U = { (cid:0) m, f ( m ) (cid:1) : m ∈ M } ⊂ M ⊕ M ′ is complemented by { ( m,
0) : m ∈ M } and satisfies 1.6(ii). Hence( M, q ) ⊥ ( M ′ , − q ′ ) ∼ = H ( M ). DD DEGREE EXTENSION THEOREM 9 (b) By (a), (
M, q ) ⊥ ( M, − q ) ∼ = H ( M ) ∼ = ( M, q ) ⊥ ( M ′ , − q ′ ). Hence( M, − q ) ∼ = ( M ′ − q ′ ) by Witt cancellation 1.2(j), which implies our claim. (cid:3) Corollary 1.7(a) is not true for nonsingular quadratic forms, even overfields. For example, let (
M, q ) = ( F, h u i q ) = ( M ′ , q ′ ) with F a field ofcharacteristic 2 and u ∈ F × . Then ( M, q ) is nonsingular by 1.2(h), but0 = (1 F , F ) ∈ rad( q ⊥ ( − q )), so that q ⊥ ( − q ) is singular, hence inparticular not hyperbolic.1.8. Unimodular and isotropic vectors.
Let M be a finite projective R –module. For x ∈ M and p ∈ Spec( R ) we put x ( p ) = x ⊗ R κ ( p ) . Recallthat u ∈ M is called unimodular if u satisfies one of the following equivalentconditions, see e.g. [Lo, 0.3]:(i) Ru is complemented and free of rank 1,(ii) there exists ϕ ∈ M ∗ satisfying ϕ ( u ) = 1,(iii) u ( p ) = 0 for all p ∈ Spec( R ),(iv) u ( m ) = 0 for all maximal m ∈ Spec( R ).Let ( M, q ) be a quadratic module. We call m ∈ M isotropic if m isunimodular and q ( m ) = 0. We say ( M, q ) is isotropic if M contains anisotropic vector. We note some useful facts.(a) If m is a unimodular (isotropic) vector of ( M, q ), then m ⊗ S is aunimodular (isotropic respectively) vector of ( M, q ) S for any S ∈ R - alg .(b) Let R [ X ] the polynomial ring over R in the variable X and let ( M, q )be a quadratic module. For v = v ( X ) ∈ M ⊗ R R [ X ] define the affine R –scheme Z v = { x ∈ G a,R : v ( x ) = 0 } , whose T –points, T ∈ R - alg , is the set Z v ( T ) = { t ∈ T : v ( t ) = 0 } where v ( t ) ∈ M ⊗ R T is obtained by substituting t for X . Then(1.8.1) Z v empty = ⇒ v unimodular.Indeed, if p ∈ Spec( R [ X ]), then v ( p ) = v ( X ⊗ κ ( p ) ) = 0 since otherwise X ⊗ κ ( p ) ∈ Z v ( κ ( p )).(c) Let ( M, q ) be a quadratic space over R and assume that q is regularor that R is semilocal. By Proposition 1.5, any isotropic vector embeds intoa hyperbolic plane H = H ( R ) and ( M, q ) = H ⊥ ( M , q ). In particular,rank p M ≥ p ∈ Spec( R ).(d) Let M be a projective R –module of constant rank 2 and let q : M → R be a nonsingular quadratic form. By 1.2(e), q is nonsingular if and only if q is regular. Hence, by (c), we have the implication “ = ⇒ ” of(1.8.2) ( M, q ) isotropic ⇐⇒ ( M, q ) ∼ = H ( R ) . Thus, in this case M is free of rank 2. The other direction in (1.8.2)holds because in H ( R ) = R ⊕ R with the hyperbolic form hyp, given byhyp( r , r ) = r r , the vector (1 ,
0) is isotropic. Springer’s odd extension theorem
Let (
M, q ) be a quadratic module. To simplify notation, we will oftenabbreviate q ( x ) = q S ( x ) for x ∈ M S if S is clear from the context. We willalso say that q is S –isotropic if q S is isotropic, cf. 1.8. We recall that aquadratic space ( M, q ) is a quadratic module with a nonsingular q .2.1. Theorem (Springer’s Theorem) . Let R be a semilocal ring and let ( M, q ) be a quadratic space. Let S be a finite R –algebra of odd degree, whichis ´etale or one-generated. If q is S –isotropic, then q is R –isotropic. The proof of this theorem will be given in 2.5. Lemma 2.2 proves Springer’sTheorem in the case of rank M = 2. It involves a much weaker conditionthan R being semilocal or that S is one-generated or ´etale. By 1.2(e), aquadratic form on such an R –module M is nonsingular if and only if it isregular.2.2. Lemma (Rank 2) . Let S ∈ R - alg be a finite R –algebra of odd degreeand let ( M, q ) be a quadratic space of constant rank , representing a unitin R . Then ( M, q ) is isotropic if and only if ( M, q ) S is isotropic.Proof. It is clear that (
M, q ) isotropic = ⇒ ( M, q ) S isotropic. Let us there-fore assume that ( M, q ) S is isotropic. Since q is isotropic if and only if uq is isotropic for some u ∈ R × , we may assume that q represents 1. It thenfollows from [Knu, V, (2.2.1)] that there exists a quadratic ´etale R –algebra A with norm n A such that ( M, q ) ∼ = ( A, n A ) (in fact A is the even Cliffordalgebra of ( M, q )). Without loss of generality, let (
M, q ) = (
A, n A ). Byassumption, ( A, n A ) S is isotropic, equivalently, ( n A ) S = n A S is hyperbolic.Now recall [Knu, V, (2.2.4)]: a quadratic ´etale S –algebra has a hyperbolicnorm if and only if A S it is split, i.e., A S ∼ = S × S as S –algebra. It followsthat A S is split. By [Knu, III, (4.1.2)], the automorphism group scheme ofthe R –algebra A is the abelian constant group scheme Z / Z . Since S hasodd degree, we are in the setting of the Example in B.4. Thus ( A, n A ) issplit, i.e., n A is isotropic. (cid:3) We recall (1.8.2): under the assumption of Lemma 2.2, (
M, q ) is isotropic ⇐⇒ ( M, q ) ∼ = H ( R ). The assumption that ( M, q ) represent a unit is alwaysfulfilled if R is semilocal, see for example [Ba, Prop. I.3.4]. Our proof ofSpringer’s Theorem also uses the following Lemma 2.3, a variation of [PR,Prop. 1.1].2.3. Lemma.
