Degenerations of Rank 3 Quadratic Bundles and Rank 4 Azumaya Bundles over Schemes

T. E. Venkata Balaji Mathematisches Institut, Georg-August-Universität Bunsenstrasse 3-5, D-37073 Göttingen, Germany

                      tevbal@uni-math.gwdg.de

Dedicated to Professor Martin Kneser

April 26, 2004

Abstract

This paper uses scheme-theoretic methods to address algebraic problems about rank 3 quadratic forms and rank 4 Azumaya algebras. It extends well-known results on semiregular rank 3 quadratic bundles and rank 4 Azumaya algebra bundles to their degenerations by studying the scheme of specialisations of Azumaya algebra bundle structures introduced in Part A of [3]. The Witt-invariant of a rank 3 quadratic bundle (V, q), which by definition is the isomorphism class (as algebra bundle) of its even-Clifford algebra C_{0}(V, q), is shown to determine the pair (V, q) upto tensoring by a discriminant line bundle. The special, usual and the general orthogonal groups of (V, q) are computed and canonically determined in terms of Aut(C_{0}(V, q)), and it is shown that the general orthogonal group is always a semidirect product. Any element of Aut(C_{0}(V, q)) can be lifted to a self-similarity, and in fact to an element of the orthogonal group provided the determinant of the automorphism is a square. The special orthogonal group and the group of determinant 1 automorphisms of C_{0}(V, q) are naturally isomorphic. If the base scheme X is integral and q is semiregular at some point of X, then every automorphism of C_{0}(V, q) has determinant 1 and is thus induced from a self-isometry; the orthogonal group is also seen to be a semidirect product in this case. If X is affine with coordinate ring a UFD, then every specialised algebra structure on a rank 4 vector bundle over it arises as the even-Clifford algebra of a global rank 3 quadratic bundle (V, q), so that the set of rank 3 quadratic bundles upto tensoring by a discriminant line bundle naturally corresponds to the set of isomorphism classes of specialised rank 4 algebra bundles. These results are seen as limiting versions of the natural bijection Ȟ1_{fppf}(X, O_{3})/Disc(X) ≅ Ȟ1_{étale}(X, PGL_{2}). The multiplication table of every specialised algebra structure on any fixed free rank 4 vector bundle with fixed unit that is part of a global basis is written down explicitly. For a connected proper scheme X of finite type over an algebraically closed field, the hypothesis of self-duality on a unital associative algebra bundle of square rank over X forces the algebra to be either globally Azumaya or to be nowhere-Azumaya. This implies that the existence of a non-Azumaya specialisation which is Azumaya at some point excludes the possibility of the existence of global Azumaya algebra structures on that bundle. The use of the nice technical notion of semiregularity introduced by Kneser in [1] allows all of the above results to be valid over an arbitrary base scheme X, some (or even all) of whose points may have residue fields of characteristic two.

Keywords: semiregular, quadratic bundle, Azumaya bundle, Witt-invariant, Clifford algebra, discriminant bundle, orthogonal group, similarity, similitude.

MSC Subject Classification Numbers: 11Exx, 11E12, 11E20, 11E88, 11R52, 14L15, 15A63, 16S, 16H05.

Degenerations of Rank 3 Quadratic Bundles and Rank 4 Azumaya Bundles over Schemes 2

Contents

1 Overview of the Main Results 2

2 Reduction of Theorems 1.1 & 1.3 to the Free Case 10

3 Investigation of Θ ; Proof of Theorem 1.4 15

4 Specialisations as Bilinear Forms: Theorems 1.9–1.12 22

5 Properties of Specialisations/Self-duality: 1.5–1.8 26

6 Stratification of the Variety of Specialisations 30

Acknowledgements 31

References 32

1 Overview of the Main Results

The smoothness of the schematic closure of Azumaya algebra structures on a fixed vector bundle of rank 4 over any scheme was obtained in Part A of [3]. Part B of that work had applied this result to obtain desingularisations (with good specialisation properties) of certain moduli spaces over fairly general base schemes. The present work is concerned with applications to the study of degenerations of rank 3 quadratic bundles over a scheme.

For a rank 3 vector bundle V over a scheme X with a quadratic form q on V (taking values in 𝒪_{X}), consider the natural functorial association of the pair (V, q) to its even Clifford algebra bundle C_{0}(V, q) of rank 4 over X. If q is a semiregular quadratic form, then C_{0}(V, q) is an Azumaya algebra bundle over X i.e., a twisted form for the the (2 × 2)-matrix algebra over 𝒪_{X} for the étale topology on X. (A generic quadratic form on a vector bundle of odd rank is semiregular and such forms are technically the “good” ones—they were introduced by Kneser in [1] and are studied in detail by Knus in his book [2]—coinciding with the usual regular quadratic forms when the residue field of every point of X has characteristic different from 2). Therefore the association is one that “generically” takes a rank 3 semiregular quadratic bundle to a rank 4 Azumaya algebra bundle. This may also be described as the natural bijection

Ȟ1_{fppf}(X, O_{3})/Disc(X) ≅ Ȟ1_{étale}(X, PGL_{2})

where the left side classifies semiregular rank 3 quadratic bundles upto tensoring by a discriminant line bundle on X, and the right side classifies Azumaya algebra bundles of rank 4 on X. If none of the residue fields of the points of X is of characteristic two, then the subscript ‘fppf’ on the left side may be replaced with ‘étale’. What follows is an attempt to extend this bijection ‘to the limit’ i.e., with degenerate (possibly non-semiregular) quadratic bundles on the left and specialised (possibly non-Azumaya) bundles on the right. We extend the left side by replacing it with the set of quadratic bundles of rank 3 on X upto tensoring by a discriminant line bundle and extend the right side by replacing it with algebraisomorphism classes of specialisations, of rank 4 Azumaya algebras, in the sense of Part A, [3]. We find that the resulting map is always an injection; we are able to prove that it is a surjection in certain cases, for example when X is affine with coordinate ring a UFD, and hope that it will always be a bijection.

A detailed analysis of the origins behind the smoothness result in Part A, [3], allows us to extend many of the known results for semiregular rank 3 quadratic bundles (and rank 4 Azumaya bundles), as in Chap.V, §3 of the book of Knus [2], to degenerate rank 3 quadratic bundles and specialisations of rank 4 Azumaya algebra bundles. For example, it turns out that the Witt-invariant of (V, q), which by definition is the isomorphism class (as algebra bundle) of C_{0}(V, q), determines the pair (V, q) upto tensoring by a discriminant line bundle. For another example, the orthogonal groups related to (V, q) viz. the special, usual and the general orthogonal groups (contained in that order inside Aut(V)×Γ(X, 𝒪_{X}^{∗}) = GL(V) × Γ(X, 𝒪_{X}^{∗})) may be computed and canonically determined in terms of Aut(C_{0}(V, q)).

Let us fix some notations and terminology before proceeding further. In the following we switch between the language of (geometric) vector bundles and the (equivalent) language of locally-free sheaves of finite constant positive rank; as a convention, sheaves corresponding to vector bundles V, W, L … are respectively denoted 𝒱, 𝒲, ℒ … Relative to any fixed base scheme B, consider the natural transformation between two natural (covariant) functors

Witt : 𝒬_{3} ⟶ 𝒜_{4}, 𝒬_{3}, 𝒜_{4} : (B − 𝔖𝔠𝔥𝔢𝔪𝔢𝔰)^{op} ⟶ 𝔖𝔢𝔱𝔰

defined as follows. If X ⟶ B is a B-scheme, recall that a pair (L, h) is said to be a discriminant line bundle on X if L is a line bundle and h : L ⊗_{X} L ≅ 𝔸_{X}^{1} is an isomorphism of line bundles (i.e., coming from an isomorphism of the associated locally-free sheaves of 𝒪_{X}-modules of rank 1). In other words h is a non-singular bilinear form on L with values in the trivial line bundle (hence necessarily symmetric). Let Disc(X) be the set of isometry classes of such discriminant bundles. It is naturally an abelian group of exponent 2 under the tensor product operation. It acts on the set of isometry classes of quadratic bundles (V, q) on X, since given (L, h) there is the natural quadratic bundle (V, q) ⊗ (L, h) viz. the quadratic form q ⊗ h on V ⊗ L. Define 𝒬_{3}(X) to be the set of equivalence classes under this action i.e., the set of rank 3 quadratic bundles on X (isometric) upto tensoring by a discriminant bundle. Also define 𝒜_{4}(X) to be the set of isomorphism classes of associative unital algebra structures on rank 4 vector bundles over X (here unital means that there exists a global nowhere-vanishing section which serves as the unit for the algebra multiplication). Associating to (V, q) its Witt-invariant i.e., the isomorphism class of its even Clifford algebra C_{0}(V, q) gives rise to a functor in X:

Witt(X) : 𝒬_{3}(X) ⟶ 𝒜_{4}(X) : [(V, q)] ↦ [C_{0}(V, q)]

since tensoring a quadratic bundle with a discriminant bundle (which includes replacing it with a similar— in particular with an isometric—quadratic bundle) does not affect the isomorphism class of its even Clifford algebra (see Prop. 2.5). Observe that by considering only semiregular quadratic bundles we obtain a subfunctor 𝒬_{3}^{sr} ↪ 𝒬_{3} (since tensoring by a discriminant bundle preserves semiregularity, see Prop.2.2). Also by considering only Azumaya algebra bundle structures we obtain a subfunctor 𝒜_{4}^{Azu} ↪ 𝒜_{4} so that the transformation Witt takes 𝒬_{3}^{sr} into 𝒜_{4}^{Azu}. In fact, Witt(X) : 𝒬_{3}^{sr}(X) ⟶ 𝒜_{4}^{Azu}(X) is bijective for each scheme X, and this may be deduced from the case of an affine X which is proved in §3, Chap.V,

[2] (see Theorem 1.14). It follows from the main result of Part A of [3] that there exists a subfunctor 𝒜_{4}^{Sp−Azu} ↪ 𝒜_{4} that contains 𝒜_{4}^{Azu} as well as the image of Witt. This subfunctor may be described as those algebra bundle structures that are scheme-theoretic specialisations of Azumaya algebra structures, or also as those that are locally (in the Zariski topology) the even Clifford algebras of quadratic bundles. It turns out that these specialisations are quaternion algebra bundles i.e., those that have a standard involution (in the sense of Knus, para.1.3, Chap.I, [2]) but of course even over an algebraically closed field there are quaternion algebras that are not even-Clifford algebras of quadratic forms.

THEOREM 1.1 For each X, the map Witt(X) : 𝒬_{3}(X) ⟶ 𝒜_{4}^{Sp−Azu}(X) is injective.

For two quadratic bundles (V, q) and (V′, q′) denote by Sim[(V, q), (V′, q′)] the set of similarities (also called similitudes) from (V, q) to (V′, q′) (with multipliers being global sections of 𝒪_{X}^{∗}) and by Iso[(V, q), (V′, q′)] the subset of isometries (i.e., similarities with trivial multipliers). When V = V′, the subset of isometries with trivial determinant is denoted S-Iso[(V, q), (V, q′)]. On taking q = q′ these sets naturally become subgroups of Aut(V) × Γ(X, 𝒪_{X}^{∗}) = GL(V) × Γ(X, 𝒪_{X}^{∗}) and we get

Sim[(V, q), (V, q)] = GO(V, q) ⊃ Iso[(V, q), (V, q)] = O(V, q) ⊃ S-Iso[(V, q), (V, q)] = SO(V, q).

Of course, O(V, q) and SO(V, q) may as usual be considered as subgroups of GL(V) ≡ GL(V) × {1}. Since in general a quadratic bundle (V, q) on a non-affine scheme X may not be induced from a global bilinear form, one is unable to identify the (ℤ/2.ℤ)– graded vector bundle underlying its Clifford algebra bundle with that underlying the exterior algebra bundle of V (which is the same as the Clifford algebra bundle of the zero quadratic form).

PROPOSITION 1.2 Every isomorphism of algebra-bundles ϕ : C_{0}(V, q) ≅ C_{0}(V′, q′) is naturally associated to an isomorphism of bundles ϕ_{Λ^{2}} : Λ^{2}(V) ≅ Λ^{2}(V′) which induces a map

ζ_{Λ^{2}} : Iso[C_{0}(V, q), C_{0}(V′, q′)] ⟶ Iso[Λ^{2}(V), Λ^{2}(V′)] : ϕ ↦ ϕ_{Λ^{2}}

where Iso[C_{0}(V, q), C_{0}(V′, q′)] is the set of algebra bundle isomorphisms. When V = V′, we may thus denote the subset of those ϕ for which det(ϕ_{Λ^{2}}) ∈ Aut[det(Λ^{2}(V))] ≅ Γ(X, 𝒪_{X}^{∗}) is a square (respectively = 1) by Iso′[C_{0}(V, q), C_{0}(V, q′)] (respectively by the smaller subset S-Iso[C_{0}(V, q), C_{0}(V, q′)]). Taking q = q′ in these sets and replacing “ Iso” by “Aut” in their notations respectively defines the groups Aut(C_{0}(V, q)) ⊃ Aut′(C_{0}(V, q)) ⊃ S-Aut(C_{0}(V, q)).

THEOREM 1.3 For rank 3 quadratic bundles (V, q) and (V, q′) with the same underlying bundle V on a scheme X, one has the following commuting diagram of natural maps of sets with the downward arrows being the canonical inclusions, the horizontal arrows being surjective and the top horizontal arrow being bijective:

S-Iso[(V, q), (V, q′)] ⟶^{≅} S-Iso[C_{0}(V, q), C_{0}(V, q′)]
inj↓  ↓inj
Iso[(V, q), (V, q′)] \underrightarrow{onto} Iso′[C_{0}(V, q), C_{0}(V, q′)]
inj↓  ↓inj
Sim[(V, q), (V, q′)] \underrightarrow{onto} Iso[C_{0}(V, q), C_{0}(V, q′)]

With respect to the surjections of the horizontal arrows in the diagram above, we further have the following (where l is the function that associates to any similitude its multiplier, det(g, l) := det(g) for a similitude with multiplier l and ζ_{Λ^{2}} is the map of Prop.1.2 above):

(a) there is a family of sections s_{2k+1} : Iso[C_{0}(V, q), C_{0}(V, q′)] ⟶ Sim[(V, q), (V, q′)] indexed by the integers such that l ∘ s_{2k+1} = det^{2k+1} ∘ ζ_{Λ^{2}} and such that (det^{2} ∘ s_{2k+1}) × (l^{−3} ∘ s_{2k+1}) = det ∘ ζ_{Λ^{2}};

(b) there is also a section s′ : Iso′[C_{0}(V, q), C_{0}(V, q′)] ⟶ Iso[(V, q), (V, q′)] such that det^{2} ∘s′ = det∘ζ_{Λ^{2}};

(c) there is a family of sections s_{2k+1}^{+} : Iso[C_{0}(V, q), C_{0}(V, q′)] ⟶ Sim[(V, q), (V, q′)] indexed by the integers which is multiplicative when followed by the natural inclusions into GL(V)×Γ(X, 𝒪_{X}^{∗}), i.e., if ϕ_{i} ∈ Iso[C_{0}(V, q_{i}), C_{0}(V, q_{i+1})] then s_{2k+1}^{+}(ϕ_{2} ∘ ϕ_{1}) = s_{2k+1}^{+}(ϕ_{2}) ∘ s_{2k+1}^{+}(ϕ_{1}) ∈ GL(V) × Γ(X, 𝒪_{X}^{∗}). Further, l ∘ s_{2k+1}^{+} = det^{2k+1} ∘ ζ_{Λ^{2}} and (det^{2} ∘ s_{2k+1}^{+}) × (l^{−3} ∘ s_{2k+1}^{+}) = det ∘ ζ_{Λ^{2}}.

(d) The maps s_{2k+1} and s′ above may not be multiplicative but are mutliplicative upto μ_{2}(Γ(X, 𝒪_{X})) i.e., these followed by the quotient map, on taking the quotient of GL(V) × Γ(X, 𝒪_{X}^{∗}) by μ_{2}(Γ(X, 𝒪_{X})) × {1}, become multiplicative.

THEOREM 1.4 For a rank 3 quadratic bundle (V, q) on a scheme X, one has the following natural commutative diagram of groups with exact rows, where the downward arrows are the canonical inclusions and where l is the function that associates to any orthogonal similitude its multiplier:

[figure: SO(V, q); inj; ≅; S-Aut(C_{0}(V, q)); inj; 1; μ_{2}(Γ(X, 𝒪_{X})); inj; O(V, q); inj; Aut_{′}(C_{0}(V, q)); inj; 1; 1; Γ(X, 𝒪_{X}^{∗}); GO(V, q); det^{2}×l^{−3}; Aut(C_{0}(V, q)); det; 1; Γ(X, 𝒪_{X}^{∗}); Γ(X, 𝒪_{X}^{∗})]

Further, we have

(a) There are splitting homomorphisms s_{2k+1}^{+} : Aut(C_{0}(V, q)) ⟶ GO(V, q) such that l∘s_{2k+1}^{+} = det^{2k+1} and (det^{2}∘s_{2k+1}^{+})×(l^{−3}∘s_{2k+1}^{+}) = det. The restriction of s_{2k+1}^{+} to Aut′(C_{0}(V, q)) does not necessarily take values in O(V, q), but the further restriction to S-Aut(C_{0}(V, q)) does take values in SO(V, q). In particular, GO(V, q) is a semidirect product. The maps s_{2k+1} and s′ of Theorem 1.3 above (under the current hypotheses) may not be homomorphisms but are homomorphisms upto μ_{2}(Γ(X, 𝒪_{X})).

(b) Suppose X is integral and q ⊗ κ(x) is semiregular at some point x of X with residue field κ(x). Then any automorphism of C_{0}(V, q) has determinant 1 so that Aut(C_{0}(V, q)) = Aut′(C_{0}(V, q)) = S-Aut(C_{0}(V, q)) and in particular, O(V, q) is the semidirect product of μ_{2}(Γ(X, 𝒪_{X})) and SO(V, q).

We next turn to the question whether the map Witt(X) : 𝒬_{3}(X) ⟶ 𝒜_{4}^{Sp−Azu}(X) is always surjective. This is the same as asking if, given a specialisation A of rank 4 Azumaya bundles on X, there exists a rank 3 quadratic bundle (V, q) such that C_{0}(V, q) ≅ A (as algebra bundles)—of course Theorem 1.1 will guarantee that (V, q) is unique upto tensoring by a discriminant bundle. It is this question of surjectivity that seems to involve the geometry of X.

THEOREM 1.5 Let X be a scheme and A a specialisation of rank 4 Azumaya algebra bundles on X. Let 𝒜 denote the locally-free 𝒪_{X}-algebra corresponding to A. Let 𝒪_{X}.1_{𝒜} ↪ 𝒜 denote the image of the canonical morphism 𝒪_{X} ⟶ 𝒜 defined by the nowhere-vanishing global section of 𝒜 corresponding to the unit for algebra multiplication.

(a) There exists a rank 3 vector bundle V on X (with corresponding 𝒪_{X}-module 𝒱) such that the following hold:

(1) det(𝒜) ⊗ Λ^{2}(𝒱) ≅ 𝒜/𝒪_{X}.1_{𝒜} , from which the following may be deduced:

(2) det(Λ^{2}(𝒱)) ≅ (det(𝒜)) ^{⊗−2};

(3) 𝒱 ≅ (𝒜/𝒪_{X}.1_{𝒜} ) ^{∨} ⊗ det(𝒱) ⊗ det(𝒜);

(4) det(𝒜^{∨}) ≅ (det(𝒜)) ^{⊗−3} ⊗ (det(𝒱))^{⊗−2} which implies that det(𝒜) ⊗ det(𝒜^{∨}) ∈ 2.Pic(X).