Let k be a field, let ( V, q ) be an isotropic quadratic k –space ofdimension r ≥ , and let P = P ( X ) ∈ k [ X ] be a monic polynomial of degree d ≥ . Then there exists v = v ( X ) ∈ V ⊗ k k [ X ] satisfying the followingconditions: (i) q ( v ( X )) ∈ k [ X ] is a polynomial of degree d − , which is divisibleby P ; (ii) the k –scheme Z v ( X ) = { x ∈ G a : v ( x ) = 0 } is empty. DD DEGREE EXTENSION THEOREM 11
In particular, v ( X ) is unimodular.Proof. Since q is isotropic, ( V, q ) contains a hyperbolic plane H and ( V, q ) = H ⊥ ( W, q | W ) for ( W, q | W ) = H ⊥ , see 1.8(c) or [EKM, 7.13]. The quadraticmodule ( W, q | W ) contains w ∈ W with q ( w ) =: a ∈ k × . In view of ourclaims, it is then no harm to replace V by H ⊕ ka . Thus q is given by q ( x, y, z ) = xy + az , which is nonsingular by 1.2(g) and 1.2(h) (but notregular in characteristic 2).By the Euclidean division algorithm there exist unique polynomials Q ( X ) ∈ k [ X ] of degree d − R ( X ) of degree ≤ d − X d − = P ( X ) Q ( X ) + R ( X ). We define v ( X ) = ( − a, R ( X ) , X d − ) ∈ V ⊗ k [ X ].Then q ( v ( X )) = − aR ( X ) + aX d − = aP ( X ) Q ( X ). Thus, the condition (i)is fulfilled with q ( v ( X )) = aP ( X ) Q ( X ). Since the first component of v ( X )is − a , the condition (ii) is satisfied too. It implies unimodularity of v ( X )by (1.8.1). (cid:3) Consequences of Lemma 2.3.
As motivation for step (III) in theproof of Theorem 2.1 below, we discuss some consequences of Lemma 2.3.Step (III) will be more technical, but uses the same ideas. Let us put S = k [ X ] / ( P ) , θ = X + ( P ) ∈ S. Then S is one-generated with θ as primitive element. The element v ( θ ) ∈ ( V ⊗ k k [ X ]) ⊗ k [ X ] S = V ⊗ k S is unimodular since it is obtained from theunimodular v ( X ) by base change, see 1.8(a), and it satisfies q (cid:0) v ( θ ) (cid:1) = 0 bycondition (i) of 2.3, i.e., v ( θ ) is an isotropic vector in V ⊗ k S .Again by condition (i) of 2.3, the exists u ∈ k × and Q ( X ) ∈ k [ X ] ofdegree d − q (cid:0) v ( X ) (cid:1) = uP ( X ) Q ( X ) ∈ k [ X ]. Let T = k [ X ] / ( Q ) , ϑ = X + ( Q ) ∈ T. Then T is one-generated of degree d − ϑ . The samearguments showing that v ( θ ) is an isotropic vector proves that v ( ϑ ) ∈ V ⊗ k T is isotropic.In step (III) of the proof of Theorem 2.1 we will know that V ⊗ k S isisotropic and use a refinement of the argument above to conclude that V ⊗ k T is isotropic, the point being that deg T = deg S − Proof of Theorem 2.1. (I)
Reduction to M free of rank r ≥ . Let R = R × · · · × R n and ( M, q ) = ( M , q ) ⊥ · · · ⊥ ( M n , q n ) be the rankdecomposition of ( M, q ) as in 1.1(i). Thus, M i is a projective R i –moduleof rank i and each q i : M i → R i is a nonsingular quadratic form. The R –algebra S decomposes correspondingly, S = S × · · · × S n where each S i isa finite R i –algebra of degree d = deg S . We have M ⊗ R S ∼ = ( M ⊗ R S ) × · · · × ( M n ⊗ R n S n )with each M i ⊗ R i S i being projective of rank i as S i –module. Since q M ⊗ R S is isotropic, so is every q M i ⊗ Ri S i . By 1.8(c), M i ⊗ R i S i = 0 for i = 0 ,
1. Since in both cases ( S one-generated or S ´etale) the R i –modules S i are faithfullyflat, we get M = 0 = M and R = 0 = R .In the decomposition R = R × · · · × R n , each R i is a semilocal ring.Wehave already observed that ( M, q ) S is isotropic if and only if every M i ⊗ R i S i is isotropic. Since the analogous fact holds for ( M, q ), it suffices to provethat every ( M i , q i ) is isotropic. Thus, without loss of generality, we canassume that M has constant rank r ≥
2. The case r = 2 has been dealtwith in Lemma 2.2. We can therefore assume that M has rank r ≥