(b) If there exists a quadratic bundle (V, q) on X such that A ≅ C_{0}(V, q) then 𝒜/𝒪_{X}.1_{𝒜} ≅ Λ^{2}(V) which implies that det(𝒜^{∨}) ∈ 2.Pic(X), and therefore using (4) of (a) above, that det(𝒜) ∈ 2.Pic(X).

A bilinear form b (with values in 𝒪_{X}) on a vector bundle V on X induces a quadratic form q_{b} given on sections by x ↦ b(x, x). Further, b also defines a (ℤ/2.ℤ)-graded linear isomorphism ψ_{b} : C(V, q_{b}) ≅ Λ(V) which is unique with respect to certain properties (see (2d), Theorem 2.1). When X is affine, every (global) quadratic form on a vector bundle is induced from a (global) bilinear form. Therefore the following theorem is optimal for affine X.

THEOREM 1.6 With X and A as in Theorem 1.5 above, there exists a rank 3 quadratic bundle (V, q) (on X) induced from a rank 3 bilinear-form bundle (V, b) such that C_{0}(V, q) ≅ A (isomorphism as algebra bundles) iff the following two conditions hold:

(1) the determinant bundle of A is an element of 2.Pic(X);

(2) 𝒪_{X}.1_{A} is an 𝒪_{X}-direct summand of A.

The hypothesis (2) is satisfied in the following cases:

(i) X is an affine scheme;

(ii) the residue field of every point of X has characteristic ≠ 2.

The hypothesis (1) is satisfied when A is itself an Azumaya algebra bundle of rank 4 over X. It is also satisfied whenever det(𝒜) ≅ 𝒪_{X} (for example, when A is (globally) free) and whenever Pic(X) = 2.Pic(X)—a condition which is trivially satisfied when Pic(X) = 0, for example when X = Spec(R) and R is a UFD or a local ring or a field. It follows therefore that if X = Spec(R) is an affine scheme such that Pic(X) = 2.Pic(X), then the map Witt(X) : 𝒬_{3}(X) ⟶ 𝒜_{4}^{Sp−Azu}(X) is surjective, and hence a bijection.

Before proceeding further, we need to recall a few concepts from Part A of [3]. For a rank n^{2} vector bundle W on a scheme X and w ∈ Γ(X, W) a nowhere-vanishing global section, recall that if Id-w-Azu_{W} is the open X–subscheme of Azumaya algebra structures on W with identity w then its schematic image (or the scheme of specialisations or the limiting scheme) in the bigger X–scheme Id-w-Assoc_{W} of associative w-unital algebra structures on W is the X–scheme Id-w-Sp-Azu_{W} . By definition, the set of distinct specialised w-unital algebra structures on W corresponds precisely to the set of global sections of this last scheme over X.

THEOREM 1.7 Let X be a connected proper scheme of finite type over an algebraically closed field and let W be a vector bundle on X of rank n^{2} for some n ≥ 2.

(a) If W is self-dual (⇔ W ≅ W^{∨}) and A is an associative unital algebra structure on W such that 𝒜 ⊗_{X} 𝒪_{X,x} is Azumaya even for a single point x ∈ X, then A is Azumaya at every point of X. This may also be stated as follows: if a section to Id-w-Assoc_{W} over X topologically meets Id-w-Azu_{W}, then it factors as a morphism through the open subscheme Id-w-Azu_{W}, where w := 1_{A} and A corresponds to the given section.

(b) Let the rank of W be 4. If there exists an associative unital algebra structure on W which is not globally Azumaya but is Azumaya atleast at one point of X, then there does not exist any global Azumaya algebra structure on W. Thus, if there is a section over X of Id-w-Assoc_{W} that topologically meets both the open subscheme Id-w-Azu_{W} and its complement (with w = 1_{A} where A corresponds to the given section), then the X-schemes Id-w′-Assoc_{W} (with w′ global nowhere-vanishing) cannot have sections that land topologically inside Id-w′-Azu_{W} and hence in particular the X-schemes Id-w′-Azu_{W} have no sections over X.

Recall that an integral separated Noetherian scheme is said to be locally-factorial if each of its local rings is a unique factorisation domain (=UFD=factorial ring).

THEOREM 1.8 Let X be a scheme and W a rank 4 vector bundle on X with a global nowhere-vanishing section w. Let D_{X} denote the closed subset Id-w-Sp-Azu_{W} \Id-w-Azu_{W} .

(a) X is irreducible iff Id-w-Sp-Azu_{W} is irreducible iff Id-w-Azu_{W} is irreducible iff D_{X} is irreducible. The set of irreducible components of X is locally finite—for example this happens when X is locally noetherian—iff the same is true of the corresponding set for Id-w-Sp-Azu_{W} or for Id-w-Azu_{W} . If X is noetherian and finite-dimensional then the same are true for Id-w-Sp-Azu_{W} and Id-w-Azu_{W} .

(b) If X′ ⟶ X is a morphism of schemes, and if (W′, w′) denotes the pullback of (W, w), then we have a canonical isomorphism

Id-w′-Azu_{W′} ≅ Id-w-Azu_{W} ×_{Id-w-Sp-Azu_{W}} Id-w′-Sp-Azu_{W′}

In particular, the topological image of D_{X′} is D_{X}. Moreover, when X′ ⟶ X is a homeomorphism onto its topological image—which is for example the case when it is a closed or an open immersion, then D_{X} ∩ Id-w′-Sp-Azu_{W′} can be identified with D_{X′}.

(c) If X is a scheme which is finite dimensional and whose set of irreducible components is locally finite, then the closed subset D_{X} is a divisor i.e., it has codimension 1 in Id-w-Sp-Azu_{W} .

(d) X is affine iff Id-w-Sp-Azu_{W} is affine iff Id-w-Azu_{W} is affine. If X is regular in codimension 1 (respectively locally-factorial) then so are Id-w-Sp-Azu_{W} and Id-w-Azu_{W} .

(e) Assume that X is locally-factorial and W is self-dual (i.e., W ≅ W^{∨}). Then the (Weil) divisor n.(D_{X}) is principal for some positive integer n, so that the natural homomorphism given by restriction of line bundles Pic(Id-w-Sp-Azu_{W}) ⟶ Pic(Id-w-Azu_{W}) is an isomorphism iff n = 1. The integer n is divisible by 2 iff det(W ⊗_{X} Id-w-Sp-Azu_{W}) ∈ 2.Pic(Id-w-Sp-Azu_{W}) so that in this case if there exists a specialised algebra structure on W with unit w, then det(W) ∈ 2.Pic(X); if in addition, 𝒪_{X}.w is an 𝒪_{X}-direct summand of W, then W is of the form Λ^{even}(V) for a rank 3 vector bundle V on X, and for every X-scheme T and for every specialisation A_{T} on W_{T} := (W ⊗_{X} T) with unit w_{T} := (w ⊗_{X} T), there exists a rank 3 quadratic bundle (V_{T} , q_{T}) induced from a rank 3 bilinear form bundle (V_{T} , b_{T}) on T such that C_{0}(V_{T} , q_{T}) ≅ A_{T} as algebra bundles. (this last conclusion holds for any affine scheme T over X, even if 𝒪_{X}.w were not an 𝒪_{X}-direct summand of W).

Notice that under the hypotheses (1) and (2) of Theorem 1.6, there exists by (1), Theorem 1.5, a rank 3 bundle V on X such that Λ^{even}(𝒱) = 𝒪_{X} ⊕ Λ^{2}(𝒱) ≅ 𝒜 with 𝒪_{X} ≅ 𝒪_{X}.1_{𝒜} . In this situation the ‘if’ part of Theorem 1.6 as well as the latter part of (e) of Theorem 1.8 are consequences of the next one which describes the specialisations as bilinear forms. We continue with the notations introduced before Theorem 1.7. If Stab_{w} ⊂ GL_{W} is the stabiliser subgroupscheme of w, recall from Theorems 3.4 and 3.8, Part A, [3], that there exists a canonical action of Stab_{w} on Id-w-Sp-Azu_{W} such that the natural inclusions

(♣) Id-w-Azu_{W} ↪ Id-w-Sp-Azu_{W} ↪ Id-w-Assoc-Alg_{W}

are all Stab_{w}-equivariant. Now let V be a rank 3 vector bundle on the scheme X and Bil_{V} be the associated rank 9 vector bundle of bilinear forms on V with values in 𝒪_{X}. Let Bil_{V}^{sr} ↪ Bil_{V} correspond to the open subscheme of semiregular bilinear forms—we say that a bilinear form is semiregular if the quadratic form it induces is semiregular (though it may turn out that a semiregular bilinear form may be degenerate). Let W := Λ^{even}(V) and let w ∈ Γ(X, W) be the nowhere-vanishing global section corresponding to the unit for the natural multiplication in the even-exterior algebra bundle. There is an obvious natural action of GL_{V} on Bil_{V} . There is also a natural morphism of groupschemes GL_{V} ⟶ Stab_{w} given on valued points by g ↦ Λ^{even}(g) and therefore the natural inclusions marked by (♣) above are GL_{V} -equivariant. Finally, note that there is an obvious involution Σ on Id-w-Assoc-Alg_{W} given by A ↦ opposite(A) which leaves the open subscheme Id-w-Azu_{W} invariant.

THEOREM 1.9

(1) Let V be a rank 3 vector bundle on the scheme X, W := Λ^{even}(V) and w ∈ Γ(X, W) correspond to 1 in the even-exterior algebra bundle. There is a natural GL_{V} -equivariant morphism of X–schemes Υ′ = Υ′_{X} : Bil_{V} ⟶ Id-w-Assoc_{W} whose schematic image is precisely the scheme of specialisations Id-w-Sp-Azu_{W} . Further if Υ′ factors canonically through Υ = Υ_{X} : Bil_{V} ⟶ Id-w-Sp-Azu_{W}, then Υ is a GL_{V} -equivariant isomorphism and it maps the GL_{V} -stable open subscheme Bil_{V}^{sr} isomorphically onto the GL_{V} -stable open subscheme Id-w-Azu_{W} .

(2) The involution Σ of Id-w-Assoc-Alg_{W} defines a unique involution (also denoted by Σ) on the scheme of specialisations Id-w-Sp-Azu_{W} leaving the open subscheme Id-w-Azu_{W} invariant, and therefore via the isomorphism Υ, it defines an involution on Bil_{V} . This involution is none other than the one on valued points given by B ↦ transpose(−B).

(3) For an X-scheme T, let V_{T} (resp. W_{T}) denote the pullback of V (resp. W) to T, and let w_{T} be the global section of W_{T} induced by w. Then the base-changes of Υ′_{X} and Υ_{X} to T, namely Υ′_{X} ×_{X} T : Bil_{V} ×_{X} T ⟶ Id-w-Assoc_{W} ×_{X} T and Υ_{X} ×_{X} T : Bil_{V} ×_{X} T ≅ Id-w-Sp-Azu_{W} ×_{X} T may be canonically identified with the corresponding ones over T namely Υ′_{T} : Bil_{V_{T}} ⟶ Id-w_{T} -Assoc_{W_{T}} and Υ_{T} : Bil_{V_{T}} ≅ Id-w_{T} -Sp-Azu_{W_{T}} .

The explicit computation of the morphism Υ locally over X is an important step in proving the above theorem. To describe this, suppose that V is free of rank 3 over X, so that we may fix a basis {e_{1}, e_{2}, e_{3}} for V, which naturally gives rise to a basis of Bil_{V} . For any X-scheme T, a T -valued point B of Bil_{V} is just a global bilinear form with values in 𝒪_{T} on the pull-back V ⊗_{X} T of V to T. Such a B is given uniquely by a (3 × 3)-matrix (b_{ij}) with the b_{ij} being global sections of the trivial line bundle 𝔸_{T}^{1} (or equivalently, elements of Γ(T, 𝒪_{T})). The chosen basis for V also gives rise to the basis {ϵ_{0} := w = 1 ; ϵ_{1} := e_{1} ∧ e_{2} , ϵ_{2} := e_{2} ∧ e_{3} , ϵ_{3} := e_{3} ∧ e_{1}} of W = Λ^{even}(V). A T -valued point A of Id-w-Assoc-Alg_{W} is just a w_{T} := (w ⊗_{X} T)-unital associative algebra structure on the bundle W_{T} := W ⊗_{X} T. Let ·_{A} denote the multiplication in the algebra bundle A, and for ease of notation, let s^{∘} denote the section s ⊗_{X} T induced from a section s (for example, (w ⊗_{X} T) = w^{∘}, ϵ_{i} ⊗_{X} T = ϵ_{i}^{∘} etc).

THEOREM 1.10 In addition to the hypothesis of Theorem 1.9, assume that V is free of rank 3. Then fixing a basis for V and adopting the notations above, the map Υ(T) takes B = (b_{ij}) to (A, 1_{A}, ·) = (W_{T} , w_{T} = w^{∘}, ·_{A}) with multiplication given as follows, where M_{ij}(B) is the determinant of the minor of the element b_{ij} in B :

• ϵ_{1}^{∘} ·_{A} ϵ_{1}^{∘} = −M_{33}(B)w^{∘} + (b_{21} − b_{12})ϵ_{1}^{∘}

• ϵ_{2}^{∘} ·_{A} ϵ_{2}^{∘} = −M_{11}(B)w^{∘} + (b_{32} − b_{23})ϵ_{2}^{∘}

• ϵ_{3}^{∘} ·_{A} ϵ_{3}^{∘} = −M_{22}(B)w^{∘} + (b_{13} − b_{31})ϵ_{3}^{∘}

• ϵ_{1}^{∘} ·_{A} ϵ_{2}^{∘} = −M_{31}(B)w^{∘} − b_{23}ϵ_{1}^{∘} − b_{12}ϵ_{2}^{∘} − b_{22}ϵ_{3}^{∘}

• ϵ_{2}^{∘} ·_{A} ϵ_{3}^{∘} = +M_{12}(B)w^{∘} − b_{33}ϵ_{1}^{∘} − b_{31}ϵ_{2}^{∘} − b_{23}ϵ_{3}^{∘}

• ϵ_{3}^{∘} ·_{A} ϵ_{1}^{∘} = +M_{23}(B)w^{∘} − b_{31}ϵ_{1}^{∘} − b_{11}ϵ_{2}^{∘} − b_{12}ϵ_{3}^{∘}

• ϵ_{1}^{∘} ·_{A} ϵ_{3}^{∘} = +M_{32}(B)w^{∘} + b_{13}ϵ_{1}^{∘} + b_{11}ϵ_{2}^{∘} + b_{21}ϵ_{3}^{∘}

• ϵ_{2}^{∘} ·_{A} ϵ_{1}^{∘} = −M_{13}(B)w^{∘} + b_{32}ϵ_{1}^{∘} + b_{21}ϵ_{2}^{∘} + b_{22}ϵ_{3}^{∘}

• ϵ_{3}^{∘} ·_{A} ϵ_{2}^{∘} = −M_{21}(B)w^{∘} + b_{33}ϵ_{1}^{∘} + b_{13}ϵ_{2}^{∘} + b_{32}ϵ_{3}^{∘}

The key to the proofs of Theorems 1.1, 1.3 and 1.4 lies in a deeper study of a different identification of the scheme of specialisations, namely one related to the scheme of quadratic forms on a trivial rank 3 bundle in the special situation when W is free and w part of a global basis. Without loss of generality we may in this situation therefore take V to be a free rank 3 vector bundle on X and (W, w) = (Λ^{even}(V), 1), so that we are in the situation of Theorem 1.10 above. This relationship with quadratic forms was shown in Theorem 5.3, Part A, [3], which we briefly recall next. Let Quad_{V} denote the bundle of quadratic forms on V (with values in 𝒪_{X}) and Quad_{V}^{sr} the open subscheme of semiregular quadratic forms. Let A_{0} denote the algebra bundle structure (with unit w = 1) on W = Λ^{even}(V) given by Λ^{even}(V) itself. Fix a basis for V and adopt the notations preceding Theorem 1.10 above. Then Stab_{w} is the semidirect product of a commutative 3-dimensional subgroupscheme L_{w} ≅ (𝔸_{X}^{3}, +) with the stabiliser subgroupscheme Stab_{A_{0}} of A_{0} in Stab_{w} (Lemma 5.1, Part A, [3]).

THEOREM 1.11 (Definition 5.2 & Theorem 5.3, Part A, [3]) There is a natural isomorphism Θ : Quad_{V} ×_{X} L_{w} ≅ Id-w-Sp-Azu_{W} which maps the open subscheme Quad_{V}^{sr} ×_{X} L_{w} isomorphically onto the open subscheme Id-w-Azu_{W} .

Section 3 is essentially devoted to studying Θ. There we compute Θ explicitly and in Theorem 3.1 we write out the multiplication table of every specialised algebra structure on any fixed free rank 4 vector bundle with fixed unit that is part of a global basis. It turns out that Θ is not equivariant with respect to GL_{V}, but nevertheless satisfies a ‘twisted’ form of equivariance (Theorem 3.4). A T -valued point q of Quad_{V} ≅ 𝔸_{X}^{6} may be identified uniquely with a 6-tuple (λ_{1}, λ_{2}, λ_{3}, λ_{12}, λ_{13}, λ_{23}) corresponding to the quadratic form (x_{1}, x_{2}, x_{3}) ↦ Σ_{i}λ_{i}x_{i}^{2} + Σ_{i<j}λ_{ij}x_{i}x_{j}. A T -valued point \underline{t} of L_{w} ≅ (𝔸_{X}^{3}, +) may be identified uniquely with a 3-tuple (t_{1}, t_{2}, t_{3}) which corresponds to the valued point of Stab_{w} given by the (4 × 4)-matrix

( \begin{matrix}1 & t_{1} & t_{2} & t_{3} \\ 0 & & I_{3} \end{matrix} )

where I_{3} is the (3 × 3)-identity matrix. With these notations, the identification of Theorems 1.9 and 1.10 may be compared with that of the above Theorem 1.11 as follows.

THEOREM 1.12 The isomorphism Υ^{−1} ∘ Θ : Quad_{V} ×_{X} L_{w} ≅ Bil_{V} takes the valued point ( q, \underline{t} ) = ( (λ_{1}, λ_{2}, λ_{3}, λ_{12}, λ_{13}, λ_{23}), (t_{1}, t_{2}, t_{3}) ) to the valued point B = (b_{ij}) given by

B = ( \begin{matrix}λ_{1} & t_{1} & λ_{13} − t_{3} \\ λ_{12} − t_{1} & λ_{2} & t_{2} \\ t_{3} & λ_{23} − t_{2} & λ_{3} \end{matrix} )

Moreover, under this identification, the involution B ↦ (−B)^{t} on Bil_{V} (induced from the isomorphism Υ of Theorem 1.9) translates into the involution on Quad_{V} ×_{X} L_{w} given by

( q, (t_{1}, t_{2}, t_{3}) ) ↦ ( −q, (t_{1} − λ_{12}, t_{2} − λ_{23}, t_{3} − λ_{13}) ).

We next make some comments relevant to the question of the surjectivity of Witt(X) : 𝒬_{3}(X) ⟶ 𝒜_{4}^{Sp−Azu}(X). Given a specialised algebra structure A (on a rank 4 vector bundle on X) which is itself an Azumaya algebra, we first indicate briefly how to naturally retrieve a rank 3 quadratic bundle (V, q) such that C_{0}(V, q) ≅ A. Let R be a unital commutative ring and A a unital associative R-algebra. Given any involution σ on A, we may define the associated trace tr_{σ} : x ↦ x + σ(x) and the associated norm n_{σ} : x ↦ x.σ(x); in para.1.3, Chap.I, [2], Knus calls σ standard if σ fixes R.1_{A} and both tr_{σ} and n_{σ} take values in R.1_{A}. In Prop.1.3.4 of the same chapter, he proves that a standard involution is unique if it exists, provided the R-module underlying A is finitely generated projective and faithful. Thereafter, in para.1.3.7, Knus defines A to be a quaternion algebra if A is a projective R-module of rank 4 and A has a standard involution. Thus we may define a rank 4 algebra bundle on a scheme X to be a quaternion algebra bundle if it is locally (in the Zariski topology) a quaternion algebra in Knus’ sense, and it would follow that the local standard involutions glue to define a unique global standard involution on the bundle.

PROPOSITION 1.13 Any specialised algebra bundle is a quaternion algebra bundle.

This result can be deduced from the following two facts:

(1) Any specialised algebra is locally (in the Zariski topology) the even Clifford algebra of a rank 3 quadratic bundle (Theorems 3.8 and 5.3, Part A, [3]).

(2) The even Clifford algebra of a quadratic module of rank 3 over a commutative ring has a standard involution which is none other than the restriction of the ‘standard’ involution on the full Clifford algebra (Prop.3.1.1, Chap.V, [2]). Q.E.D.

THEOREM 1.14 For each scheme X, the map Witt(X) : 𝒬_{3}^{sr}(X) ⟶ 𝒜_{4}^{Azu}(X) is bijective.

The proof of the above Theorem follows from Prop.3.2.3 and Prop.3.2.4, Chap.V, [2] generalised to the scheme-theoretic setting. We recall how the surjectivity is established. Let A be a specialised algebra bundle on the scheme X. By the results just quoted if A is Azumaya, or more generally by Prop.1.13, we have the existence of a unique standard involution σ_{A} on A, to which are associated the norm n_{σ_{A}} : A ⟶ 𝔸_{X}^{1} given on sections by x ↦ x.σ_{A}(x) and the trace tr_{σ_{A}} : A ⟶ 𝔸_{X}^{1} given on sections by x ↦ x + σ_{A}(x). Let A′ := kernel(tr_{σ_{A}}) ↪ A be the subsheaf of trace zero elements. As the calculations in para.3.2, Chap.V, [2] show, the trace map is surjective if A is itself an Azumaya algebra; if this is the case, then it is further shown there that the rank 3 quadratic bundle (V, q) := (A′, n_{σ_{A}}|A′) is semiregular and its even Clifford algebra C_{0}(V, q) ≅ A.

However the above method of retrieving a canonical rank 3 quadratic bundle fails badly for specialised non-Azumaya algebras. Consider even the case of X = Spec(k) where k is a field of characteristic two and the Clifford algebra A = C_{0}(V, q) of a quadratic form q on V = k^{⊕3} which is a perfect square (i.e., a square of a linear form or equivalently a sum of squares). In this case an easy computation shows that the subspace A′ of trace zero elements is the full space A. However, the underlying module of A trivially satisfies the hypotheses of Theorem 1.6, and so the existence of (V, q) follows from that Theorem.

The good algebraic properties of Azumaya algebras are reflected as good geometric properties of the scheme of Azumaya algebra structures on a fixed vector bundle: this scheme is separated, of finite type and smooth relative to the base scheme (over which the vector bundle is fixed) and also base-changes well relative to the base scheme (Theorem 3.4, Part A, [3]). When the vector bundle is of rank 4, the nice thing that happens is that all these good properties also pass over to the limit i.e., to the scheme of specialisations, defined to be the schematic image of the scheme of Azumaya algebra structures (Theorem 3.8, Part A, [3]). If A is Azumaya over X, then as seen above, A ≅ C_{0}(V, q) with (V, q) semiregular. So by (b), Theorem 1.5, we have that det(A) ∈ 2.Pic(X). On the other hand, by the same Theorem, this is a necessary condition for a specialisation to arise as C_{0}(V, q) for a global q. Therefore the author expects this condition to hold in general (the assertions in (e), Theorem 1.8, arose after an attempt to investigate this viewpoint). Of course, if this holds and also if 𝒪_{X}.1_{A} is an 𝒪_{X}-direct summand of A, then by Theorem 1.6 we get something more, i.e., even a bilinear form bundle (V, b) such that C_{0}(V, q_{b}) ≅ A. But the latter hypothesis is seemingly strong for non-affine X. The author hopes that the surjectivity of Witt(X) is valid for general X, and believes that this may follow from a deeper understanding of the geometry of the scheme of specialisations. We next state some results on rank 3 quadratic forms and specialised algebras in certain particular cases.

PROPOSITION 1.15 Let S be a commutative semilocal ring that is 2-perfect i.e., such that the square map S ⟶ S : s ↦ s^{2} is surjective, and V a free rank 3 S-module. Then the set of semiregular quadratic S-forms on V forms a single GL(V)–orbit; in other words, upto isometry, ∃ only one semiregular quadratic S-module structure on V.

Corollary 1.16 Let S be a commutative local ring that is 2-perfect. Then any two rank 4 Azumaya Salgebras are isomorphic. If S were only semilocal, the conclusion still holds provided the identity elements for multiplication for each of the two Azumaya S-algebras can be completed to an S-basis.

Since Witt(X) : 𝒬_{3}^{sr}(X) ≅ 𝒜_{4}^{Azu}(X) is bijective, taking X = Spec(S) with S as in the Prop.1.15 proves the first assertion of the above corollary. The second may be deduced by an application of Theorem 1.11 alongwith Prop.1.15.

Let W be a rank 4 vector bundle on a scheme X, w ∈ Γ(X, W) a nowhere-vanishing global section and Stab_{w} ⊂ GL_{W} the stabiliser subgroupscheme of w. Recall that the natural inclusion Id-w-Azu_{W} ↪ Id-w-Sp-Azu_{W} is Stab_{w}-equivariant (see (♣), page 6). When X = Spec(k) where k is an algebraically closed field, there is a canonical Stab_{w}-stratification of the k-variety underlying Id-w-Sp-Azu_{W} as follows.

THEOREM 1.17

(1) Let k be a quadratically closed field and X = Spec(k). Then 𝒬_{3}(X) has 4 elements which correspond to (a) semiregular quadratic modules; (b) rank 2 quadratic modules i.e., those that are not semiregular but which are regular on a two-dimensional subspace; (c) nonzero perfect squares and (d) the zero form. If V is a 3-dimensional vector space over k and {e_{1}, e_{2}, e_{3}} a k-basis for V, then representatives for these 4 GL_{V} -orbits in the space Quad_{V} of quadratic forms on V can respectively be taken to be: (a) q^{(1)}(Σ_{i=1}^{3}x_{i}e_{i}) = x_{1}x_{2} + x_{3}^{2} ; (b) q^{(2)}(Σ_{i=1}^{3}x_{i}e_{i}) = x_{1}x_{2} ; (c) q^{(3)}(Σ_{i=1}^{3}x_{i}e_{i}) = x_{3}^{2} ; (d) q^{(4)} = 0.

(2) In addition to the hypotheses and notations of (1) above, assume that k is an algebraically closed field. Then the four orbits Quad_{V}^{(i)} := GL_{V} · q^{(i)} for 1 ≤ i ≤ 4 form a stratification of the k-variety Quad_{V} in the sense that we have

\overline{Quad_{V}^{(1)}} = Quad_{V} and \overline{Quad_{V}^{(i+1)}} = \overline{Quad_{V}^{(i)}} \ Quad_{V}^{(i)} for 1 ≤ i ≤ 3

and further we also have

Sing(\overline{Quad_{V}^{(i+1)}}) = \overline{Quad_{V}^{(i+1)}} \ Quad_{V}^{(i+1)} for 1 ≤ i ≤ 2

unless the characteristic of k is 2 in which case \overline{Quad_{V}^{(3)}} is itself smooth (the notation \overline{T} denotes the orbit closure and Sing(T) denotes the subset of singular (non-smooth) points of T, each given the canonical reduced induced closed subscheme structure).

(3) Continuing with the notations and hypotheses of (2) above, set (W, w) := (Λ^{even}(V), 1). For ease of notation denote Id-w-Sp-Azu_{W} by SpAzu and Stab_{w} by H. Then the four orbits SpAzu^{(i)} := H · Θ(q^{(i)}, I_{4}) for 1 ≤ i ≤ 4 form a stratification of the k-variety SpAzu in the sense that we have

\overline{SpAzu^{(1)}} = SpAzu and \overline{SpAzu^{(i+1)}} = \overline{SpAzu^{(i)}} \ SpAzu^{(i)} for 1 ≤ i ≤ 3

and further we also have

Sing(\overline{SpAzu^{(i+1)}}) = \overline{SpAzu^{(i+1)}} \ SpAzu^{(i+1)} for 1 ≤ i ≤ 2

unless the characteristic of k is 2 in which case \overline{SpAzu^{(3)}} is itself smooth.

2 Reduction of Theorems 1.1 & 1.3 to the Free Case

In this section, we prove Prop.1.2 and reduce the proof of Theorem 1.1 to Theorem 1.3. Thereafter we reduce the proof of Theorem 1.3 to the case when V is free. Our means towards this end are several standard results which we recall below. Bourbaki’s tensor operations are described first, followed by basic facts on tensoring quadratic bundles with discriminant bundles, and thereafter by some facts about similitudes between quadratic bundles. On the way we justify the definitions of the functors 𝒬_{3}^{sr} ↪ 𝒬_{3}, and the natural transformation Witt introduced in §1 above.

THEOREM 2.1 (Bourbaki’s Tensor Operations, §9, Chap.9, [4]; para.1.7, Chap.IV, [2]) In the following, let R be a commutative ring (with 1) and V an R-module.

(1) Let q : V ⟶ R be a quadratic form on V and f ∈ V^{∨} := Hom_{R}(V, R) be a functional on V. Then there exists an R-linear endomorphism t_{f} of the tensor algebra T V which is unique with respect to the first three of the following properties it satisfies:

(a) t_{f} (1) = 0;

(b) t_{f} (x ⊗ y) = f(x).y − x ⊗ t_{f} (y) for every x ∈ V and every y ∈ T V;

(c) If J(q) is the two-sided ideal of T V generated by the set {x⊗x−q(x).1 | x ∈ V }, then t_{f} (J(q)) ⊂ J(q);

(d) t_{f} is homogeneous of degree -1 (for elements it does not annihilate);

(e) (Recall that the Clifford algebra of q is C(V, q) := T V/J(q)). By (c) above, t_{f} induces a (ℤ/2.ℤ)- antigraded endomorphism d_{f}^{q} : C(V, q) ⟶ C(V, q);

(f) t_{f} ∘ t_{f} = 0;

(g) if g ∈ V^{∨} is also a functional, then t_{f} ∘ t_{g} + t_{g} ∘ t_{f} = 0;

(h) if α ∈ End_{R}(V), then t_{f} ∘T (α) = T (α)∘t_{α^{∗}f} where α^{∗}f ∈ End_{R}(V) is defined by x ↦ f(α(x));

(i) t_{f} ≡ 0 on the subalgebra of T V generated by kernel(f).

(2) Let q, q′ : V ⟶ R be two quadratic forms whose difference is the quadratic form q_{b} induced by a bilinear form b ∈ Bil_{R}(V) := Hom_{R}(V ⊗_{R} V, R) i.e., q′(x)−q(x) = q_{b}(x) := b(x, x) ∀x ∈ V. Further, for any x ∈ V denote by b_{x} the functional on V given by y ↦ b(x, y). Then there exists an R-linear automorphism Ψ_{b} of T V which is unique with respect to the first three of the following properties it satisfies:

(a) Ψ_{b}(1) = 1;

(b) Ψ_{b}(x ⊗ y) = x ⊗ Ψ_{b}(y) + t_{b_{x}}(Ψ_{b}(y)) for any x ∈ V and any y ∈ T V;

(c) Ψ_{b}(J(q′)) ⊂ J(q);

(d) by the previous property, Ψ_{b} induces an isomorphism of (ℤ/2.ℤ)-graded R-modules

ψ_{b} : C(V, q′) ≅ C(V, q);

in particular, given a quadratic form q_{1} : V ⟶ R, we may take a bilinear form b_{1} that induces q_{1} (i.e., such that q_{1}(x) = q_{b_{1}}(x) := b_{1}(x, x) ∀x ∈ V), and get a (ℤ/2.ℤ)-graded linear isomorphism ψ_{b_{1}} : C(V, q_{1} = q_{b_{1}}) ≅ C(V, 0) = Λ(V);

(e) Ψ_{b}(T^{2n}V) ⊂ ⊕_{(i≤n)}T^{2i}V and Ψ_{b}(T^{2n+1}V) ⊂ ⊕_{(odd i≤2n+1)}T^{i}V;

(f) in particular, for x, x′ ∈ V, Ψ_{b}(x ⊗ x′) = x ⊗ x′ + b(x, x′).1_{T V} so that for ψ_{b} : C_{0}(V, q_{b}) ≅ C_{0}(V, 0) = Λ^{even}(V) we have ψ_{b}(x.x′) = x ∧ x′ + b(x, x′).1 where x.x′ denotes the product in C(V, q_{b}).

(g) if f ∈ V^{∨} and t_{f} is given by (1) above, then Ψ_{b} ∘ t_{f} = t_{f} ∘ Ψ_{b};

(h) if b_{i} are bilinear forms on V, then Ψ_{b_{1}+b_{2}} = Ψ_{b_{1}} ∘ Ψ_{b_{2}} and Ψ_{0} = Identity on T V;

(i) for any α ∈ End_{R}(V), Ψ_{b} ∘ T (α) = T (α) ∘ Ψ_{(b.α)} where (b.α)(x, x′) := b(α(x), α(x′)) ∀x, x′ ∈ V;

(j) by property (h), one has a homomorphism of groups (Bil_{R}(V), +) ⟶ (Aut_{R}(T V), ∘) : b ↦ Ψ_{b}; the associative unital monoid (End_{R}(V), ∘) acts on Bil_{R}(V) on the right by b′ ⇝ b′.α and acts on the left (resp. on the right) of End_{R}(T V) by α.Φ := T (α) ∘ Φ (resp. by Φ.α := Φ ∘ T (α)), and the homomorphism b ↦ Ψ_{b} satisfies α.Ψ_{(b.α)} = Ψ_{b}.α; the group Aut_{R}(V) = GL_{R}(V) acts on the left of Bil_{R}(V) by g.b : (x, x′) ↦ b(g^{−1}(x), g^{−1}(x′)) and on the left of Aut_{R}(T V) by conjugation via the natural group homomorphism GL_{R}(V) ⟶ Aut_{R}(T V) : g ↦ T (g) i.e., g.Φ := T (g) ∘ Φ ∘ T (g^{−1}), and the homomorphism b ↦ Ψ_{b} is GL_{R}(V)-equivariant: Ψ_{g.b} = g.Ψ_{b}.

(3) For a commutative R-algebra S (with 1), let (q ⊗_{R} S), (q′ ⊗_{R} S) : (V ⊗_{R} S =: V_{S}) ⟶ S be the quadratic S-forms induced from the quadratic forms q, q′ of (2) above and (b ⊗_{R} S) ∈ Bil_{S}(V_{S}) the bilinear S-form induced from the bilinear form b of (2) above. Then as a result of the uniqueness properties (2a)–(2c) satisfied by Ψ_{b} and Ψ_{(b⊗_{R}S)}, the S-linear automorphisms (Ψ_{b} ⊗_{R} S) and Ψ_{(b⊗_{R}S)} may be canonically identified. In particular, the (ℤ/2.ℤ)-graded S-linear isomorphism (ψ_{b} ⊗_{R} S) : C(V_{S}, (q′ ⊗_{R} S)) ≅ C(V_{S}, (q ⊗_{R} S)) induced from ψ_{b} of (2d) above may be canonically identified with ψ_{(b⊗_{R}S)}.

Tensoring by Discriminant Bundles. We recall that a bilinear form (resp. alternating form, resp. quadratic form) with values in 𝒪_{X} on a vector bundle V over an open set U ↪ X of the scheme X is by definition a section over U of the vector bundle Bil_{V} (resp. of Alt_{V}^{2} , resp. of Quad_{V}), or equivalently, an element of Γ (U, Bil_{𝒱} := (T_{𝒪_{X}}^{2} (𝒱))^{∨}) (resp. of Γ (U, Alt_{𝒱}^{2} := (Λ_{𝒪_{X}}^{2} (𝒱))^{∨}), resp. of Γ (U, Quad_{𝒱} ) ), where the sheaf Quad_{𝒱} —the (coherent locally-free) sheaf of 𝒪_{X}-modules corresponding to the bundle Quad_{V} of quadratic forms on V —is defined by the exactness of the following sequence:

0 ⟶ Alt_{𝒱}^{2} ⟶ Bil_{𝒱} ⟶ Quad_{𝒱} ⟶ 0.

In terms of the corresponding (geometric) vector bundles over X, the above translates into the following sequence of morphisms of vector bundles, with the first one a closed immersion and the second one a Zariski locally-trivial principal Alt_{V}^{2} -bundle:

Alt_{V}^{2} ↪ Bil_{V} ↠ Quad_{V} .

Given a quadratic form q ∈ Γ(U, Quad_{𝒱} ), recall that the usual ‘associated’ bilinear form b_{q} ∈ Γ(U, Bil_{𝒱} ) is given on sections (over open subsets of U) by v ⊗ v′ ↦ q(v + v′) − q(v) − q(v′). This association in general does not lead to a bijective correspondence between quadratic forms and symmetric bilinear forms (which is nevertheless correct when 2 ∈ Γ(X, 𝒪_{X}^{∗}) or equivalently when the residue field of each point of X is of characteristic ≠ 2). Given a (not-necessarily symmetric!) bilinear form b, we also have the induced quadratic form q_{b} given on sections by v ↦ b(v ⊗ v). Since a surjection of sheaves does not necessarily imply a surjection on global sections, a global quadratic form may not be induced from a global bilinear form (unless we assume something more, for e.g., that the scheme is affine, or more generally that the sheaf cohomology group H^{1}(X, Alt_{𝒱}^{2} ) = 0).

PROPOSITION 2.2 Let (V, q) be a quadratic bundle on X and (M, b) be a symmetric bilinear form bundle on X (i.e., M is a vector bundle on X and b is a global symmetric bilinear form on M with values in 𝒪_{X}).

(1) Then we can tensor (V, q) with (M, b) to get a unique quadratic bundle (V ⊗ M, q ⊗ b). The quadratic form on V ⊗ M is given on sections by v ⊗ m ↦ q(v).b(m ⊗ m) and has associated bilinear form b_{q⊗b} = b_{q} ⊗ b.

(2) When the rank of M is 1 i.e., M is a line bundle, (M, b) is regular iff (M, q_{b}) is semiregular iff b : ℳ ⊗ ℳ ≅ 𝒪_{X} is an isomorphism (i.e., iff (M, b) is a discriminant bundle).

(3) Let V be of odd rank and (M, b) a discriminant bundle. Then (V, q) is semiregular iff (V, q)⊗(M, b) = (V ⊗ M, q ⊗ b) is semiregular.

By taking an affine open cover for X (which always exists by definition since X is a scheme), the proof of the assertions in the above proposition may be reduced to the case when X is itself an affine scheme. In this case we may further assume that V and M are themselves free. Then (1) follows from para.8.4, Chap.I, [2], while (2) and (3) are easy consequences of the definition and basic properties of semiregularity (§3, Chap.IV, [2]). Note that statments (1) and (3) above justify respectively the definition of the functor 𝒬_{3} and its subfunctor 𝒬_{3}^{sr} introduced in §1.