3. Since R is semilocal, this implies that M is free of rank r .(II) R = k is a field and S is one–generated. In this case S ∼ = k [ X ] /P forsome monic P ∈ k [ X ]. Let P = P e · · · P e n n be the prime factor decomposi-tion of P in k [ X ], and put L i = k [ X ] /P i . Since d = dim k S = P i e i [ L i : k ] isodd, one of the [ L i : k ] is odd. Then L = L i ∈ S - alg and thus q L = ( q S ) ⊗ S L is isotropic. Since L/k has odd degree, the classical Springer Odd DegreeExtension Theorem [EKM, Cor. 18.5] says that q is isotropic.(III) R semilocal and S is one-generated. As before, let S = R [ X ] / ( P )where P ∈ R [ X ] is a monic polynomial opf degree d . We denote by κ , . . . , κ c the residue fields of R . For each i , 1 ≤ i ≤ c , the κ i –algebra S κ i = S ⊗ R κ i is one-generated, namely S κ i = κ i [ X ] /P κ i for P κ i = P ⊗ R κ i , and of odddegree d . Since q is S –isotropic, it is also S ⊗ R κ i -isotropic. The case (II) thenshows that q κ i is isotropic. Now Lemma 2.3 provides unimodular elements v i ( X ) ∈ ( M ⊗ R κ i ) ⊗ κ i κ i [ X ] = M ⊗ R κ i [ X ] such that q ( v i ( X )) is the productof a unit in κ × i and a monic polynomial of degree 2 d − P κ i and has the property that the κ i –scheme Z v i = { x ∈ G a,κ i : v i ( x ) = 0 } is empty.Let θ = X + ( P ) ∈ S and denote by θ i its image in S κ i . Then v i ( θ i )is obtained from the unimodular v i ( X ) by base change and is thereforeunimodular. It also satisfies q ( v i ( θ i )) = 0 since q ( v i ( X )) is divisible by P κ i .In other words, v i ( θ i ) is an S κ i –isotropic vector.According to Corollary A.6 of the appendix, the v i ( θ i ) ′ s ∈ M ⊗ R S κ i lift toan isotropic v ∈ M ⊗ R S . We decompose v = m + m θ + · · · + m d − θ d − wherethe m j ’s belong to M , and define v ( X ) = m + m X + · · · + m d − X d − ∈ M ⊗ R R [ X ]. By construction q ( v ( X )) ∈ P ( X ) R [ X ] is a polynomial of degree ≤ d −
2. Since the specialization to each κ i [ X ] is of degree 2 d −
2, it followsthat q ( v ( X )) is the product of a unit u ∈ R × and a monic polynomial ofdegree 2 d −
2. Summarizing, we have that q ( v ( X )) = u P ( X ) Q ( X ) with Q ( X ) monic of degree d − u ∈ R × .Since Z v i = { t ∈ G a,κ i : v i ( t ) = 0 } is empty for each i , it follows that Z v = { x ∈ G a,R : v ( x ) = 0 } is empty too. We define T = R [ X ] /Q ( X ). Then w = v ( X ) modulo Q is an isotropic vector of M T . Thus, q is T -isotropicwith T one-generated of degree d −
2. We continue the induction until wereach d − M is isotropic. DD DEGREE EXTENSION THEOREM 13 (IV) S is ´etale. According to [BFP, Prop. 7.3] there exists a finite ´etale R –algebra T of odd degree such that S ⊗ R T is one-generated as R –algebra.The paper [BFP] assumes throughout that 2 ∈ R × , but the proof of thequoted proposition works for arbitrary R .Since q is S -isotropic, it is a fortiori S ⊗ R T -isotropic. Since S ⊗ R T hasodd degree, the preceding case (III) shows that q is isotropic. (cid:3) As in the case of fields, see e.g. [Sch, II], Theorem 2.1 has a number ofconsequences worth stating. First, by 1.8(c), Springer’s Theorem says: if(
M, q ) S contains a hyperbolic plane H , then so does ( M, q ). Corollary 2.6says that this is true for arbitrary hyperbolic spaces. We say a quadraticmodule ( M, q ) contains a quadratic module ( M , q ) if there exists a comple-mented submodule N ⊂ M such that ( M , q ) ∼ = ( N, q | N ). In this case, weusually identify ( M , q ) = ( N, q | N ). We recall that if ( M , q ) is regular,e.g., a hyperbolic space, then ( M, q ) = ( M , q ) ⊥ ( M , q ) ⊥ by 1.1(e).2.6. Corollary.