PROPOSITION 2.3 Let V and V′ be vector bundles of the same rank on the scheme X, with associated locally-free sheaves 𝒱 and 𝒱′ respectively. Let α : V′ ≅ V ⊗ L be an isomorphism of bundles where (L, h) is a discriminant bundle on X.

(1) Over any open subset U ↪ X, given a bilinear form b ∈ Γ (U, Bil_{𝒱} ), we can define a bilinear form b′ ∈ Γ (U, Bil_{𝒱′} ) using α and h as follows: we let b′ := b ∘ ζ_{(α,h)} where ζ_{(α,h)} : 𝒱′ ⊗ 𝒱′ ≅ 𝒱 ⊗ 𝒱 is the 𝒪_{X}-module isomorphism given by the composition of the following natural morphisms:

𝒱′ ⊗ 𝒱′ ⟶^{α⊗α (≅)} 𝒱 ⊗ ℒ ⊗ 𝒱 ⊗ ℒ ⟶^{SWAP(2,3) (≡)} 𝒱 ⊗ 𝒱 ⊗ ℒ ⊗ ℒ ⟶^{Id⊗h (≅)} 𝒱 ⊗ 𝒱 ⊗ 𝒪_{X} ⟶^{CANON (≡)} 𝒱 ⊗ 𝒱.

Then the association b ↦ b′ induces 𝒪_{X}-linear isomorphisms shown by vertical downward arrows in the following diagram (with exact rows) making it commutative:

0 ⟶ Alt_{𝒱}^{2} ⟶ Bil_{𝒱} ⟶ Quad_{𝒱} ⟶ 0
≅↓  ≅↓  ≅↓
0 ⟶ Alt_{𝒱′}^{2} ⟶ Bil_{𝒱′} ⟶ Quad_{𝒱′} ⟶ 0

Therefore one also has the following commutative diagram of vector bundle morphisms with the vertical downward arrows being isomorphisms:

Alt_{V}^{2} closed ; ⟶ ; immersion Bil_{V} locally ; ⟶ ; trivial Quad_{V}
≅↓  ≅↓  ≅↓
Alt_{V′}^{2} closed ; ⟶ ; immersion Bil_{V′} locally ; ⟶ ; trivial Quad_{V′}

(2) Let b ∈ Γ (X, Bil_{𝒱} ) be a global bilinear form and let it induce b′ ∈ Γ (X, Bil_{𝒱′} ) via α and h as defined in (1) above. Let Ψ_{b} ∈ Aut_{𝒪_{X}} (T 𝒱) (resp. Ψ_{b′} ∈ Aut_{𝒪_{X}} (T 𝒱′)) be the (ℤ/2.ℤ)-graded linear automorphism of the tensor algebra (with even elements given degree 0 and odd elements given degree 1) induced by b (resp. by b′) defined locally (and hence globally) as in (2e) of Theorem 2.1 above. Let Z_{(α,h)} : T_{𝒪_{X}}^{even}(𝒱′) ≅ T_{𝒪_{X}}^{even}(𝒱) be the 𝒪_{X}-algebra isomorphism induced via the isomorphism ζ_{(α,h)} : T_{𝒪_{X}}^{2} (𝒱′) ≅ T_{𝒪_{X}}^{2} (𝒱) defined in (1) above. Then the following diagram commutes:

T_{𝒪_{X}}^{even}(𝒱′) ⟶_{≅}^{Z_{(α,h)}} T_{𝒪_{X}}^{even}(𝒱)
Ψ_{b′} ↓≅  ≅↓^{Ψ_{b}}
T_{𝒪_{X}}^{even}(𝒱′) ⟶_{Z_{(α,h)}}^{≅} T_{𝒪_{X}}^{even}(𝒱)

thereby inducing (see (2d), Theorem 2.1) the following commutative diagram of 𝒪_{X}-linear isomorphisms

C_{0}(𝒱′ , q_{b′}) ⟶_{≅}^{via Z_{(α,h)}} C_{0}(𝒱, q_{b})
ψ_{b′} ↓≅  ≅↓^{ψ_{b}}
Λ_{𝒪_{X}}^{even}(𝒱′) ⟶_{via Z_{(α,h)}}^{≅} Λ_{𝒪_{X}}^{even}(𝒱)

(3) Let b and b′ be as in (2) above. Then α : V′ ≅ V ⊗ L induces an isometry of bilinear form bundles α : (V′, b′) ≅ (V, b) ⊗ (L, h) and also an isometry of the induced quadratic bundles α : (V′, q_{b′}) ≅ (V, q_{b}) ⊗ (L, h). Moreover, if we are just given a global quadratic form q on V (resp. q′ on V′), then we may define the global quadratic form q′ on V′ (resp. q on V) via q′ := (q ⊗ h) ∘ α (resp. via q := (q′ ⊗ (h^{∨})^{−1}) ∘ (α ⊗ L^{−1})^{−1}) and again α : (V′, q′) ≅ (V, q) ⊗ (L, h) becomes an isometry of quadratic bundles.

Similitudes. Let (V, q) and (V′, q′) be quadratic bundles on the scheme X. We recall that a morphism of vector bundles g : V ⟶ V′ is called a similitude (or similarity) with multiplier l ∈ Γ(X, 𝒪_{X}^{∗}), and we write g : (V, q) ≅_{l} (V′, q′), if g is an isomorphism of vector bundles and the following diagram involving their associated locally-free sheaves commutes (where q and q′ are considered as morphisms of sheaves of sets):

𝒱 ⟶_{≅}^{g} 𝒱′
q↓  ↓^{q′}
𝒪_{X} ⟶_{MULT. BY l}^{≅} 𝒪_{X}

Moreover, when the multiplier l = 1, we write g : (V, q) ≅ (V′, q′) and call g an isometry.

PROPOSITION 2.4 Let g : (V, q) ≅_{l} (V′, q′) be a similarity with multiplier l ∈ Γ(X, 𝒪_{X}^{∗}).

(1) There exists a unique isomorphism of 𝒪_{X}-algebra bundles C_{0}(g, l) : C_{0}(V, q) ≅ C_{0}(V′, q′) such that C_{0}(g, l)(v.v′) = l^{−1}g(v).g(v′) on sections.

(2) There exists a unique vector bundle isomorphism C_{1}(g, l) : C_{1}(V, q) ≅ C_{1}(V′, q′) such that

(a) C_{1}(g, l)(v.c) = g(v).C_{0}(g, l)(c) and

(b) C_{1}(g, l)(c.v) = C_{0}(g, l)(c).g(v)

for any section v of V and any section c of C_{0}(V, q). Thus C_{1}(g, l) is C_{0}(g, l)-semilinear.

(3) If g_{1} : (V′, q′) ≅_{l_{1}} (V′′, q′′) is another similarity with multiplier l_{1}, then the composition g_{1} ∘ g : (V, q) ≅_{ll_{1}} (V′′, q′′) is also a similarity with multiplier given by the product of the multipliers. Further C_{i}(g_{1} ∘ g, ll_{1}) = C_{i}(g_{1}, l_{1}) ∘ C_{i}(g, l) for i = 0, 1.

The proof of the above follows from the case of an affine scheme which is established in Prop.7.1.1, Chap.IV, [2]. A local computation shows that tensoring by a discriminant bundle amounts to (locally) applying a similarity. In this case also one gets a global isomorphism of even Clifford algebras:

PROPOSITION 2.5 Let (V, q) be a quadratic bundle on a scheme X and (L, h) be a discriminant bundle. There exists a unique isomorphism of algebra bundles γ_{(L,h)} : C_{0} ((V, q) ⊗ (L, h)) ≅ C_{0}(V, q) given by γ_{(L,h)} ((v ⊗ λ).(v′ ⊗ λ′)) = h(λ ⊗ λ′)v.v′ for any sections v, v′ of V and λ, λ′ of L.

The proof of the above may be reduced to the case of an affine scheme, in which case, we may further assume that L is free, so that tensoring by (L, h) is the same as applying a similarity. In this case, the result follows from Prop.2.4. The above result justifies the definition of the natural transformation Witt introduced in §1.

Proof of Prop.1.2: Start with an isomorphism of algebra-bundles ϕ : C_{0}(V, q) ≅ C_{0}(V′, q′). Let {U_{i}}_{i∈I} be an affine open covering of X (which may also be chosen so as to trivialise the rank 3 vector bundles V and V′ if needed). Choose bilinear forms b_{i} ∈ Γ(U_{i}, Bil_{V}) and b′_{i} ∈ Γ(U_{i}, Bil_{V′}) such that q|U_{i} = q_{b_{i}} and q′|U_{i} = q_{b′_{i}} for each i ∈ I. By (2d), Theorem 2.1, we have isomorphisms of vector bundles ψ_{b_{i}} and ψ_{b′_{i}}, which preserve 1 by (2a) of the same Theorem, and we define the isomorphism of vector bundles ϕ_{Λ_{i}^{ev}} so as to make the following diagram commute:

C_{0}(V |U_{i}, q|U_{i}) ⟶_{≅}^{ϕ|U_{i}} C_{0}(V′|U_{i}, q′|U_{i})
ψ_{b_{i}} ↓≅  ≅↓^{ψ_{b′_{i}}}
Λ^{even}(V |U_{i}) ⟶_{ϕ_{Λ{ev \atop i}}}^{≅} Λ^{even}(V′|U_{i})

The linear isomorphism ϕ_{Λ_{i}^{ev}} preserves 1 and therefore it induces a linear isomorphism from Λ^{2}(V |U_{i}) to Λ^{2}(V′|U_{i}), which we denote by (ϕ_{Λ^{2}})_{i}. Observe that (ϕ_{Λ^{2}})_{i} is independent of the choice of the bilinear forms b_{i} and b′_{i}. For, replacing these respectively by b̂_{i} and b̂′_{i}, it follows from (2f), Theorem 2.1, that ψ_{b_{i}}∘(ψb̂_{i})^{−1} (resp. ψ_{b′_{i}}∘(ψb̂′_{i})^{−1}) followed by the canonical projection onto Λ^{2}(V |U_{i}) (resp. onto Λ^{2}(V′|U_{i})) is the same as the projection itself. By this observation, it is also clear that the isomorphisms {(ϕ_{Λ^{2}})_{i}}_{i∈I} agree on (any open affine subscheme of, and hence on all of) any intersection U_{i} ∩ U_{j}. Therefore they glue to give a global isomorphism of vector bundles ϕ_{Λ^{2}} : Λ^{2}(V) ≅ Λ^{2}(V′). Q.E.D, Prop.1.2.

Reduction of Proof of Theorem 1.1 to Theorem 1.3. We start with an isomorphism of

algebra-bundles ϕ : C_{0}(V, q) ≅ C_{0}(V′, q′), construct the isomorphism of vector bundles ϕ_{Λ^{2}} : Λ^{2}(V) ≅ Λ^{2}(V′) and keep the notations introduced in the proof of Prop.1.2. Firstly we deduce a linear isomorphism det((ϕ_{Λ^{2}})^{∨})^{−1} : det((Λ^{2}(V))^{∨}) ≅ det((Λ^{2}(V′))^{∨}). Since V and V′ are of rank 3, there are canonical isomorphisms η : Λ^{2}(V) ≡ V^{∨} ⊗ det(V) and η′ : Λ^{2}(V′) ≡ (V′)^{∨} ⊗ det(V′). It follows therefore that if we set L := det(V′) ⊗ (det(V))^{−1} then we get a discriminant line bundle (L, h) and a vector bundle isomorphism α : V′ ≅ V ⊗ L. Now for each i ∈ I, the bilinear form b_{i} ∈ Γ(U_{i}, Bil_{V}) induces, via α|U_{i} and (L|U_{i}, h|U_{i}) and (1), Prop.2.3, a bilinear form b′′_{i} ∈ Γ(U_{i}, Bil_{V′}). By (3) of the same Proposition, over each U_{i} we get an isometry of bilinear form bundles α|U_{i} : (V′|U_{i}, b′′_{i}) ≅ (V |U_{i}, b_{i}) ⊗ (L|U_{i}, h|U_{i}) and also an isometry of quadratic bundles α|U_{i} : (V′|U_{i}, q_{b′′_{i}}) ≅ (V |U_{i}, q_{b_{i}} = q|U_{i}) ⊗ (L|U_{i}, h|U_{i}). On the other hand, by an assertion in (3), Prop.2.3, we could also define the global quadratic bundle (V′, q′′) using (V, q), α and (L, h), so that we have an isometry of quadratic bundles α : (V′, q′′) ≅ (V, q) ⊗ (L, h). It follows therefore that the q_{b′′_{i}} glue to give q′′. Notice that in general the b′′_{i} (resp. the b_{i}) need not glue to give a global bilinear form b′′ (resp. b) such that q_{b′′} = q′′ (resp. q_{b} = q). By (1), Prop.2.4, there exists a unique isomorphism of 𝒪_{X}-algebra bundles C_{0}(α, 1) : C_{0}(V′, q′′) ≅ C_{0} ((V, q) ⊗ (L, h)) and by Prop.2.5 we have a unique isomorphism of algebra bundles γ_{(L,h)} : C_{0} ((V, q) ⊗ (L, h)) ≅ C_{0}(V, q). Therefore the composition of the following sequence of isomorphisms of algebra bundles on X

C_{0}(V′, q′′) ⟶^{C_{0}(α,1) (≅)} C_{0} ((V, q) ⊗ (L, h)) ⟶^{γ_{(L,h)} (≅)} C_{0}(V, q) ⟶^{ϕ (≅)} C_{0}(V′, q′)

is an element of Iso[C_{0}(V′, q′′), C_{0}(V′, q′)], which, granting Theorem 1.3, is induced by a similarity in Sim[(V′, q′′), (V′, q′)]. Therefore, we would have that (V′, q′′) and (V′, q′) are globally similar (which means that they differ by a discriminant bundle with underlying line bundle being trivial), and this combined with the fact that (V, q) and (V′, q′′) are isometric upto the discriminant bundle (L, h) (by the construction above), we would have that (V, q) and (V′, q′) also differ by a discriminant bundle. Therefore the proof of Theorem 1.1 reduces to the proof of Theorem 1.3.

Reduction of Theorem 1.3 to the Free Case: For a similarity g with multiplier l, we have C_{0}(g, l) given by (1), Prop.2.4, so that the map Sim[(V, q), (V, q′)] ⟶ Iso[C_{0}(V, q), C_{0}(V, q′)] mentioned in the statement is the natural g ↦ C_{0}(g, l). We shall show (locally in the Zariski topology and hence globally) the equality det ((C_{0}(g, l))_{Λ^{2}}) = l^{−3}det^{2}(g) so that Iso[(V, q), (V, q′)] and S-Iso[(V, q), (V, q′)] are respectively mapped into Iso′[C_{0}(V, q), C_{0}(V, q′)] and S-Iso[C_{0}(V, q), C_{0}(V, q′)] as claimed. This equality will be verified in Lemma 3.11 following the proof of the surjectivity of Sim[(V, q), (V, q′)] ⟶ Iso[C_{0}(V, q), C_{0}(V, q′)]. We start with an isomorphism of algebra-bundles ϕ : C_{0}(V, q) ≅ C_{0}(V, q′), which by the above discussion leads to the automorphism of vector bundles ϕ_{Λ^{2}} : Λ^{2}(V) ≅ Λ^{2}(V). Firstly, define the global bundle automorphism g′ ∈ GL (V ⊗ (det(V))^{−1}) so that the following diagram commutes:

(Λ^{2}(V))^{∨}  ⟶_{≅}^{((ϕ_{Λ^{2}} )^{∨})^{−1}} (Λ^{2}(V))^{∨}
(η^{∨})^{−1}↓^{≡}  ^{≡}↓_{(η^{∨})}^{−1}
V ⊗ (det(V))^{−1}  ⟶_{g′}^{≅} V ⊗ (det(V))^{−1}

where η : Λ^{2}(V) ≡ V^{∨} ⊗ det(V) is the canonical isomorphism (since V is of rank 3). Now let g ∈ GL(V) ⟵^{≅} GL(V ⊗ (det(V))^{−1}) be the image of g′ i.e., the image of g′ ⊗ det(V) under the canonical identification GL(V ⊗ (det(V))^{−1} ⊗ det(V)) ≡ GL(V). Next, let l ∈ Γ(X, 𝒪_{X}^{∗}) be a global section such that γ(l) := (l^{3}).det(ϕ_{Λ^{2}}) has a square root in Γ(X, 𝒪_{X}^{∗}). For example, we have the following special cases when this is true:

Case 1. If det(ϕ_{Λ^{2}}) is itself a square, set l := 1. If further det(ϕ_{Λ^{2}}) = 1, set \sqrt{γ(l)} = 1, otherwise let \sqrt{γ(l)} denote any fixed square root of det(ϕ_{Λ^{2}}).

Case 2. If det(ϕ_{Λ^{2}}) is not a square, given an integer k, take l = (det(ϕ_{Λ^{2}}))^{2k+1} and let \sqrt{γ(l)} denote any fixed square root of (det(ϕ_{Λ^{2}}))^{6k+4}.

For each integer k, we now associate to ϕ the element g_{l}^{ϕ} := (l^{−1}\sqrt{γ(l)} )g with g as defined above. We shall show the following locally for the Zariski toplogy on X (more precisely, for each open subscheme of X over which V is free):

(1) that g_{l}^{ϕ} is a similitude from q to q′ with multiplier l (Lemma 3.9);

(2) that g_{l}^{ϕ} induces ϕ i.e., with the notations of (1), Prop.2.4, that C_{0}(g_{l}^{ϕ}, l) = ϕ (Lemma 3.10);

(3) that det(g_{l}^{ϕ}) = \sqrt{γ(l)} so that det^{2}(g_{l}^{ϕ}) = det(ϕ_{Λ^{2}}) when det(ϕ_{Λ^{2}}) is itself a square (Lemma 3.8) and

(4) that the map S-Iso[(V, q), (V, q′)] ⟶ S-Iso[C_{0}(V, q), C_{0}(V, q′)] is injective (Lemma 3.12).

It would follow then that these statements are also true globally. The maps s_{2k+1} : ϕ ↦ g_{l}^{ϕ} with l as in Case 2 and s′ : ϕ ↦ g_{l}^{ϕ} with l as in Case 1 will then give the sections to the maps (which would imply their surjectivities) as mentioned in Theorem 1.3. But these maps are not necessarily multiplicative since a computation reveals that if ϕ_{i} ∈ Iso[C_{0}(V, q_{i}), C_{0}(V, q_{i+1})] is associated to g_{l_{i}}^{ϕ_{i}} ∈ Sim[(V, q_{i}), (V, q_{i+1})], and ϕ_{2} ∘ ϕ_{1} to g_{l_{21}}^{ϕ_{2}∘ϕ_{1}}, then g_{l_{21}}^{ϕ_{2}∘ϕ_{1}} = δg_{l_{2}}^{ϕ_{2}} ∘ g_{l_{1}}^{ϕ_{1}} for δ ∈ μ_{2}(Γ(X, 𝒪_{X}^{∗})) because of the ambiguity in the initial global choices of square roots for γ(l_{i}) and γ(l_{21}). However this can be remedied as follows. For any given ϕ ∈ Iso[C_{0}(V, q), C_{0}(V, q′)], irrespective of whether or not det(ϕ_{Λ^{2}}) is a square, take

l = (det(ϕ_{Λ^{2}}))^{2k+1}, γ(l) = l^{3}det(ϕ_{Λ^{2}}), \sqrt{γ(l)} := (det(ϕ_{Λ^{2}}))^{3k+2} and s_{2k+1}^{+}(ϕ) := g_{l}^{ϕ} = (l^{−1}\sqrt{γ(l)}) g.