Let R , S and ( M, q ) be as in Theorem . If ( M, q ) S contains a hyperbolic space H ( N ′ ) with N ′ projective of constant rank r ,then ( M, q ) contains a hyperbolic space H ( N ) with N projective of rank r .Proof. Recall ([Knu, V, (1.1.1)]) that S is semilocal and that a projectivemodule of constant rank over a semilocal ring is free. We can of courseassume that r > M has constant rank, cf. 1.2(i). It followsthat ( M, q ) S is isotropic. Hence, by Springer’s Theorem 2.1, the quadraticmodule ( M, q ) contains an isotropic vector e . By 1.8(c) we then have adecomposition ( M, q ) = H ( Re ) ⊥ ( M , q ) where ( M , q ) is a quadraticspace by 1.2(g). If ( M , q ) = 0, we are done. Otherwise, M has positiverank. Also, H ( Se ) ⊥ ( M , q ) S = ( M, q ) S ∼ = H ( N ′ ) ⊥ H ( N ′ ) ⊥ . Since N ′ is free, it decomposes as N ′ ∼ = Re ′ ⊕ N ′ where e ′ is isotropic and N ′ is atotally isotropic submodule N ′ of rank r −
1. Therefore H ( Se ) ⊥ ( M , q ) S = ( M, q ) S ∼ = H ( Se ′ ) ⊥ ( H ( N ′ ) ⊥ H ( N ′ ) ⊥ ) . Since H ( Se ) is regular and ( M , q ) S is nonsingular and of positive rank,we can cancel H ( Se ) ∼ = H ( Se ′ ) using 1.2(j). We find that the nonsingularquadratic module ( M , q ) satisfies the assumption of the corollary with r replaced by r −
1. We then continue by induction on r . (cid:3) Corollary.
Let R , S and ( M, q ) be as in Theorem , let ( M , q ) bea regular quadratic R –module such that ( M , q ) S is contained in ( M, q ) S .Then ( M , q ) is contained in ( M, q ) .Proof. We apply the rank decomposition 1.2(i) to ( M , q ) and can thenwithout loss of generality assume that ( M , q ) has constant rank. Since( M , q ) S is regular, the nonsingular quadratic S –space ( M, q ) S decomposes,( M, q ) S ∼ = ( M , q ) S ⊥ ( M ′ , q ′ )where ( M ′ , q ′ ) is a nonsingular S –space by 1.2(g). By Corollary 1.7, thequadratic S –module q S ⊥ ( − q ) S contains a hyperbolic S –space isometric to H ( M ,S ). Hence by Corollary 2.6, there exists a nonsingular quadratic R –module ( M , q ) such that q ⊥ ( − q ) ∼ = H ( M ) ⊥ q ∼ = q ⊥ ( − q ) ⊥ q . Canceling − q by Witt cancellation 1.2(j), yields the result. (cid:3) For easier reference, we explicitly state the case ( M , q ) S ∼ = ( M , q ) S inthe following Corollary 2.8. The case (iii) is obtained by writing S/R as afinite tower of odd one-generated (= simple) field extensions.2.8.
Corollary.
Let R be a semilocal ring, let q and q ′ be regular R –quadraticforms, and let S ∈ R - alg be a finite R –algebra of odd degree which satisfiesone of the following conditions, (i) S is a one-generated R –algebra, or (ii) S is an ´etale R –algebra, or (iii) R is a field and S/R is a field extension.Then q S ∼ = q ′ S ⇐⇒ q ∼ = q ′ . In Corollary 2.9, c W q ( R ) and W q ( R ) denote the Witt-Grothendieck ringand Witt ring of regular quadratic R –modules respectively, as for exampledefined in [Ba, I, § § / ∈ R × . The proof of 2.9 is standard, usingCorollary 2.7.2.9. Corollary.
Let R be a semilocal ring and let S ∈ R - alg be a finite R –algebra of odd degree, which is one-generated or ´etale. Then the maps (2.9.1) c W q ( R ) −→ c W q ( S ) and W q ( R ) −→ W q ( S ) , induced by [ q ] [ q S ] , are monomorphisms. Corollary 2.9 was established in [Ba, V, Thm.6.9] for Frobenius exten-sions, based on a detailed study of torsion in the kernels of the maps in(2.9.1) and in this way avoiding Springer’s Theorem, which is not proven in[Ba].2.10.
Set of values.
By definition, the set of values D ( q ) of a quadraticmodule ( M, q ) is D( q ) = q ( M ) ∩ R × . We will use the following elementary facts.(a) The set of values is stable under base change: if S ∈ R - alg , thenD( q ) ⊗ S ⊂ D( q S ).(b) If q contains a hyperbolic plane, then D( q ) = R × . Recall from 1.8(c)that q contains a hyperbolic plane whenever ( M, q ) is an isotropic quadraticspace and R is semilocal or q is regular.(c) ( Direct products ) Let R = R × R be a direct product. By 1.2(i),the quadratic module ( M, q ) uniquely decomposes as (
M, q ) = ( M , q ) × DD DEGREE EXTENSION THEOREM 15 ( M , q ) where ( M i , q i ), i = 1 ,
2, is a quadratic R i –module, which is nonsin-gular if ( M, q ) is so. We have D( q ) = D( q ) × D( q ). An S ∈ R - alg whichis projective of rank d ∈ N + uniquely decomposes as S = S × S whereeach S i is a projective R i –module of rank d . In view of 1.2(i), this showsthat the determination of D( q ) can often be reduced to that of D( q ) where( M, q ) has constant rank.(d) Let (
M, q ) = ( R, h u i ) with u ∈ R × . Then D( q ) = uR × . For any S ∈ R - alg which is projective of odd rank d we have(2.10.1) a ⊗ S ∈ D( q S ) ⇐⇒ a ∈ D( q ) . By (a) we only need to prove “ = ⇒ ”. We have D( q S ) = ( u ⊗ S ) S × , and if a ⊗ S = ( u ⊗ S ) s for some s ∈ S , then a d = N S/R ( a ⊗ S ) = u d N S/R ( s ) .Since a d ∈ aR × and u d ∈ uR × , we get a ∈ uR × = D( q ).In the following Corollary 2.11 we will prove (2.10.1) for more generalnonsingular forms.2.11. Corollary.
Let R be a semilocal ring, let a ∈ R × and let ( M, q ) be aquadratic space for which q ′ = q ⊥ h− a i is nonsingular, cf. . Further-more, let S ∈ R - alg be a finite R –algebra of odd degree which is ´etale orone-generated. Then (2.11.1) a ⊗ S ∈ D( q S ) ⇐⇒ a ∈ D( q ) . This is a well-known result in case R is a field of characteristic = 2 and S/R is an extension field, see for example [Lam, VII, Cor. 2.9]. In this case,no assumption on q ′ is necessary. Proof.
We will of course only prove “ = ⇒ ”. By 2.10(c) and 2.10(d) we canassume that M has constant rank ≥
2. The assumption implies that q ′ S contains an isotropic vector ( x,
1) for some x ∈ M S . Hence, by Theorem 2.1for q ′ , we get that q ′ is R –isotropic, i.e., there exists m ∈ M and r ∈ R suchthat ( m, r ) ∈ M ⊕ R is unimodular and satisfies q ( m ) = ar . At this point,two cases are clear:(i) r = 0: Then m ∈ M is an isotropic vector of q . Hence, by 1.8(c),( M, q ) contains a hyperbolic plane and then we are done by 2.10(b).(ii) r ∈ R × : Then a = q ( r − m ) and we are again done.In particular, (2.11.1) holds in case R is a field. For general R we will choosean isotropic vector of q ′ more carefully.Let m ⊳ R be a maximal ideal of R with residue field κ = R/ m , and put S ⊗ R κ = S/ m S . We thus have extensions S / / S ⊗ R κR O O / / κ O O . Since nonsingularity is inherited by extensions and since ( S ⊗ R κ ) /κ has odddegree and is ´etale or one-generated, the field case applies and yields the existence of m κ ∈ M ⊗ R κ satisfying q κ ( m κ ) = a ⊗ κ . In this way we obtaina family ( v m ) m ∈ Specmax( R ) of isotropic vectors v m = ( m κ , κ ) ∈ ( M ⊕ R ) κ .By Corollary A.6, we then get an isotropic element v = ( x, r ) ∈ M ⊕ R thatlifts the v m ’s. In particular, r κ = 1 κ for every residue field κ of R . Thisimplies r ∈ R × . So we are done by (ii) above. (cid:3) Appendix A. Lifting isotropic elements
The goal of this appendix is Corollary A.6, which gives a criterion forlifting isotropic elements from localisations. We obtain this as a consequenceof a surjectivity result for localizations of quadrics (Proposition A.5), whichin turn is a special case of Demazure’s fundamental Conjugacy Theorem,proven in [SGA3, XXVI] and re-stated below.If C ∈ R - alg and m ⊳ R is a maximal ideal of R we abbreviate C/ m := C/ m C . We use the terms reductive (semisimple) group scheme and parabolicsubgroup scheme as defined in [SGA3, Tome III], but refer to a group schemeover Spec( R ) as an R –group scheme. We furthermore use the setting andthe results of [SGA3, XXVI.3] for schemes of parabolic subgroup schemes.A.1. Theorem ( Demazure’s Conjugacy Theorem).
Let R be a semilocalring and let G be a reductive R -group scheme. Denote by Dyn( G ) its Dynkin R -scheme (which is finite ´etale) and by Of(Dyn( G )) the R -scheme of clopensubsets of Dyn( G ) . Let t ∈ Of(Dyn( G ))( R ) be a type of parabolic subgroupsand denote by X = Par( G ) t the R -scheme of parabolic subgroups of type t .Then the following hold. (a) G ( R ) acts transitively on X ( R ) . (b) If S is a finite R -algebra such that X ( S ) = ∅ , the map X ( S ) −→ Y m ∈ Specmax( S ) X ( S/ m ) is onto.Proof. (a) If X ( R ) = ∅ , the statement is obvious. We can thus assume that X ( R ) = ∅ and pick a point x ∈ X ( R ); this corresponds to an R -parabolicsubgroup P of G of type t . According to [SGA3, XXVI.4.3.5], G admits aparabolic subgroup P ′ opposite to P . Corollary 5.2. of loc. cit. then says inparticular that X ( R ) = G ( R ) / P ( R ). Thus G ( R ) acts transitively on X ( R ).(b) Recall that S is semilocal, for example by [Knu, VI, (1.1.1)]. Ourassumption is that G S admits an S -parabolic subgroup scheme Q of type t .As noted in (a), it admits an opposite parabolic S -subgroup Q ′ . Accordingto [SGA3, XXVI.5.2], the product rad u ( Q )( S ) × rad u ( Q ′ )( S ) → X ( S ) issurjective (here rad u ( . ) denotes the unipotent radical). Applying this forthe semilocal ring S as well as for the semilocal ring S/ m , m ∈ Specmax( R ), DD DEGREE EXTENSION THEOREM 17 shows that the horizontal maps in the commutative diagram below are sur-jective: rad u ( Q )( S ) × rad u ( Q ′ )( S ) / / (cid:15) (cid:15) X ( S ) (cid:15) (cid:15) Q m rad u ( Q )( S/ m ) × rad u ( Q ′ )( S/ m ) / / Q m X ( S/ m )Since Q m S/ m ∼ = Q m S ⊗ R ( R/ m ) ∼ = S ⊗ R ( R/ Jac( R ) ∼ = S/ Jac( R ) S , whereJac( R ) denotes the Jacobson radical of R , the map S → Q m S/ m is onto. Onthe other hand, the S –scheme rad u ( Q ) (respectively rad u ( Q ′ )) is isomorphicto a vector S –group scheme [SGA3, XXVI.2.5], so that the left vertical mapis onto. Hence, by a simple diagram chase the right vertical map is ontotoo. (cid:3) A.2.