Then it is clear that each s_{2k+1}^{+} is multiplicative with the properties as claimed in the statement. We thus reduce the proof of Theorem 1.3 to the case when the rank 3 vector bundle V is free. This will be taken up next in §3.

3 Investigation of Θ ; Proof of Theorem 1.4

In this section, we conclude the proofs of Theorems 1.1 and 1.3 which were begun in §2 and also prove Theorem 1.4. As means to these ends, we carry out two explicit computations. Firstly we compute the isomorphism Θ of Theorem 1.11. This provides us with the multiplication table of every specialised algebra structure on any fixed free rank 4 vector bundle with fixed unit which is part of a global basis (Theorem 3.1 below). This result will also be used in §4 in the proof of Theorem 1.12. It turns out that Θ is not equivariant with respect to GL_{V}, but nevertheless satisfies a ‘twisted’ form of equivariance (Theorem 3.4). Secondly, we explicitly compute the algebra bundle isomorphism C_{0}(g, l) : C_{0}(V, q) ≅ C_{0}(V, q′) of (1), Prop.2.4 induced by a similarity g : (V, q) ≅_{l} (V, q′) with multiplier l ∈ Γ(X, 𝒪_{X}^{∗}) in the case when V is free of rank 3 (Theorem 3.5).

The Action of GL_{V} . Let V be a vector bundle over a scheme X with associated locally-free sheaf 𝒱. The X-smooth X-groupscheme GL_{V} acts naturally on the left on the sheaves Alt_{𝒱}^{2} , Bil_{𝒱} and Quad_{𝒱} of alternating, bilinear and quadratic forms on 𝒱. Namely, for U ↪ X an open subscheme, and for b ∈ Γ(U, Bil_{𝒱} ) (resp. a ∈ Γ(U, Alt_{𝒱}^{2} ), resp. q ∈ Γ(U, Quad_{𝒱} )), and for g ∈ Γ(U, GL_{V}) = GL(V |U), the corresponding form of the same type g.b (resp. g.a, resp. g.q) is defined on sections (over open subsets of U) by (g.b)(v, v′) := b(g^{−1}(v), g^{−1}(v′)) (resp.(g.a)(v, v′) := a(g^{−1}(v), g^{−1}(v′)), resp. (g.q)(v) := q(g^{−1}(v))). It is immediate that the following short-exact-sequence of sheaves, indicated in the discussion before Prop.2.2, is equivariant with respect to this action:

(♠) 0 ⟶ Alt_{𝒱}^{2} ⟶ Bil_{𝒱} ⟶ Quad_{𝒱} ⟶ 0.

Equivalently, the X-groupscheme GL_{V} acts on the corresponding geometric vector bundles such that both of the X-morphisms of X-vector bundles in the following sequence are GL_{V} -equivariant:

Alt_{V}^{2} ↪ Bil_{V} ↠ Quad_{V} .

Notice that it is one and the same thing to require that GL(V |U) ∋ g : (V |U, q) ≅_{l} (V |U, q′) be a similitude with multiplier l ∈ Γ(U, 𝒪_{X}), and to require that g.q = l^{−1}q′.

Definition of the isomorphism Θ. We briefly recall the definition of Θ from Part A of [3]. We keep the notations introduced just before Theorem 1.11; for ease of notation, the pullback of a section s (of a vector bundle or its associated sheaf) is denoted by s^{∘}. Since V is free of rank 3 on X, we choose an identification 𝒱 ≡ 𝒪_{X}.e_{1} ⊕ 𝒪_{X}.e_{2} ⊕ 𝒪_{X}.e_{3}. This gives the identification of the dual bundle as 𝒱^{∨} ≡ 𝒪_{X}.f_{1} ⊕𝒪_{X}.f_{2} ⊕𝒪_{X}.f_{3} (defined uniquely by f_{i}(e_{j}) = δ_{ij}, the Kronecker delta). Therefore the dual of the sheaf of quadratic forms on V, which is (Quad_{𝒱} )^{∨} := (Bil_{𝒱} /Alt_{𝒱}^{2} )^{∨} = ((T^{2}𝒱)^{∨}/(Λ^{2}𝒱)^{∨})^{∨} has global 𝒪_{X}- basis given by {e_{i}⊗e_{i}; (e_{i}⊗e_{j}+e_{j}⊗e_{i})}. This leads to an identification of the associated sheaf of symmetric algebras Sym_{𝒪_{X}} (Quad_{𝒱}^{∨}) ≡ 𝒪_{X}[Y_{1}, Y_{2}, Y_{3}, Y_{12}, Y_{13}, Y_{23}], where e_{i} ⊗ e_{i} ≡ Y_{i} and e_{i} ⊗ e_{j} + e_{j} ⊗ e_{i} ≡ Y_{ij}, and therefore Quad_{V} := Spec (Sym_{𝒪_{X}} (Quad_{𝒱}^{∨})) ≡ Spec(𝒪_{X}[Y_{1}, Y_{2}, Y_{3}, Y_{12}, Y_{13}, Y_{23}]) = 𝔸_{X}^{6}. Consider the universal quadratic bundle (V, q) where V is the pullback of V by Quad_{V} ⟶ X. The universal quadratic form q is given by (x_{1}, x_{2}, x_{3}) ↦ Σ_{i}Y_{i}.(x_{i})^{2} + Σ_{i<j}Y_{ij}.x_{i}.x_{j} and moreover the global bilinear form on V given by

b(q) : ((x_{1}, x_{2}, x_{3}), (x′_{1}, x′_{2}, x′_{3})) ↦ Σ_{i}Y_{i}.x_{i}.x′_{i} + Y_{12}.x_{2}.x′_{1} + Y_{23}.x_{3}.x′_{2} + Y_{13}.x_{1}.x′_{3}

induces q (the bilinear form ‘associated in the usual sense’ to q, viz. b_{q} is not b(q) but in fact its symmetrisation). Therefore, by (2d), Theorem 2.1, we get an isomorphism of vector bundles ψ_{b(q)} : C_{0}(V, q = q_{b(q)}) ≅ Λ^{even}(V) =: W which, according to (2a) and (2f) of the same Theorem, carries the ordered Poincaré-Birkhoff-Witt basis {1; e_{1}^{∘}.e_{2}^{∘}, e_{2}^{∘}.e_{3}^{∘}, e_{3}^{∘}.e_{1}^{∘}} onto the corresponding ordered basis of the even exterior algebra (=even Clifford algebra of the zero quadratic form on V) given by {w^{∘} = 1^{∘} = 1; e_{1}^{∘}∧e_{2}^{∘}, e_{2}^{∘}∧e_{3}^{∘}, e_{3}^{∘}∧e_{1}^{∘}}. The choices e_{3}^{∘}.e_{1}^{∘} and e_{3}^{∘}∧e_{1}^{∘} instead of the usual e_{1}^{∘}.e_{3}^{∘} and e_{1}^{∘}∧e_{3}^{∘} are deliberate— for example, ψ_{b(q)} would carry {1; e_{1}^{∘}.e_{2}^{∘}, e_{2}^{∘}.e_{3}^{∘}, e_{1}^{∘}.e_{3}^{∘}} onto {w^{∘} = 1^{∘} = 1; e_{1}^{∘} ∧ e_{2}^{∘}, e_{2}^{∘} ∧ e_{3}^{∘}, e_{1}^{∘} ∧ e_{3}^{∘} + Y_{13}.w^{∘}} which depends on Y_{13}. Thus the even Clifford algebra bundle C_{0}(V, q = q_{b(q)}) induces via ψ_{b(q)} a w^{∘}-unital algebra structure on the pullback bundle W of W := Λ^{even}(V) (where w corresponds to 1 in Λ^{even}(V)). But by definition, this algebra structure corresponds precisely to an X-morphism θ : Quad_{V} ⟶ Id-w-Sp-Azu_{W} . The isomorphism Θ is now given by the composition of the following Xmorphisms (cf. Def.5.2, Part A, [3]):

Quad_{V} ×_{X} L_{w} ⟶^{θ×ID} Id-w-Sp-Azu_{W} ×_{X} L_{w} ⟶^{SWAP (≅)} L_{w} ×_{X} Id-w-Sp-Azu_{W} ⟶^{ACTION} Id-w-Sp-Azu_{W} .

The association of q with b(q) also defines a splitting of the exact sequence (♠) above, so that more generally, given a valued point q ∈ (Quad_{V})(T), we may associate uniquely a valued point b(q) ∈ (Bil_{V})(T) which induces it. That this association is not GL_{V} -equivariant is reflected in the lack of equivariance of the isomorphism Θ (Theorem 3.4).

THEOREM 3.1 Let T be an X-scheme. Let q be a T -valued point of Quad_{V} ≡ 𝔸_{X}^{6} which is identified uniquely with a 6-tuple (λ_{1}, λ_{2}, λ_{3}, λ_{12}, λ_{13}, λ_{23}) corresponding to the quadratic form (x_{1}, x_{2}, x_{3}) ↦ Σ_{i}λ_{i}x_{i}^{2} + Σ_{i<j}λ_{ij}x_{i}x_{j}. Let \underline{t} be a T -valued point of L_{w} ≡ (𝔸_{X}^{3}, +) which is identified uniquely with a 3-tuple (t_{1}, t_{2}, t_{3}) that corresponds to the T -valued point of Stab_{w} given by the (4 × 4)-matrix

( \begin{matrix}1 & t_{1} & t_{2} & t_{3} \\ 0 & & I_{3} \end{matrix} )

where I_{3} is the (3 × 3)-identity matrix. Then the multiplication table for the specialised algebra structure Θ(q, \underline{t}) = \underline{t}.θ(q) on the pullback bundle W_{T} with unit w^{∘} = w_{T} is given as follows (in terms of the global basis {w^{∘} = 1^{∘} = 1 ; ϵ_{1}^{∘} := e_{1}^{∘} ∧ e_{2}^{∘} , ϵ_{2}^{∘} := e_{2}^{∘} ∧ e_{3}^{∘} , ϵ_{3}^{∘} := e_{3}^{∘} ∧ e_{1}^{∘}} induced from that of W = Λ^{even}(V) :)

• ϵ_{1}^{∘}.ϵ_{1}^{∘} = (t_{1}λ_{12} − λ_{1}λ_{2} − t_{1}^{2}).w^{∘} + (λ_{12} − 2t_{1}).ϵ_{1}^{∘};

• ϵ_{2}^{∘}.ϵ_{2}^{∘} = (t_{2}λ_{23} − λ_{2}λ_{3} − t_{2}^{2}).w^{∘} + (λ_{23} − 2t_{2}).ϵ_{2}^{∘};

• ϵ_{3}^{∘}.ϵ_{3}^{∘} = (t_{3}λ_{13} − λ_{1}λ_{3} − t_{3}^{2}).w^{∘} + (λ_{13} − 2t_{3}).ϵ_{3}^{∘};

• ϵ_{1}^{∘}.ϵ_{2}^{∘} = (λ_{2}λ_{13} − λ_{2}t_{3} − t_{1}t_{2}).w^{∘} − t_{2}ϵ_{1}^{∘} − t_{1}ϵ_{2}^{∘} − λ_{2}ϵ_{3}^{∘};

• ϵ_{2}^{∘}.ϵ_{3}^{∘} = (λ_{3}λ_{12} − λ_{3}t_{1} − t_{2}t_{3}).w^{∘} − λ_{3}ϵ_{1}^{∘} − t_{3}ϵ_{2}^{∘} − t_{2}ϵ_{3}^{∘};

• ϵ_{3}^{∘}.ϵ_{1}^{∘} = (λ_{1}λ_{23} − λ_{1}t_{2} − t_{1}t_{3}).w^{∘} − t_{3}ϵ_{1}^{∘} − λ_{1}ϵ_{2}^{∘} − t_{1}ϵ_{3}^{∘};

• ϵ_{2}^{∘}.ϵ_{1}^{∘} = (λ_{2}t_{3} − (λ_{12} − t_{1})(λ_{23} − t_{2})).w^{∘} + (λ_{23} − t_{2})ϵ_{1}^{∘} + (λ_{12} − t_{1})ϵ_{2}^{∘} + λ_{2}ϵ_{3}^{∘};

• ϵ_{3}^{∘}.ϵ_{2}^{∘} = (λ_{3}t_{1} − (λ_{13} − t_{3})(λ_{23} − t_{2})).w^{∘} + λ_{3}ϵ_{1}^{∘} + (λ_{13} − t_{3})ϵ_{2}^{∘} + (λ_{23} − t_{2})ϵ_{3}^{∘};

• ϵ_{1}^{∘}.ϵ_{3}^{∘} = (λ_{1}t_{2} − (λ_{12} − t_{1})(λ_{13} − t_{3})).w^{∘} + (λ_{13} − t_{3})ϵ_{1}^{∘} + λ_{1}ϵ_{2}^{∘} + (λ_{12} − t_{1})ϵ_{3}^{∘}.

Proof of Theorem 3.1: For clarity, let ∗_{q} denote the multiplication in C_{0}(V_{T} , q), and for uniformity, let ϵ_{0} := w. Since q = q_{b(q)}, we have by (2d), Theorem 2.1, the isomorphism ψ_{b(q)} : C_{0}(V_{T} , q) ≅ Λ^{even}(V_{T}) = W_{T} . Let ∗_{b(q)} denote the product in the algebra structure θ(q) thus induced on W_{T} . Since the ϵ_{i}^{∘} are a basis for W_{T}, it is enough to compute the products ϵ_{i}^{∘} ∗_{b(q)} ϵ_{j}^{∘} for 1 ≤ i, j ≤ 3. For example, consider the product ϵ_{2}^{∘} ∗_{b(q)} ϵ_{1}^{∘}. Using the properties of the multiplication in C(V_{T} , q), and the properties of the isomorphism ψ_{b(q)} from (2), Theorem 2.1, we get the following:

ϵ_{2}^{∘} ∗_{b(q)} ϵ_{1}^{∘} = ψ_{b(q)} ({ψ_{b(q)}^{−1}(e_{2}^{∘} ∧ e_{3}^{∘})} ∗_{q} {ψ_{b(q)}^{−1}(e_{1}^{∘} ∧ e_{2}^{∘})})
= ψ_{b(q)} ((e_{2}^{∘} ∗_{q} e_{3}^{∘}) ∗_{q} (e_{1}^{∘} ∗_{q} e_{2}^{∘}))
= ψ_{b(q)} ((λ_{23}(1^{∘}) − e_{3}^{∘} ∗_{q} e_{2}^{∘}) ∗_{q} (λ_{12}(1^{∘}) − e_{2}^{∘} ∗_{q} e_{1}^{∘}))
= ψ_{b(q)} (λ_{23}λ_{12}(1^{∘}) − λ_{23}e_{2}^{∘} ∗_{q} e_{1}^{∘} − λ_{12}e_{3}^{∘} ∗_{q} e_{2}^{∘} + (e_{3}^{∘} ∗_{q} e_{2}^{∘}) ∗_{q} (e_{2}^{∘} ∗_{q} e_{1}^{∘}))

= ψ_{b(q)} (λ_{23}λ_{12}(1^{∘}) − λ_{23}(λ_{12}(1^{∘}) − e_{1}^{∘} ∗_{q} e_{2}^{∘}) − λ_{12}(λ_{23}(1^{∘}) − e_{2}^{∘} ∗_{q} e_{3}^{∘}) + e_{3}^{∘} ∗_{q} (e_{2}^{∘} ∗_{q} e_{2}^{∘}) ∗_{q} e_{1}^{∘})

= (−λ_{12}λ_{23})w^{∘} + λ_{23}ϵ_{1}^{∘} + λ_{12}ϵ_{2}^{∘} + λ_{2}ϵ_{3}^{∘}.

In a similar fashion, the other products may be computed; this amounts to computing θ on T -valued points. The following result is needed to compute Θ from θ.

Lemma 3.2 Let ∗_{(b(q),\underline{t})} denote the multiplication in the algebra Θ(q, \underline{t}) = \underline{t}.θ(q) and as before, ∗_{b(q)} denote the multiplication in θ(q). Then we have

1. \underline{t}(ϵ_{i}^{∘}) = t_{i}w^{∘} + ϵ_{i}^{∘} for 1 ≤ i ≤ 3;

2. (\underline{t})^{−1}(ϵ_{i}^{∘}) = −t_{i}w^{∘} + ϵ_{i}^{∘} for 1 ≤ i ≤ 3;

3. ϵ_{i}^{∘} ∗_{(b(q),\underline{t})} ϵ_{j}^{∘} = \underline{t}(ϵ_{i}^{∘} ∗_{b(q)} ϵ_{j}^{∘}) − t_{j}ϵ_{i}^{∘} − t_{i}ϵ_{j}^{∘} − t_{i}t_{j}w^{∘}.

While the first two of the above formulae follow easily by direct computation, the third follows by using the first two alongwith the following:

ϵ_{i}^{∘} ∗_{(b(q),\underline{t})} ϵ_{j}^{∘} = \underline{t} ( (\underline{t}^{−1}(ϵ_{i}^{∘})) ∗_{b(q)} (\underline{t}^{−1}(ϵ_{j}^{∘})) ) .

We may now compute the multiplication in the algebra Θ(q, \underline{t}) = \underline{t}.θ(q) by making use of the formulas listed in the above lemma and the expressions for the products of the form ϵ_{i}^{∘} ∗_{b(q)} ϵ_{j}^{∘} whose computation had already been illustrated before the lemma. Q. E. D., Theorem 3.1.

Computation of C_{0}(g, l) in the free case. We continue with the notations introduced above. In the following we study the lack of equivariance of the isomorphism Θ relative to GL_{V} and show that it satisfies a curious ‘twisted’ version of equivariance. Firstly we consider the morphism of X-groupschemes Λ^{even} : GL_{V} ⟶ Stab_{w} given on valued points by g ↦ Λ^{even}(g). Recall that Λ^{even}(V) =: A_{0} ∈ Id-w-Sp-Azu_{W} (X) is the even graded part of the Clifford algebra of the zero quadratic form on V. A simple computation reveals the following result. Lemma 3.3 For each X-scheme T, define the map

GL(V_{T}) ⟶ Stab((A_{0})_{T}) : g ↦ (\begin{matrix}1 & 0 \\ 0 & B(g) \end{matrix}) ∈ Stab(w_{T})

where B(g) := det(g) (E_{12}E_{23}(g^{−1})^{t}E_{23}E_{12}) with

E_{12} = (\begin{matrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix}) and E_{23} = (\begin{matrix}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix}) .

Then the above maps define a morphism of X-groupschemes which in fact is none other than Λ^{even} : GL_{V} ⟶ Stab_{w}; in other words: B(g) = Λ^{2}(g). Recall from Lemma 5.1, Part A, [3], that Stab_{w} is the semidirect product of Stab_{A_{0}} and L_{w}, so that Stab_{A_{0}} naturally acts on L_{w} by “outer conjugation”. Let GL_{V} act on L_{w} via the homomorphism Λ^{even} i.e., for g ∈ GL(V_{T}) and \underline{t} ∈ L_{w}(T),

g.\underline{t} := Λ^{even}(g).\underline{t} := Λ^{even}(g) \underline{t} Λ^{even}(g^{−1}).

Any element h ∈ Stab(w_{T}) can be uniquely written as h = h_{s}h_{l} = h′_{l}h_{s} where h_{s} ∈ Stab((A_{0})_{T}) and h_{l}, h′_{l} ∈ L_{w}(T). Then the relation between h_{l} and h′_{l} can be written as h′_{l} = h_{s}.h_{l} or h_{l} = h_{s}^{−1}.h′_{l} where “.” stands for the action of Stab_{A_{0}} on L_{w}. Thus one has a GL_{V} -action on Quad_{V} ×_{X} L_{w} induced by

the diagonal embedding GL_{V} ↪^{Δ} GL_{V} ×_{X} GL_{V} . Since Id-w-Sp-Azu_{W} comes with a canonical action of Stab_{w} on it, we let GL_{V} act on Id-w-Sp-Azu_{W} via Λ^{even}. The following result describes the lack of GL_{V} -equivariance of the isomorphism Θ.

THEOREM 3.4 Let T be an X-scheme. For T -valued points g, q, \underline{t} respectively of GL_{V}, Quad_{V}, and L_{w}, there exists a unique T -valued point of L_{w} given by an isomorphism h′_{l}(g, q) of 𝒪_{T} -algebra bundles

h′_{l}(g, q) : g.Θ(q, \underline{t}) ⟶^{≅} Θ(g.q, g.\underline{t}).

Further, h′_{l}(g, q) satisfies the formula

h′_{l}(gg′, q) = h′_{l}(g, g′.q)(g.h′_{l}(g′, q)).

Therefore Θ satisfies a ‘twisted’ version of GL_{V} -equivariance. The next theorem, which was originally motivated by the proof of this ‘twisted equivariance’, will be of central importance to us for the rest of this section. THEOREM 3.5 Given a similarity g : (V_{T} , q) ≅_{l} (V_{T} , q′) with multiplier l ∈ Γ(T, 𝒪_{T}^{∗}), let h(g, l, q, q′) be the automorphism of (W_{T} , w_{T}) given by the composition of the following isomorphisms:

W_{T} ⟶^{(ψ_{b(q)})^{−1} (≅)} C_{0}(V_{T} , q) ⟶^{C_{0}(g,l) (≅)} C_{0}(V_{T} , q′) ⟶^{ψ_{b(q′)} (≅)} W_{T}

where the algebra bundle isomorphism C_{0}(g, l) comes from (1), Prop.2.4 and the linear isomorphisms ψ_{b(q)} and ψ_{b(q′)} come from (2d), Theorem 2.1. In terms of actions, this means that h(g, l, q, q′).θ(q) = θ(q′). Write h(g, l, q, q′) ∈ Stab(w_{T}) uniquely as a product

h(g, l, q, q′) = h_{s}(g, l, q, q′)h_{l}(g, l, q, q′)

with the first factor in Stab_{A_{0}}(T) and the second in L_{w}(T) as explained earlier. Then h_{s}(g, l, q, q′) depends only on g and l and not on q or q′. In fact, one has

h_{s}(g, l, q_{1}, q_{2}) = h_{s}(g, l) := (\begin{matrix}1 & 0 \\ 0 & l^{−1}Λ^{2}(g) \end{matrix}) ∀q_{1}, q_{2} ∈ Quad(V_{T}).

Proof: We directly compute the 𝒪_{T} -linear automorphism h(g, l, q, q′) of W_{T} as follows. Of course, this automorphism fixes w^{∘} = w_{T} . So we need to only compute the images of the three remaining basis elements ϵ_{1}^{∘} = e_{1}^{∘} ∧ e_{2}^{∘}, ϵ_{2}^{∘} = e_{2}^{∘} ∧ e_{3}^{∘} and ϵ_{3}^{∘} = e_{3}^{∘} ∧ e_{1}^{∘} in terms of the basis elements w^{∘} and ϵ_{i}^{∘}. Let q and l(g.q) = q′ respectively correspond to the 6-tuples (μ_{1}, μ_{2}, μ_{3}, μ_{12}, μ_{13}, μ_{23}) and (μ′_{1}, μ′_{2}, μ′_{3}, μ′_{12}, μ′_{13}, μ′_{23}) ∈ Γ(T, 𝒪_{T} ^{⊕6}). (We caution the reader that l(g.q) ≠ (lg).q = l^{−2}(g.q)!) Let g ∈ GL(V_{T}) ≡ GL_{3}(Γ(T, 𝒪_{T})) be given by the matrix (g_{ij}). Observe that the μ′ are polynomials in the μ and g_{ij}. In the following computation, for the sake of clarity, we denote the product in C(V_{T} , q) by ∗_{q}. For example, we have

h(g, l, q, q′)ϵ_{1} = ψ_{b(q′)} ∘ C_{0}(g, l) ((ψ_{b(q)})^{−1}(e_{1}^{∘} ∧ e_{2}^{∘})) = ψ_{b(q′)} (C_{0}(g, l) (e_{1}^{∘} ∗_{q} e_{2}^{∘})) (by (2f), Theorem 2.1)
= ψ_{b(q′)} (l^{−1}(g(e_{1}^{∘}) ∗_{q′} g(e_{2}^{∘}))) (by (1), Prop.2.4)
= l^{−1}ψ_{b(q′)} ((g_{11}e_{1}^{∘} + g_{21}e_{2}^{∘} + g_{31}e_{3}^{∘}) ∗_{q′} (g_{12}e_{1}^{∘} + g_{22}e_{2}^{∘} + g_{32}e_{3}^{∘}))

= l^{−1}ψ_{b(q′)} ( (g_{11}g_{12}μ′_{1} + g_{21}g_{22}μ′_{2} + g_{31}g_{32}μ′_{3} + g_{21}g_{12}μ′_{12} + g_{31}g_{22}μ′_{23} + g_{11}g_{32}μ′_{13})w^{∘}+ + (g_{11}g_{22} − g_{21}g_{12})e_{1}^{∘} ∗_{q′} e_{2}^{∘} + (g_{21}g_{32} − g_{31}g_{22})e_{2}^{∘} ∗_{q′} e_{3}^{∘} + (g_{31}g_{12} − g_{11}g_{32})e_{3}^{∘} ∗_{q′} e_{1}^{∘} )

= l^{−1} (P_{1}(g, l, q, q′)w^{∘} + C_{33}(g)ϵ_{1}^{∘} + C_{13}(g)ϵ_{2}^{∘} + C_{23}(g)ϵ_{3}^{∘}) (by (2f), Theorem 2.1)

where P_{1}(g, l, q, q′) is the polynomial in the μ′ and g_{ij} (as computed in the previous step) and where C_{ij}(g) represents the cofactor determinant of the element g_{ij} of the matrix g = (g_{ij}). Similarly one computes the values of h(g, l, q, q′)ϵ_{2} and h(g, l, q, q′)ϵ_{3}. Then the matrix of h(g, l, q, q′) is given by

h(g, l, q, q′) = [\begin{matrix}1 & l^{−1}P_{1}(g, l, q, q′) & l^{−1}P_{2}(g, l, q, q′) & l^{−1}P_{3}(g, l, q, q′) \\ 0 & l^{−1}C_{33}(g) & l^{−1}C_{31}(g) & l^{−1}C_{32}(g) \\ 0 & l^{−1}C_{13}(g) & l^{−1}C_{11}(g) & l^{−1}C_{12}(g) \\ 0 & l^{−1}C_{23}(g) & l^{−1}C_{21}(g) & l^{−1}C_{22}(g) \end{matrix}]

which implies that

h_{s}(g, l, q, q′) = [\begin{matrix}1 & 0 & 0 & 0 \\ 0 & l^{−1}C_{33}(g) & l^{−1}C_{31}(g) & l^{−1}C_{32}(g) \\ 0 & l^{−1}C_{13}(g) & l^{−1}C_{11}(g) & l^{−1}C_{12}(g) \\ 0 & l^{−1}C_{23}(g) & l^{−1}C_{21}(g) & l^{−1}C_{22}(g) \end{matrix}] depends only on g and l.

Next define the matrix

ĝ = [\begin{matrix}g_{33} & g_{13} & g_{23} \\ g_{31} & g_{11} & g_{21} \\ g_{32} & g_{12} & g_{22} \end{matrix}] so that h_{s}(g, l, q, q′) = [\begin{matrix}1 & 0 \\ 0 & l^{−1}C(ĝ)^{t} \end{matrix}]

where C(ĝ) is the cofactor matrix of ĝ. Now if E_{12} and E_{23} are the matrices defined in Lemma 3.3 above, premultiplying by E_{ij} has the effect of interchanging the ith and jth rows, while postmultiplying has a similar effect on the columns. Thus we get ĝ = E_{12}E_{23}(g^{t})E_{23}E_{12} from which it follows that

C(ĝ)^{t} = Adjoint (ĝ) = det (ĝ).(ĝ)^{−1} = det (g). (E_{12}E_{23}(g^{−1})^{t}E_{23}E_{12}) ,

showing that C(ĝ)^{t} = Λ^{2}(g) by Lemma 3.3. Q.E.D., Theorem 3.5.

Proof of Theorem 3.4. Note that g is an isometry from (V_{T} , q) to (V_{T} , g.q) and hence according to (1), Prop.2.4, induces the algebra isomorphism C_{0}(g, l = 1) : C_{0}(V_{T} , q) ≅ C_{0}(V_{T} , g.q). Let h(g, q) := h(g, 1, q, g.q) where h(g, l, q, q′) was defined in Theorem 3.5 above. As explained in page 17, there are two canonical decompositions of Stab(w_{T}), one leading to the unique (ordered) decomposition of h(g, q) as h′_{l}(g, q)h_{s}(g, q), and the other leading to the unique ordered decomposition h_{s}(g, q)h_{l}(g, q). By Theorem 3.5 above, h_{s}(g, q_{1}) = h_{s}(g, q_{2}) = h_{s}(g, 1) = Λ^{even}(g) =: h_{s}(g) ∀q_{1}, q_{2} ∈ Quad(V_{T}) and hence we get:

Θ(g.q, g.\underline{t}) := (g.\underline{t}).θ(g.q)
= (Λ^{even}(g)\underline{t}Λ^{even}(g^{−1})).(h(g, 1, q, g.q).θ(q))
= (h_{s}(g)\underline{t}h_{s}^{−1}(g)).(h(g, q).θ(q))
= ((h_{s}(g)\underline{t}h_{s}^{−1}(g))(h_{s}(g, q)h_{l}(g, q))).θ(q)
= (h_{s}(g)\underline{t}h_{l}(g, q)).θ(q)
= (h_{s}(g)h_{l}(g, q)).(\underline{t}.θ(q))
= (h_{s}(g, q)h_{l}(g, q)).(Θ(q, \underline{t}))
= (h′_{l}(g, q)h_{s}(g, q)).(Θ(q, \underline{t}))
= h′_{l}(g, q).(h_{s}(g, q).Θ(q, \underline{t}))
= h′_{l}(g, q).(g.Θ(q, \underline{t})).

Note that h_{l}(g, q) was explicitly computed in the proof of Theorem 3.5 above to be

h_{l}(g, q) = \left[{1 P_{1}(g, 1, q, g.q) P_{2}(g, 1, q, g.q) P_{3}(g, 1, q, g.q) \atop 0 I_{3}}\right] ∈ L_{w}(T).

The formula for h′_{l}(g_{1}g_{2}, q) stated in the theorem is gotten thus:

\begin{aligned}h′_{l}(g_{1}g_{2}, q) & = (h′_{l}(g_{1}g_{2}, q)h_{s}(g_{1}g_{2}))h_{s}^{−1}(g_{1}g_{2}) \\ & = h(g_{1}g_{2}, q)h_{s}^{−1}(g_{1}g_{2}). \end{aligned}

Now by (3) of Prop.2.4 it follows that

h′_{l}(g_{1}g_{2}, q) = (h(g_{1}, g_{2}.q)h(g_{2}, q))h_{s}^{−1}(g_{1}g_{2})
= h′_{l}(g_{1}, g_{2}.q)h_{s}(g_{1}, g_{2}.q)h_{s}(g_{2}, q)h_{l}(g_{2}, q)h_{s}^{−1}(g_{1}g_{2})
= h′_{l}(g_{1}, g_{2}.q)h_{s}(g_{1})h_{s}(g_{2})h_{l}(g_{2}, q)h_{s}^{−1}(g_{1}g_{2})
= h′_{l}(g_{1}, g_{2}.q)h_{s}(g_{1}g_{2})h_{l}(g_{2}, q)h_{s}^{−1}(g_{1}g_{2})
= h′_{l}(g_{1}, g_{2}.q)((g_{1}g_{2}).h_{l}(g_{2}, q)).

But g_{2}^{−1}.h′_{l}(g_{2}, q) = h_{s}^{−1}(g_{2})(h′_{l}(g_{2}, q)h_{s}(g_{2})) = h_{s}^{−1}(g_{2})(h_{s}(g_{2})h_{l}(g_{2}, q)) = h_{l}(g_{2}, q) and therefore

\begin{aligned}h′_{l}(g_{1}g_{2}, q) & = h′_{l}(g_{1}, g_{2}.q).((g_{1}g_{2}).(g_{2}^{−1}.h′_{l}(g_{2}, q))) \\ & = h′_{l}(g_{1}, g_{2}.q).(g_{1}.h′_{l}(g_{2}, q)). \end{aligned}

Finally, one has to show the uniqueness of h′_{l}(g, q) ∈ L_{w}(T). Suppose h_{l} ∈ L_{w}(T) is also an algebra isomorphism h_{l} : g.Θ(q, \underline{t}) ⟶^{≅} Θ(g.q, g.\underline{t}), i.e., h_{l}.(g.Θ(q, \underline{t})) = Θ(g.q, g.\underline{t}). Notice that while showing the ‘twisted’ equivariance of Θ above, we have also proved that Θ(g.q, g.\underline{t}) = h(g, q)Θ(q, \underline{t}). Therefore we get

\begin{aligned}(h_{l}h_{s}(g)).(\underline{t}.θ(q)) & = h(g, q).Θ(q, \underline{t}) \\ ⇒ (h_{l}h_{s}(g)).(\underline{t}.θ(q)) & = (h_{s}(g)h_{l}(g, q)).Θ(q, \underline{t}) \\ ⇒ (h_{s}^{−1}(g)h_{l}h_{s}(g)).(\underline{t}.θ(q)) & = (h_{l}(g, q)\underline{t}).θ(q) \\ ⇒ Θ(q, (g^{−1}.h_{l})\underline{t}) & = Θ(q, h_{l}(g, q)\underline{t}). \end{aligned}

But since Θ is an isomorphism by Theorem 1.11, this implies that (g^{−1}.h_{l})\underline{t} = h_{l}(g, q)\underline{t} which gives

h_{l} = h_{s}(g)h_{l}(g, q)h_{s}^{−1}(g) = h′_{l}(g, q).

Q.E.D., Theorem 3.4.

Proofs of Theorems 1.3 and 1.1: We remind the reader that towards the end of §2, we had reduced the proof of Theorem 1.1 to that of Theorem 1.3, and had indicated in page 14 that it would be enough to prove the latter in the case when V is free—which has been the case in this section so far. Starting with an isomorphism of algebra bundles ϕ : C_{0}(V, q) ≅ C_{0}(V, q′) we arrive at the element g_{l}^{ϕ} ∈ GL(V) as defined in page 14; to briefly recall this, firstly g ∈ GL(V) was defined by the following commuting diagram:

(C_{0}(V, q))^{∨} ⟶_{≅}^{((ψ_{b(q)})^{∨})^{−1}} (Λ^{even}(V))^{∨} ⟶^{surjection} (Λ^{2}(V))^{∨} ⟶_{≡}^{(η^{∨})^{−1}} V ⊗ (det(V))^{−1} ⟶_{≡}^{⊗det(V )} V
ϕ^{∨}↑≅  ^{(ϕ_{Λ^{ev}})}^{∨}↑≅  ^{(ϕ_{Λ^{2}} )}^{∨}↑≅  _{(g′)}^{−1}↑≅  _{g^{−1}}↑≅
(C_{0}(V, q′))^{∨} ⟶_{≅}^{((ψ_{b(q′)})^{∨})^{−1}} (Λ^{even}(V))^{∨} ⟵^{inclusion} (Λ^{2}(V))^{∨} ⟶_{≡}^{(η^{∨})^{−1}} V ⊗ (det(V))^{−1} ⟶_{≡}^{⊗det(V )} V

Secondly, we had defined g_{l}^{ϕ} := (l^{−1}\sqrt{γ(l)} )g. Our current special choices of bilinear forms b(q) and b(q′) that induce q and q′ respectively do not affect the generality, as was observed in the proof of Prop.1.2. We shall now show that g_{l}^{ϕ} is a similitude from (V, q) to (V, q′) with multiplier l and that this similitude induces ϕ i.e., with the notations of (1), Prop. 2.4, that C_{0}(g_{l}^{ϕ}, l) = ϕ. We shall also check that det^{2}(g_{l}^{ϕ}) = det(ϕ_{Λ^{2}}) when det(ϕ_{Λ^{2}}) is itself a square. We proceed with the proof which will follow from several lemmas.

Lemma 3.6 Write the element h := ϕ_{Λ^{ev}} ∈ (Stab_{w})(X) uniquely as an ordered product h = h_{s}h_{l} = h′_{l}h_{s} where h_{l}, h′_{l} ∈ (L_{w})(X) and h_{s} ∈ (Stab_{A_{0}})(X) as explained in page 17; let B be the matrix corresponding to ϕ_{Λ^{2}}, and let the matrices E_{ij} be as defined in Lemma 3.3. Then we have matrix representations:

h_{s} = (\begin{matrix}1 & 0 \\ 0 & B \end{matrix}) and g_{l}^{ϕ} = (l^{−1}\sqrt{γ(l)}) E_{23}E_{12}((B)^{t})^{−1}E_{12}E_{23}

The proof of the above lemma follows from the fact that the matrix of the canonical isomorphism η : Λ^{2}(V) ≡ V^{∨} ⊗ det(V) is given by E_{23}E_{12}, which can be verified by a simple computation.

Lemma 3.7 h_{s}(Identity, l^{−1}, q′, l^{−1}q′) = (\begin{matrix}1 & 0 \\ 0 & l × I_{3} \end{matrix}) and h_{s}(Identity, l^{−1}, q′, l^{−1}q′).θ(q′) = θ(l^{−1}q′).

The identity map on V is obviously a similarity with multiplier l^{−1} from (V, q′) to (V, l^{−1}q′). Hence the above lemma follows by taking T = X, g = Identity, and the l^{−1} and the q′ at hand for the l and the q of Theorem 3.5 (caution: the q′ there would have to be replaced by l^{−1}q′!). This can also be seen directly from the multiplication tables for θ(l^{−1}q′) = Θ(l^{−1}q′, I_{4}) and θ(q′) = Θ(q′, I_{4}) written out in Theorem 3.1, where we must take T = X and \underline{t} = I_{4} i.e., t_{i} = 0 ∀ i. We observe from the multiplication table that each of the coefficients of ϵ_{i} for 1 ≤ i ≤ 3 is a single λ, whereas each coefficient of w = 1 = ϵ_{0} is a product of two λ s, and this observation implies the lemma above. As the reader might have noticed, there are two crucial facts about the identifications in this section; namely, firstly, for any X-scheme T, each of the maps ψ_{b(q)} (for different q) identify (C_{0}(V_{T} , q), 1) with the same (W_{T} , w_{T}) and secondly, relative to the bases chosen, all these identifying maps have trivial determinant. The latter is also true of the identification η, since it is given by the matrix E_{23}E_{12} (as was noted after Lemma 3.6). It therefore follows that det(ϕ) = det(ϕ_{Λ^{ev}}) = det(ϕ_{Λ^{2}}) = det(g′) = det(g) = det(B^{−1}). But we had chosen l ∈ Γ(X, 𝒪_{X}^{∗}) such that γ(l) := (l^{3}).det(ϕ_{Λ^{2}}) = l^{3}det(B). Using these facts alongwith Lemma 3.6 above, a straightforward computation gives the following.