Corollary.
We use the notation of
A.1 , except that R need not besemilocal, but can be arbitrary. In particular, G is a reductive R –groupscheme and x , y are parabolic subgroups in X ( R ) .Then there exist f , . . . , f n ∈ R satisfying f + · · · + f n = 1 and y R fi ∈ G ( R f i ) . x R fi for i = 1 , .., n . In other words, x and y are locally G -conjugatedfor the Zariski topology on R .Proof. The group G is an R –group of type (RR) by [SGA3, XXII.5.1.3] andthe points x , y of X ( R ) are parabolic subgroups, hence subgroups of type(R) by 5.2.3 of loc. cit. . It then follows from Theorem 5.3.9 of loc. cit. thatthe strict transporter T , defined by T ( S ) = { g ∈ G ( S ) : g · x S = y S } ( S ∈ R - alg ) , is a finitely presented affine R –scheme (among other properties). Since T ( R m ) = ∅ for any maximal m ∈ Spec( R ) by A.1(a), the claim followsfrom Lemma A.3 below. (cid:3) A.3.
Lemma.
Let R be arbitrary and let T be an R –scheme which is locallyof finite presentation. If T ( R m ) = ∅ for all maximal m ∈ Spec( R ) , thenthere exists a Zariski cover ( f , . . . , f n ) of R for which T ( R f i ) = ∅ for all i , ≤ i ≤ n .Proof. Fix a maximal m ∈ Spec( R ). Then R m = lim −→ f m R f , hence Spec( R ) =lim ←− f m Spec( R f ) and so T ( R m ) = lim −→ f m T ( R f ), according to [St, Tag01ZC]. It follows that there exists f m ∈ R \ m such that T ( R f m ) = ∅ .The f m ’s for m running over the maximal ideals of R generate R as ideal.Hence, there exists finitely many maximal ideals m , . . . , m n such that R = Rf m + · · · + Rf m n . (cid:3) A.4. An important special case of Theorem A.1 and Corollary A.2 is thatof quadrics elaborated below. Let us explain some notation. We let (
M, q )be a quadratic space with M of constant rank n . The associated special orthogonal group scheme G = SO ( q ) is defined in [Co1, C.2.10]. It is asemisimple R –group scheme of type A for n = 3, of type B ( n − / for odd n ≥ n/ for even n ≥
4. We use Bourbaki’s enumeration ofthe corresponding Dynkin diagrams: r α . . . . . . > r r r r r r r r α α α n − . . . . . . . . . ❍❍✟✟ r r r r r r r r r r rr α α α n − α n We also use the projective space P ( M ∨ ) with Grothendieck’s convention,i.e., P ( M ∨ )( S ), S ∈ R - alg , corresponds to the direct summands D of M S which are locally free of rank 1. The quadric Q defined by q = 0 consistsof those D in P ( M ∨ )( S ) with q ( D ) = 0. For D ∈ Q ( R ) we let P be the R -subgroup scheme of G which stabilizes D . Finally, we remind the readerof our abbreviation S/ m = S/ m S for S ∈ R - alg and m ∈ Specmax( R ).A.5. Proposition.