Lemma 3.8 det(g_{l}^{ϕ}) = \sqrt{γ(l)} from which it follows that B(g_{l}^{ϕ}) = l × B where B(g_{l}^{ϕ}) and B are as defined in Lemmas 3.3 and 3.6 respectively. In particular, det^{2}(g_{l}^{ϕ}) = det(ϕ_{Λ^{2}}) when det(ϕ_{Λ^{2}}) is itself a square, since in this case we had chosen l := 1.

Lemma 3.9 g_{l}^{ϕ} is a similitude from (V, q) to (V, q′) with multiplier l.

The hypothesis ϕ : C_{0}(V, q) ≅ C_{0}(V, q′) is an algebra isomorphism translates in terms of actions into h.θ(q) = θ(q′) where h = ϕ_{Λ^{ev}} ∈ (Stab_{w})(X). Let h(g_{l}^{ϕ}, q) := h(g_{l}^{ϕ}, 1, q, g_{l}^{ϕ}.q) where h(g, l, q, q′) was defined in Theorem 3.5 above. Then we have

Θ(g_{l}^{ϕ}.q, I_{4}) := θ(g_{l}^{ϕ}.q) = h(g_{l}^{ϕ}, q).θ(q) = h(g_{l}^{ϕ}, q).(h^{−1}.θ(q′)) = (h′_{l}(g_{l}^{ϕ}, q)h_{s}(g_{l}^{ϕ}, q)h_{s}^{−1}(h′_{l})^{−1}) .θ(q′)

= (h′_{l}(g_{l}^{ϕ}, q) (\begin{matrix}1 & 0 \\ 0 & B(g_{l}^{ϕ}) \end{matrix}) (\begin{matrix}1 & 0 \\ 0 & B^{−1} \end{matrix}) (h′_{l})^{−1}) .θ(q′) (by Theorem 3.5; Lemmas 3.3 & 3.6)

= (h′_{l}(g_{l}^{ϕ}, q) (\begin{matrix}1 & 0 \\ 0 & l × I_{3} \end{matrix}) (h′_{l})^{−1}) .θ(q′) (by Lemma 3.8)

= (h′_{l}(g_{l}^{ϕ}, q) h_{s}(Identity, l^{−1}, q′, l^{−1}q′) (h′_{l})^{−1}) .θ(q′) (by Lemma 3.7)

= (h′_{l}(g_{l}^{ϕ}, q)h′′_{l}). (h_{s}(Identity, l^{−1}, q′, l^{−1}q′).θ(q′)) (since Stab_{w} is a semidirect product)

= (h′_{l}(g_{l}^{ϕ}, q)h′′_{l}).θ(l^{−1}q′)) (by Lemma 3.7)

=: Θ(l^{−1}q′, (h′_{l}(g_{l}^{ϕ}, q)h′′_{l}))

But since Θ is an isomorphism, this implies the claim of the above Lemma i.e., that g_{l}^{ϕ}.q = l^{−1}q′ (and further that h′_{l}(g_{l}^{ϕ}, q) = (h′′_{l})^{−1}).

Lemma 3.10 g_{l}^{ϕ} : (V, q)≅_{l}(V, q′) induces ϕ i.e., with the notations of (1), Prop.2.4, C_{0}(g_{l}^{ϕ}, l) = ϕ.

We have C_{0}(g_{l}^{ϕ}, l) = ϕ ⟺ h(g_{l}^{ϕ}, l, q, q′) := ψ_{b(q′)} ∘ C_{0}(g_{l}^{ϕ}, l) ∘ ψ_{b(q)}^{−1} = ψ_{b(q′)} ∘ ϕ ∘ ψ_{b(q)}^{−1} =: ϕ_{Λ^{ev}} =: h. Now using successively Theorem 3.5, Lemma 3.3, Lemma 3.8 and Lemma 3.6, we get the following sequence of equalities:

h_{s}(g_{l}^{ϕ}, l, q, q′) = (\begin{matrix}1 & 0 \\ 0 & l^{−1} × Λ^{2}(g_{l}^{ϕ}) \end{matrix}) = (\begin{matrix}1 & 0 \\ 0 & l^{−1} × B(g_{l}^{ϕ}) \end{matrix}) = (\begin{matrix}1 & 0 \\ 0 & B \end{matrix}) = h_{s}

Therefore the present hypotheses translated in terms of actions give

\begin{aligned}h(g_{l}^{ϕ}, l, q, q′).θ(q) = θ(q′) & = h.θ(q) \\ ⇒ h_{s}(g_{l}^{ϕ}, l, q, q′).(h_{l}(g_{l}^{ϕ}, l, q, q′).θ(q)) & = h_{s}.(h_{l}.θ(q)) \\ ⇒ Θ(q, h_{l}(g_{l}^{ϕ}, l, q, q′)) & = Θ(q, h_{l}). \end{aligned}

But Θ being an isomorphism, the last equality implies that h_{l}(g_{l}^{ϕ}, l, q, q′) = h_{l} which gives h(g_{l}^{ϕ}, l, q, q′) = h as wanted.

Lemma 3.11

(1) For a similarity g ∈ Sim[(V, q), (V, q′)] with multiplier l and the induced isomorphism C_{0}(g, l) ∈ Iso[C_{0}(V, q), C_{0}(V, q′)] given by (1), Prop.2.4, we have the equality det((C_{0}(g, l))_{Λ^{2}}) = l^{−3}det^{2}(g). Therefore the map

Sim[(V, q), (V, q′)] ⟶ Iso[C_{0}(V, q), C_{0}(V, q′)] : g ↦ C_{0}(g, l)

maps the subset Iso[(V, q), (V, q′)] into the subset Iso′[C_{0}(V, q), C_{0}(V, q′)] and S-Iso[(V, q), (V, q′)] into S-Iso[C_{0}(V, q), C_{0}(V, q′)].

(2) In the case q′ = q, if C_{0}(g, l) is the identity on C_{0}(V, q), then g = l^{−1}det(g) × Id_{V}, and further if g ∈ O(V, q) then g = det(g) × Id_{V} with det^{2}(g) = 1.

By definition, (C_{0}(g, l))_{Λ^{ev}} = ψ_{b(q′)} ∘ C_{0}(g, l) ∘ ψ_{b(q)}^{−1}, and the latter isomorphism is h(g, l, q, q′) from Theorem 3.5 which further gives a formula for h_{s}(g, l, q, q′). Now using the facts that ψ_{b(q)} and ψ_{b(q′)} have trivial determinant (as noted before Lemma 3.8) we get assertion (1):

det ((C_{0}(g, l))_{Λ^{2}}) = det ((C_{0}(g, l))_{Λ^{ev}} ) = det(h(g, l, q, q′)) = det(h_{s}(g, l, q, q′)) = l^{−3}det^{2}(g).

If q = q′ and C_{0}(g, l) is the identity, then the same argument in fact shows that l^{−1}Λ^{2}(g) = I_{3} and by using the formula in Lemma 3.3 for B(g) = Λ^{2}(g), we get g = l^{−1}det(g)I_{3}; taking determinants in the last equality gives det^{2}(g) = l^{3}, so that when g ∈ O(V, q) i.e., l = 1, det^{2}(g) ∈ μ_{2}(Γ(X, 𝒪_{X})) and assertion (2) follows.

Lemma 3.12 The map S-Iso[(V, q), (V, q′)] ⟶ S-Iso[C_{0}(V, q), C_{0}(V, q′)] : g ↦ C_{0}(g, 1, q, q′) is a bijection.

Given ϕ ∈ S-Iso[C_{0}(V, q), C_{0}(V, q′)], by the definition in Prop.1.2 we have det(ϕ_{Λ^{2}}) = 1, so by Lemma 3.8 det(g_{l}^{ϕ}) = \sqrt{γ(l)} := 1 by our earlier choice (see Case 1 on page 14). Therefore g_{l}^{ϕ} ∈ S-Iso[(V, q), (V, q′)] and is, according to Lemma 3.10, such that C_{0}(g_{l}^{ϕ}, l = 1, q, q′) = ϕ which gives the surjectivity. As for the injectivity, if g_{1}, g_{2} : (V, q)≅_{1}(V, q′) are isometries with determinant 1 such that C_{0}(g_{1}, 1, q, q′) = C_{0}(g_{2}, 1, q, q′), then we have h(g_{1}, 1, q, q′) = h(g_{2}, 1, q, q′) so that h_{s}(g_{1}, 1, q, q′) = h_{s}(g_{2}, 1, q, q′) whence by Theorem 3.5 and Lemma 3.3 B(g_{1}) = B(g_{2}) ⇒ g_{1} = g_{2}.

To sum up the above discussion: Let now V be a not-necessarily globally-trivial rank 3 vector bundle on the scheme X, and given ϕ, we define l, γ(l), \sqrt{γ(l)} and g_{l}^{ϕ} globally as before. Then the statements of Lemmas 3.9, 3.10, 3.11 and 3.12 are again valid, for the objects involved are defined globally, but the assertions are of a local nature. Given an isomorphism of algebra bundles ϕ ∈ Iso[C_{0}(V, q), C_{0}(V, q′)], when det(ϕ_{Λ^{2}}) ∉ (Γ(X, 𝒪_{X}^{∗}))^{2}, we have thus lifted ϕ to a family of similarities s_{2k+1}(ϕ) := g_{l}^{ϕ} ∈ Sim[(V, q), (V, q′)] with multipliers l := (det(ϕ_{Λ^{2}}))^{2k+1} parametrised by integers k; when 1 ≠ det(ϕ_{Λ^{2}}) ∈ (Γ(X, 𝒪_{X}^{∗}))^{2}, we have lifted ϕ to a similarity with multiplier l = 1 i.e., to an isometry s′(ϕ) := g_{l}^{ϕ} ∈ Iso[(V, q), (V, q′)] such that det(ϕ_{Λ^{2}}) = det^{2}(g_{l}^{ϕ}), and moreover, when ϕ_{Λ^{2}} has determinant 1, we have lifted it to a unique isometry g_{l}^{ϕ} with determinant 1. Of course, s_{2k+1} and s′ are multiplicative only upto an element of μ_{2}(Γ(X, 𝒪_{X}^{∗})), but this can be remedied by considering s_{2k+1}^{+} as noted at the end of section 2. Q.E.D., Theorems 1.3 & 1.1.

Proof of Theorem 1.4. Taking q′ = q in Theorem 1.3 gives the commutative diagram of groups and homomorphisms as asserted in the statement of the theorem. We continue with the notations used in the proof of Theorem 1.3. For g ∈ GO(V, q) with multiplier l, the equality det(C_{0}(g, l)) = det ((C_{0}(g, l))_{Λ^{2}}) = l^{−3}det^{2}(g) was proved in (1), Lemma 3.11. Assertion (2) of the same Lemma gives exactness at GO(V, q) and at O(V, q).

We proceed to prove assertion (b). Let ϕ ∈ Aut(C_{0}(V, q)), and consider the self-similarity s_{2k+1}^{+}(ϕ) = g_{l}^{ϕ} with multiplier l = det(ϕ)^{2k+1}. For the moment, assume that V is trivial over X with global basis {e_{1}, e_{2}, e_{3}}, and set e′_{i} = g_{l}^{ϕ}(e_{i}). It follows from Kneser’s definition of the half-discriminant d_{0}—see formula (3.1.4), Chap.IV, [2]—that d_{0}(q, {e_{i}}) = d_{0}(q, {e′_{i}}) det^{2}(g_{l}^{ϕ}). Since we have g_{l}^{ϕ}.q = l^{−1}q, a simple computation shows that d_{0}(q, {e′_{i}}) = l^{3}d_{0}(q, {e_{i}}). The hypothesis that q ⊗ κ(x) is semiregular means that the image of the element d_{0}(q, {e_{i}}) ∈ Γ(X, 𝒪_{X}) in κ(x) is nonzero. Since X is integral, we therefore deduce that det^{2}(g_{l}^{ϕ}) = l^{−3}. On the other hand, we know that det^{2}(g_{l}^{ϕ})l^{−3} = det(ϕ). It follows that det^{12k+7}(ϕ) = 1 ∀ k ∈ ℤ, which forces det(ϕ) = 1 as claimed. Q.E.D., Theorem 1.4.

4 Specialisations as Bilinear Forms: Theorems 1.9–1.12

In this section we reduce the proof of Theorem 1.9 to Theorem 1.10. We prove the latter and using it alongwith Theorem 3.1, deduce Theorem 1.12.

Preliminaries on the notion of Schematic Image.

Definition 4.1 (Defs.6.10.1-2, Chap.I, EGA I [6]) Let f : X ⟶ Y be a morphism of schemes. If there exists a smallest closed subscheme Y′ ↪ Y such that the inverse image scheme f^{−1}(Y′) := Y′ ×_{Y} (_{f} X) is equal to X, one calls Y′ the schematic image of f (or of X in Y under f). If X were a subscheme of Y and f the canonical immersion, and if f has a schematic image Y′, then Y′ is called the

schematic limit or the limiting scheme of the subscheme X ↪^{f} Y.

PROPOSITION 4.2 (Prop.6.10.5, Chap.I, EGA I) The schematic image Y′ of X by a morphism f : X ⟶ Y exists in the following two cases: (1) f_{∗}(𝒪_{X}) is a quasi-coherent 𝒪_{Y} -module, which is for example the case when f is quasi-compact and quasi-separated; (2) X is reduced.

PROPOSITION 4.3 In each of the following statements whenever a schematic image is mentioned, we assume that one of the two hypotheses of the above Prop. is true so that the schematic image does exist.

1. Let Y′ be the schematic image of X under a morphism f : X ⟶ Y and let f factor as X ⟶^{g} Y′ ↪^{j} Y. Then Y′ is topologically the closure of f(X) in Y, the morphism g is schematically dominant (i.e., g^{#} : 𝒪_{Y′} ⟶ g_{∗}(𝒪_{X}) is injective) and the schematic image of X in Y′ (under g) is Y′ itself. If X is reduced (respectively integral) then the same is true of Y′.

2. The schematic image of a closed subscheme under its canonical closed immersion is itself.

3. (Transitivity of Schematic Image) Let there be given morphisms X ⟶^{f} Y ⟶^{g} Z, such that the schematic image Y′ of X under f exists, and further such that if g′ is the restriction of g to Y′, the schematic image Z′ of Y′ by g′ exists. Then the schematic image of X under g ∘ f exists and equals Z′.

4. Let f : X ⟶ Y be a morphism which factors through a closed subscheme Y_{1} of Y by a morphism f_{1} : X ⟶ Y_{1}. Then the scheme-theoretic image Y′ of X in Y is the same as the scheme-theoretic image Y′_{1} of X in Y_{1} considered canonically as closed subscheme of Y.

5. If f : X ⟶ Y has a schematic image Y′ then f is schematically dominant iff Y′ = Y.

6. The formation of schematic image commutes with flat base change: i.e., if f : X ⟶ Y is a morphism of S-schemes which has a schematic image Y′ then for a flat morphism S′ ⟶ S, one has that the induced S′-morphism f ×_{S} S′ : X ×_{S} S′ ⟶ Y ×_{S} S′ has a schematic image and it may be canonically identified with Y′ ×_{S} S′. In particular this means that the formation of schematic image is local over the base.

Assertions (1) and (3) are respectively Prop.6.10.5 and Prop.6.10.3 in EGA I. The defining property of schematic image gives (2), while (4) can be deduced from the first three. As for (5), from (1) it follows that Y′ = Y implies f = g is schematically dominant. For the other way around, one uses the following characterisation of a schematically dominant morphism in Prop.5.4.1 of EGA I: if f : X ⟶ Y is a morphism of schemes, then f is schematically dominant iff for every open subscheme U of Y and every closed subscheme Y_{1} of U such that there exists a factorisation f^{−1}(U) ⟶^{g_{1}} Y_{1} ↪^{j_{1}} U, of the restriction f^{−1}(U) ⟶ U of f (where j_{1} is the canonical closed immersion), one has Y_{1} = U—given f is schematically dominant, one just has to take U = Y, Y_{1} = Y′ and g_{1} = g. Assertion (6) follows from statement (ii) (a) of Theorem 11.10.5 of EGA IV [7].

Preliminaries from Part A of [3]. Until further notice we assume that W is a vector bundle of fixed positive rank on the scheme X. Given any X-scheme T, by a T -algebra structure on W_{T} := W ×_{X} T (also referred to as T -algebra bundle), we mean a morphism W_{T} ×_{T} W_{T} ⟶ W_{T} of vector bundles on T arising from a morphism of the associated locally-free sheaves. Given such a T -algebra structure and T′ ⟶ T an X-morphism, it is clear that one gets by pullback (i.e., by base-change) a canonical T′- algebra structure on W_{T′}. Thus one has a contravariant “functor of algebra structures on W ” from {X − Schemes} to {Sets} denoted Alg_{W} whose set of T -valued points is the set of T -algebra structures on W_{T} viz. Hom_{𝒪_{T}} (𝒲_{T} ⊗ 𝒲_{T} , 𝒲_{T}) . By Prop.9.6.1, Chap.I of EGA I [6], it follows that the functor Alg_{W} is represented by the X-scheme

Alg_{W} := Spec (Sym_{X} [(𝒲_{X} ^{∨} ⊗_{X} 𝒲_{X} ^{∨} ⊗_{X} 𝒲_{X} )^{∨}]) .

Hence Alg_{W} is affine (hence separated), of finite presentation over X and in fact smooth of relative dimension rank_{X}(W)^{3}. If X′ ⟶ X is an extension of base, then the construction Alg_{W} base-changes well i.e., one may canonically identify Alg_{W} ×_{X} X′ with Alg_{W′} where W′ = W ×_{X} X′ (cf. Prop.9.4.11, Chap.I, EGA I [6]). We remark that an algebra structure may fail to be associative and may fail to have a (two-sided) identity element for multiplication. However, a multiplicative identity for an associative algebra structure must be a nowhere vanishing section (Lemma 2.3, and (2)⇒(4) of Lemma 2.4, Part A, [3]).

The general linear groupscheme associated to W viz GL_{W} naturally acts on Alg_{W} on the left, so that for each X-scheme T, Alg_{W} (T) mod GL_{W} (T) is the set of isomorphism classes of T -algebra structures on W ×_{X} T.

Let w ∈ Γ(X, 𝒲) be a nowhere vanishing section. For any X-scheme T, let Id-w-Assoc_{W} (T) denote the subset of Alg_{W} (T) consisting of associative algebra structures with multiplicative identity the nowhere vanishing section w_{T} over T induced from w. Thus we obtain a contravariant subfunctor Id-w-Assoc_{W} of Alg_{W} . Let Stab_{w}(T) ⊂ GL_{W} (T) denote the stabiliser subgroup of w_{T}, so that one gets a subfunctor in subgroups Stab_{w} ⊂ GL_{W} . It is in fact represented by a closed subgroupscheme (also denoted by) Stab_{w} and further behaves well under base change relative to X i.e., Stab_{w} ×_{X} T can be canonically identified with Stab_{w_{T}} for any X-scheme T. These follow from para 9.6.6 of Chap.I, EGA I [6]. It is clear that the natural action of GL_{W} on Alg_{W} induces one of Stab_{w} on Id-w-Assoc_{W} . It is easy to check (p.489, Part A, [3]) that the functor Id-w-Assoc_{W} is a sheaf in the big Zariski site over X and further that this functor is represented by a natural closed subscheme Id-w-Assoc_{W} ↪ Alg_{W} which is Stab_{w}-invariant. Further the construction Id-w-Assoc_{W} behaves well with respect to base-change (relative to X). Consider the subfunctor Id-w-Azu_{W} ↪ Id-w-Assoc_{W} corresponding to Azumaya algebras.