We use the notation of
A.4 . Then the following hold. (a) (i) P is a parabolic R –subgroup of G . (ii) The orbit map G → P ( M ∨ ) , g → g. [ D ] , induces an isomor-phism G / P ∼ −→ Q of R –schemes. (iii) Q is G -isomorphic to Par( G ) t where the type t is constantand of value Dyn( G )( R ) \ { α } R with the enumeration of therespective Dynkin diagrams displayed above. (b) Let R be semilocal. Then (i) G ( R ) acts transitively on Q ( R ) , and (ii) if S is a finite R -algebra such that Q ( S ) = ∅ , then (A.5.1) Q ( S ) −→ Y m ∈ Specmax( R ) Q ( S/ m ) is onto.Proof. We prove only the even rank case, the odd case being analogous.(a) We observe that the claims hold over a field [Co2, Th. 3.9.(i)] and alsothat they are local for the flat topology. According to [Co1, Lemma C.2.1]or [Knu, IV, (3.2.1)], we may assume that (
M, q ) is the standard hyperbolicquadratic R -form over R n ( n ≥ q ( x , . . . , x n , y , . . . , y n ) = x y + · · · + x n y n .We first consider the case D = Rp where p = (1 , , . . . , , , . . . , P p the corresponding subgroup scheme. This permits us to as-sume R = Z . Then P p is an affine finitely presented Z -group scheme whosealgebraically closed fibers are smooth connected and of dimension 2 n −
2. By
DD DEGREE EXTENSION THEOREM 19 [AG, Lemma B.1], P p is smooth. Since the geometric fibers of P p are para-bolic subgroups, P p is parabolic too. The induced orbit map f : G / P p → Q is a monomorphism. The field case ensures that this is a fiberwise isomor-phism. Since G / P p is flat and of finite presentation, the fiberwise iso-morphism criterion [EGA, IV , 17.9.5] enables us to conclude that f is anisomorphism. It follows that Q is homogeneous under G . Thus (i) and (ii)hold for the special p .Let us now deal with the general case. Since G -homogeneity is a localproperty with respect to the flat topology, Q is homogeneous under G . Thecase of a general D then follows from the observation that D is locally G -conjugated to Rp in the Zariski topology by applying Corollary A.2 to Q .This proves (i) and (ii) in general.(iii) We first deal with the special p above. We have an isomorphism G / P ∼ = Q . On the other hand, according to [SGA3, XXVI.3.6], we havea G -isomorphism G / P ∼ −→ Par( G ) t ( P ) , hence an G -isomorphism Q ∼ −→ Par( G ) t ( P ) . This isomorphism applies a point x ∈ Q ( R ) to the stabilizer G x , so is a canonical isomorphism. Here t ( P ) ∈ Of(Dyn( G ))( R ) is the typeof P . Checking that it is t , reduces to the field case which is [Co2, Lemma3.12].For the general case, let S be a flat cover of R such that Q ( S ) = ∅ .We have an isomorphism Q S ∼ −→ Par( G ) t ,S which is canonical and G S -equivariant. By faithfully flat descent, it descends to a G -equivariant iso-morphism Q ∼ −→ Par( G ) t .(b) follows from Theorem A.1 applied to Q ∼ = Par( G ) t . (cid:3) A.6.
Corollary.
We assume that R is a semilocal ring. Let ( M, q ) be aquadratic space of constant rank ≥ , let S be a finite R -algebra such that q S is isotropic, and let ( v m ) m ∈ Specmax( R ) be a family of isotropic elements v m ∈ M ⊗ R S/ m . Then there exists an isotropic v ∈ M ⊗ R S that lifts the v m ’s.Proof. Let Q ⊂ P ( M ∨ ) be the projective quadric associated with q in Propo-sition A.5. Our assumption is that Q ( S ) = ∅ .Each S/ m –module ( S/ m ) .v m is a direct summand of M ⊗ R S/ m which isfree of rank one, so defines a point x m ∈ Q ( S/ m ). Proposition A.5(b)(ii)provides an element x ∈ Q ( S ) which lifts the x m ’s. Since S is semilocal, x is represented by an S -module D which is a direct summand of rank oneof M ⊗ R S and satisfies q ( D ) = 0. We write D = Sv where v is an S –unimodular element of M ⊗ R S satisfying q ( v ) = 0. Since S × → Q m ( S/ m ) × is onto, we can modify v by an unit of S to ensure that v lifts the v m ’s. (cid:3) Appendix B. Trace for torsors
B.1.
Weil restriction ( [BLR, 7.6] , [CGP, A.5] , [DG, I, §
1, 6.6] ). Let S ∈ R - alg . Given an S –functor Y ′ , the Weil restriction of Y ′ is the R –functor R S/R ( Y ′ ) defined by R S/R ( Y ′ )( A ) = Y ′ ( A ⊗ R S ) , ( A ∈ R - alg ) . It is uniquely determined by the following universal property: for every R –functor X there exists a bijection(B.1.1) ξ = ξ X,Y ′ : Mor S - alg ( X S , Y ′ ) ∼ −→ Mor R - alg ( X, R S/R ( Y ′ ))where X S is the S –functor obtained from X by base change, thus satisfying X S ( B ) = X ( R B ) for B ∈ S - alg . Here and sometimes in the followingwe write R B to denote the R –algebra obtained from the S –algebra B byrestriction of scalars. The bijection ξ is functorial in X and Y ′ . It maps g ∈ Mor S - alg ( X S , Y ′ ) to the composition X ( A ) X (inc ) −−−−−→ X (cid:0) R ( A ⊗ R S ) (cid:1) g ( A ⊗ R S ) −−−−−−→ Y ′ ( A ⊗ R S )where A ∈ R - alg and inc is the R –algebra homomorphisminc = inc ,A : A −→ R ( A ⊗ R S ) , a a ⊗ S . We consider two special cases of (B.1.1). First, for Y ′ = X S and g = Id X S we get the morphism j = j X = ξ (Id X S ) : X −→ R S/R ( X S )of R –functors, determined by j X ( A ) = X (inc ,A ), A ∈ R - alg . Second,putting X = R S/R ( Y ′ ) in (B.1.1), there exists a unique morphism q = q Y ′ : R S/R ( Y ′ ) S −→ Y ′ of S –functors satisfying ξ ( q Y ′ ) = Id R S/R ( Y ′ ) . For B ∈ S - alg we have R S/R ( Y ′ ) S ( B ) = R S/R ( Y ′ )( R B ) = Y ′ ( R B ⊗ R S ) and so q Y ′ ( B ) : Y ′ ( R B ⊗ R S ) −→ Y ′ ( B ) . In fact, q Y ′ ( B ) = Y ′ ( m B ) where m B is the S –algebra homomorphism m B : ( R B ) ⊗ R S −→ B, b ⊗ s bs ([CGP, A.5.7]). For Y ′ = X S we now have constructed morphisms X S j X,S −−−→ R S/R ( X S ) S q XS −−−→ X S between S –functors. Untangling the constructions above, we find(B.1.2) q X S ◦ j X,S = Id X S because B inc ,B −−−−→ R B ⊗ R S m B −−→ B equals Id B . DD DEGREE EXTENSION THEOREM 21
B.2.
Cohomology and restriction.
Let G be a flat R –group sheaf. Wedenote by H ( R, G ) = H ( R, G )the pointed set of isomorphism classes of G –torsors over Spec( R ) in the flattopology. Let S ∈ R - alg . The base change X S = X × Spec( R ) Spec( S ) of a G –torsor X is a G S –torsor, giving rise to the restriction map res = res S/R, G : H ( R, G ) → H ( S, G S ) , [ X ] [ X S ] . A homomorphism f : G → H of flat R –group sheaves induces a map incohomology f ∗ : H ( R, G ) → H ( R, H ) , [ X ] [ X ∧ G H ]where X ∧ G H = ( X × Spec( R ) H ) /G is the contracted product with respectto the G –action on H via f . Contracted products are special cases of fppfquotients and as such allow base change [GM, (4.30)]. Hence ( X ∧ G H ) S and X S ∧ G S H S are isomorphic H S –torsors, giving rise to a commutativediagram(B.2.1) H ( R, G ) f ∗ / / res (cid:15) (cid:15) H ( R, H ) res (cid:15) (cid:15) H ( S, G S ) f S, ∗ / / H ( S, H S )of pointed sets. It is known that H = R S/R ( G S ) is again a flat R –groupsheaf if G is so and that the maps j : G → R S/R ( G S ) and q : R S/R ( G S ) S → G S of B.1 are homomorphisms of R –group sheaves.Hence, passing to cohomology, the maps in the diagram are well-defined:(B.2.2) H ( R, G ) j ∗ / / res G (cid:15) (cid:15) H ( R, R S/R ( G S )) res R ( GS ) (cid:15) (cid:15) i u u ❦ ❦ ❦ ❦ ❦ ❦ ❦ H ( S, G S ) H ( S, R S/R ( G S ) S ) q ∗ o o We claim that (B.2.2) is a commutative diagram. Indeed, since q ◦ j S = Id G S by (B.1.2), this follows from commutativity of (B.2.1):(B.2.3) res G = q ∗ ◦ ( j S ) ∗ ◦ res S/R,G = q ∗ ◦ res R S/R ( G S ) ◦ j ∗ = i ◦ j ∗ . The map i = q ∗ ◦ res R S/R ( G S ) is injective by [SGA3, XXIV.8.2].In particular, assume that G is an abelian affine R –group scheme . Then soare G S , R S/R ( G S ) and R S/R ( G S ) S . Moreover, the cohomology sets and mapsused in (B.2.2) are abelian groups and group homomorphisms respectively.Since i is injective, we get from (B.2.3) that(B.2.4) Ker(res G ) = Ker(i ◦ j ∗ ) = Ker( j ∗ ) . B.3.
Deligne trace homomorphism.
Let S ∈ R - alg be locally free offinite rank d ∈ N + , i.e., the R –module S is projective of constant rank d ,and let G be an abelian affine R –group scheme. We then have Deligne’strace homomorphism(B.3.1) tr : R S/R ( G ) → G, satisfying tr ◦ j = × d, where × d : G → G is the group homomorphism given on T –points, T ∈ R - alg , by g g d [SGA4 , XVII, 6.3.13–6.3.15]. It induces an endomorphism( × d ) ∗ : H ( R, G ) −→ H ( R, G )given by the analogous formula. Since ( × d ) ∗ = tr ∗ ◦ j ∗ by (B.3.1), we obtainfrom (B.2.4) that(B.3.2) Ker(res S/R,G ) = Ker( j ∗ ) ⊂ Ker (cid:0) ( × d ) ∗ (cid:1) . In particular, this implies the following.B.4.
Lemma.
In the setting of
B.3 assume that × d : G → G is an iso-morphism. Then the restriction homomorphism res S/R, G : H ( R, G ) → H ( S, G ) is injective. Example.
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Stacks project , http://stacks.math.columbia.edu/ UMR 5208 du CNRS - Institut Camille Jordan - Universit´e Claude BernardLyon 1, 43 boulevard du 11 novembre 1918, 69622 Villeurbanne cedex - France
Email address : [email protected] Department of Mathematics and Statistics, University of Ottawa, Ottawa,Ontario, Canada, K1N 6N5
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