THEOREM 4.4 (Theorem 3.4, Part A, [3])

(1) Id-w-Azu_{W} is represented by a Stab_{w}-stable open subscheme Id-w-Azu_{W} ↪ Id-w-Assoc_{W} and the canonical open immersion is an affine morphism.

(2) Id-w-Azu_{W} is affine (hence separated) and of finite presentation over X, and the construction Id-w-Azu_{W} behaves well with respect to base-change (relative to X).

(3) Further, Id-w-Azu_{W} is smooth of relative dimension (m^{2} −1)^{2} and geometrically irreducible over X, where m^{2} := rank_{X}(𝒲).

THEOREM 4.5 (Theorem 3.8, Part A, [3])

(1) The open immersion Id-w-Azu_{W} ↪ Id-w-Assoc_{W} has a schematic image denoted Id-w-Sp-Azu_{W} which is affine (hence separated) and of finite type over X and is naturally a Stab_{w}-stable closed subscheme of Id-w-Assoc_{W}, the action extending the natural one on the open subscheme Id-w-Azu_{W}.

(2) When the rank of W over X is 4, Id-w-Sp-Azu_{W} is locally (over X) isomorphic to relative 9dimensional affine space; in fact over every open affine subscheme U of X where W becomes trivial and w becomes part of a global basis, Id-w-Sp-Azu_{W} |_{U} ≅ 𝔸_{U}^{9}. Thus Id-w-Sp-Azu_{W} is smooth of relative dimension 9 and geometrically irreducible over X. In particular, it is of finite presentation over X.

(3) When rank_{X}(W) = 4, the construction Id-w-Sp-Azu_{W} ⟶ X base changes well.

Definition of the Morphism Υ′. We adopt the notations of Theorem 1.9. Let T be an Xscheme. Given a bilinear form b ∈ Bil_{V} (T) = Bil(V_{T}), consider the linear isomorphism ψ_{b} : C_{0}(V_{T} , q_{b}) ≅ Λ^{even}(V_{T}) = W_{T} of (2d), Theorem 2.1. Let A_{b} denote the algebra bundle structure on W_{T} with unit w_{T} = 1 induced via ψ_{b} from the even Clifford algebra C_{0}(V_{T} , q_{b}). By definition, A_{b} ∈ Id-w-Assoc_{W} (T) and we get a map of T -valued points

Υ′(T) : Bil_{V} (T) ⟶ Id-w-Assoc_{W} (T) : b ↦ A_{b}.

This is functorial in T because of (3), Theorem 2.1, and hence defines an X-morphism Υ′ : Bil_{V} ⟶ Id-w-Assoc_{W} . The morphism Υ′ is GL_{V} -equivariant due to (2j), Theorem 2.1.

Semiregular Bilinear forms. Fundamental problems in dealing with quadratic forms over arbitrary base schemes arise essentially from two abnormalities in characteristic two: firstly, the mapping that associates a quadratic form to its symmetric bilinear form is not bijective and secondly, there do not exist regular quadratic forms on any odd-rank bundle. The remedy for this is to consider semiregular quadratic forms, a concept due to M.Kneser [1] and elaborated upon by Knus in [2], which in fact works over an arbitrary base scheme (and hence in a characteristic-free way) and further reduces to the usual notion of regular form in characteristics ≠ 2. Let Spec(R) = U ↪ X be an open affine subscheme of X such that V |U is trivial. Consider a quadratic form q ∈ Γ(U, Quad_{V}) on V |U and its associated symmetric bilinear form b_{q}. The matrix of this bilinear form relative to any fixed basis is a symmetric matrix of odd rank and in particular, if U is of pure characteristic two (i.e., the residue field of any point of U is of characteristic two), then this matrix is also alternating and is hence singular, immediately implying that q cannot be regular. However, computing the the determinant of such a matrix in formal variables {ζ_{i}, ζ_{ij}} shows that it is twice the following polynomial

P_{3}(ζ_{i}, ζ_{ij}) = 4ζ_{1}ζ_{2}ζ_{3} + ζ_{12}ζ_{13}ζ_{23} − (ζ_{1}ζ_{23}^{2} + ζ_{2}ζ_{13}^{2} + ζ_{3}ζ_{12}^{2}).

The value P_{3}(q(e_{i}), b_{q}(e_{i}, e_{j})) corresponding to a basis {e_{1}, e_{2}, e_{3}} is called the half-discriminant of q relative to that basis, and q is said to be semiregular if its half-discriminant is a unit. It turns out that this definition is independent of the basis chosen (§3, Chap.IV, [2]). Even if V |U were only projective (i.e., locally-free but not free), the semiregularity of q may be defined as the semiregularity of q ⊗_{R} R_{𝔪} for each maximal ideal 𝔪 ⊂ R, and it turns out that with this definition, the notion of a quadratic form being semiregular is local and is well-behaved under base-change (Prop.3.1.5, Chap.IV, [2]). We may thus define the subfunctor of Quad_{V} of semiregular quadratic forms. This subfunctor is represented by a GL_{V} -invariant open subscheme i : Quad_{V}^{sr} ↪ Quad_{V} because, over each affine open subscheme U ↪ X which trivialises V, it corresponds to localisation by the non-zerodivisor P_{3}. Note that this canonical open immersion is affine and schematically dominant as well. We next turn to semiregular bilinear forms. Recall that we had defined a bilinear form b to be semiregular iff its induced quadratic form q_{b} is semiregular. Thus by definition, Bil_{V}^{sr} is the fiber product:

Bil_{V} ⟶^{p} Quad_{V}
i′↑  ↑^{i}
Bil_{V}^{sr} ⟶_{p′} Quad_{V}^{sr}

Since p is a Zariski-locally-trivial principal Alt_{V}^{2} -bundle, it is smooth and surjective (in particular faithfully flat). It therefore follows that the affineness and schematic dominance of i imply those of i′. We record these facts below. PROPOSITION 4.6 The open immersion Bil_{V}^{sr} ↪ Bil_{V} is a GL_{V} -equivariant schematically dominant affine morphism. Further this open immersion behaves well under base-change (relative to X).

Reduction of Proof of Theorem 1.9 to Theorem 1.10. We first recall the following crucial

fact (see (1), Prop.3.2.4, Chap.IV [2]): The even Clifford algebra of a semiregular quadratic form is an Azumaya algebra. Using this fact and the definition of Υ′, we see that the morphism Υ′ restricted to Bil_{V}^{sr} factors through Id-w-Azu_{W} by a morphism Υ^{sr} such that the following diagram is commutative

Bil_{V} ⟶^{Υ′} Id-w-Assoc_{W}
↑  ↑
Bil_{V}^{sr} ⟶_{Υ^{sr}} Id-w-Azu_{W}

where the vertical arrows are the canonical open immersions. The above diagram base changes well in view of (2), Theorem 4.4, Prop.4.6 and (3), Theorem 2.1. Notice that since the structure morphism Bil_{V} ⟶ X is an affine morphism, and since the same is true of Id-w-Assoc_{W} ⟶ X, it is also true of Υ′. In particular, Υ′ is quasi-compact and separated, and therefore has a schematic image by case (1) of Prop.4.2. The same is true of each of the two vertical arrows and of Υ^{sr} in view of Prop.4.6 and (1) of Theorem 4.4. Further, as noted in Prop.4.6, Bil_{V}^{sr} ↪ Bil_{V} is schematically dominant and therefore by (5), Prop.4.3, the limiting scheme of the former in the latter is the latter itself. So using the commutative diagram above, the transitivity of the schematic image (assertion (3), Prop.4.3), and the definition of Id-w-Sp-Azu_{W} (assertion (1), Theorem 4.5), we see that in order to prove (1), Theorem 1.9, it is enough to show that

(*) Υ^{sr} is schematically dominant and surjective, and Υ′ is a closed immersion.

We now claim that the above properties are equivalent to

(**) Υ^{sr} is proper and Υ′ is a closed immersion.

Suppose (**) holds. To show (*), we only need to show that Υ^{sr} is surjective and schematically dominant. From (**) it follows that Υ_{K}^{sr} := Υ^{sr} ⊗_{X} K is functorially injective and proper for each algebraically closed field K with an X-morphism Spec(K) ⟶ X.

That both the K-schemes Bil_{V}^{sr} ⊗_{X} K and Id-w-Azu_{W} ⊗_{X} K are integral and smooth of the same dimension follows from the smoothness of relative dimension 9 and geometric irreducibility /X of Bil_{V}^{sr} (which is obvious), and of Id-w-Azu_{W} from (3), Theorem 4.4. Since Υ_{K}^{sr} is differentially injective at each closed point, it has to be a smooth morphism by Theorem 17.11.1 of EGA IV [7] and thus has to be an open map. But by (**) it is also proper and hence a closed map. Thus Υ_{K}^{sr} is bijective etale, and hence an isomorphism. This also gives that Υ^{sr} is surjective. Now from Cor.11.3.11 of EGA IV and from the flatness of Bil_{V}^{sr} over X, it follows that Υ^{sr} is itself flat, and hence schematically dominant since it is faithfully flat (being already surjective). Therefore (**) ⟹ (*).

The property of a morphism being proper is local on the target (see for e.g., (f), Cor.4.8, Chap.IV, [9]) and the same is true of the property of being a closed immersion. Therefore, in verifying (**), we may assume that V is free over X (so that W = Λ^{even}(V) is also free over X and and w is part of a global basis). We are now in the situation of Theorem 1.10. Granting it, we see immediately from the multiplication table that (**) holds. For the table shows that the composition of the following X-morphisms

Bil_{V} ⟶^{Υ′} Id-w-Assoc_{W} ↪^{CLOSED} Alg_{W}

is a closed immersion, which implies that Υ′ is also a closed immersion. Further, the multiplication table also shows that both Υ′ and Υ^{sr} satisfy the valuative criterion for properness, and are therefore proper. Thus the conditions (**) are verified. So we have reduced the proof of (1), Theorem 1.9 to Theorem 1.10. As for statement (2) of Theorem 1.9, firstly, the involution Σ of Id-w-Assoc-Alg_{W} defines a unique involution (also denoted by Σ) on the scheme of specialisations Id-w-Sp-Azu_{W} (leaving the open subscheme Id-w-Azu_{W} invariant) because of the defining property of the schematic image involved; for we may verify that an automorphism of a scheme T which leaves an open subscheme U stable will also leave stable the limiting scheme of U in T (of course we assume here that the canonical open immersion U ↪ T is a quasi-compact open immersion, which ensures the existence of the limiting scheme). Secondly, a glance at the multiplication table of Theorem 1.10 keeping in view the definition of opposite algebra shows that indeed the induced Σ ∈ Aut_{X} (Bil_{V}) takes the T -valued point B = (b_{ij}) to transpose(−B) = (−b_{ji}). Finally, assertion (3) of Theorem 1.9 is a consequence of (1) taking into account (3), Theorem 4.5.

Proof of Theorem 1.10. Given B = (b_{ij}) ∈ Bil_{V} (T), by our definition above, (Υ′(T))(B) = A_{B} is the algebra structure induced from the linear isomorphism ψ_{B} : C_{0}(V_{T} , q_{B}) ≅ Λ^{even}(V_{T}) of (2d), Theorem 2.1. The stated multiplication table for A = A_{B} is a consequence of straightforward calculation, keeping in mind (2f), Theorem 2.1 and the standard properties of the multiplication in the even Clifford algebra C_{0}(V_{T} , q_{B}). Q.E.D., Theorems 1.10 & 1.9.

Proof of Theorem 1.12. The proof follows by comparing the multiplication table relative to Θ as computed in Theorem 3.1 with the multiplication table relative to Υ of Theorem 1.10 computed above. Q.E.D., Theorem 1.12.

5 Properties of Specialisations/Self-duality: 1.5–1.8

In this section, we prove Theorem 1.5 and use it alongwith Theorem 1.9 to prove Theorem 1.6. Thereafter we investigate the specialised algebras when the underlying bundle is self-dual and prove Theorems 1.7 and 1.8.

Proof of Theorem 1.5. Let W be the rank 4 vector bundle underlying the specialised algebra A and w ∈ Γ(X, W) be the global section corresponding to 1_{𝒜} . We choose an affine open covering {U_{i}}_{i∈I} of X such that W |U_{i} is trivial and w|U_{i} is part of a global basis ∀ i. Therefore on the one hand, for each i ∈ I, we can find a linear isomorphism ζ_{i} : Λ^{even} (𝒪_{X}^{⊕3}|U_{i}) ≅ W |U_{i} taking 1_{Λ^{even}} onto w|U_{i}. The (w|U_{i})-unital algebra structure A|U_{i} induces via ζ_{i} an algebra structure A_{i} on Λ^{even} (𝒪_{X}^{⊕3}|U_{i}) (so that ζ_{i} becomes an algebra isomorphism). Since A_{i} is also a specialised algebra structure ((3), Theorem 4.5), by Theorem 1.9 we can on the other hand also find a quadratic form q_{i} on 𝒪_{X}^{⊕3}|U_{i} induced from a bilinear form b_{i} so that the algebra structure A_{i} is precisely the one induced by the linear isomorphism ψ_{b_{i}} : C_{0} (𝒪_{X}^{⊕3}|U_{i}, q_{i}) ≅ Λ^{even} (𝒪_{X}^{⊕3}|U_{i}) given by (2d) of Theorem 2.1. For each pair of indices (i, j) ∈ I×I, let ζ_{ij} and ϕ_{ij} be defined so that the following diagram commutes:

C_{0}(𝒪_{X}^{⊕3}|U_{ij}, q_{i}|U_{ij}) ⟶_{≅}^{ψ_{b_{i}} |U_{ij}} Λ^{ev}(𝒪_{X}^{⊕3}|U_{ij}) ⟶_{≅}^{ζ_{i}|U_{ij}} A|U_{ij}
ϕ_{ij} ↓≅  ^{ζ_{ij}} ↓≅  ^{=}↓
C_{0}(𝒪_{X}^{⊕3}|U_{ij}, q_{j}|U_{ij}) ⟶_{ψ_{b_{j}} |U_{ij}}^{≅} Λ^{ev}(𝒪_{X}^{⊕3}|U_{ij}) ⟶_{ζ_{j} |U_{ij}}^{≅} A|U_{ij}

The above diagram means that the algebras A_{i} glue along U_{ij} := U_{i}∩U_{j} via ζ_{ij} to give (an algebra bundle isomorphic to) A, and in the same vein, the even Clifford algebras C_{0}(𝒪_{X}^{⊕3}|U_{i}, q_{i}) glue along the U_{ij} via ϕ_{ij} to give A as well. Now consider the similarity g_{l_{ij}}^{ϕ_{ij}} = s_{−1}^{+}(ϕ_{ij}) : (𝒪_{X}^{⊕3}|U_{ij}, q_{i}|U_{ij}) ≅_{l_{ij}} (𝒪_{X}^{⊕3}|U_{ij}, q_{j}|U_{ij}) with multiplier l_{ij} := det(ϕ_{ij})^{−1} given by (c), Theorem 1.3. Since s_{−1}^{+} is multiplicative, and since ϕ_{ij} satisfy the cocycle condition, it follows that s_{−1}^{+}(ϕ_{ij}) also satisfy the cocycle condition and therefore glue the 𝒪_{X}^{⊕3}|U_{i} along the U_{ij} to give a rank 3 vector bundle V on X. (Observe that unfortunately the q_{i} do not glue !) We shall now revert to the notations of Section 3. By Theorem 3.5, we have

h_{s}(g_{l_{ij}}^{ϕ_{ij}} , l_{ij}, q_{i}|U_{ij}, q_{j}|U_{ij}) = (\begin{matrix}1 & 0 \\ 0 & l_{ij}^{−1}Λ^{2}(g_{l_{ij}}^{ϕ_{ij}}) \end{matrix})

which means that (ϕ_{ij})_{Λ^{2}} = det(ϕ_{ij}) × Λ^{2}(g_{l_{ij}}^{ϕ_{ij}}). This immediately implies part (1) of assertion (a) of Theorem 1.5, from which parts (2)—(4) can be deduced using the standard properties of the determinant and the perfect pairings between suitable exterior powers of a bundle. To prove (b), let A = C_{0}(V, q). Using a trivialisation {U_{i}}_{i∈I} for the quadratic bundle (V, q) with transition functions g_{ij}, and remembering that by definition the g_{ij} are isometries, we have the following commutative diagram on the intersections U_{ij}:

C_{0}(𝒪_{X}^{⊕3}|U_{ij}, q_{i}|U_{ij}) ⟶_{≅}^{ψ_{b(q_{i})}|U_{ij}} Λ^{ev}(𝒪_{X}^{⊕3}|U_{ij})
C_{0}(g_{ij} ,1,q_{i}|U_{ij} ,q_{j} |U_{ij})↓≅  ≅↓^{h(g_{ij} ,1,q_{i}|U_{ij} ,q_{j} |U_{ij})}
C_{0}(𝒪_{X}^{⊕3}|U_{ij}, q_{j}|U_{ij}) ⟶_{ψ_{b(q_{j})}|U_{ij}}^{≅} Λ^{ev}(𝒪_{X}^{⊕3}|U_{ij})

But by the formula in Theorem 3.5, we have

h_{s}(g_{ij}, 1, q_{i}|U_{ij}, q_{j}|U_{ij}) = (\begin{matrix}1 & 0 \\ 0 & Λ^{2}(g_{ij}) \end{matrix})

which immediately implies that C_{0}(V, q)/𝒪_{X}.1_{C_{0}} ≅ Λ^{2}(V). Q.E.D., Theorem 1.5.

Proof of ‘if’ part of Theorem 1.6. Suppose that the hypotheses hold; by (1) ∃L ∈ Pic(X) such that det(A) ≅ L^{⊗2} and by (2), we may choose a splitting A ≅ 𝒪_{X}.1_{A} ⊕ (A/𝒪_{X}.1_{A}). Using assertion (1) of (a), Theorem 1.5, we see that there exists a rank 3 vector bundle V′ on X such that

A ≅ 𝒪_{X}.1_{A} ⊕ (A/𝒪_{X}.1_{A}) ≅ 𝒪_{X}.1_{A} ⊕ (L^{⊗2} ⊗ Λ^{2}(V′)) ≅ 𝒪_{X}.1_{A} ⊕ Λ^{2}(V′ ⊗ L) ≅ Λ^{even}(V)

where V := V′ ⊗ L and the last isomorphism is chosen so as to map 𝒪_{X}.1_{A} isomorphically onto 𝒪_{X}.1_{Λ^{ev}(V )}. Therefore if (W, w) := (Λ^{even}(V), 1), then by the above identification A induces an element of Id-w-Sp-Azu_{W} (X), and since Υ : Bil_{V} ≅ Id-w-Sp-Azu_{W} is an X-isomorphism by (1), Theorem 1.9, it follows that there exists a global quadratic form q = q_{b} induced from a bilinear form b such that the algebra structure Υ(b) ≅ A. To finish the proof, we only have to remember that Υ(b) is the algebra structure induced from the linear isomorphism ψ_{b} : C_{0}(V, q = q_{b}) ≅ Λ^{even}(V) =: W of (2d), Theorem 2.1, which preserves 1 by (2a) of the same Theorem.

Proof of ‘only if’ part of Theorem 1.6. Suppose ∃ (V, q) such that C_{0}(V, q) ≅ A, and that q is induced from a bilinear form b. Then by (2d), Theorem 2.1, we have a linear isomorphism ψ_{b} : C_{0}(V, q = q_{b}) ≅ Λ^{even}(V). Since ψ_{b} preserves 1 by (2a) of the same Theorem, we get (2). Further (1) follows from assertion (b) of Theorem 1.5.

Proofs of assertions on validity of hypotheses (1) & (2). If (i) holds, the validity of (2) is

a simple exercise in commutative algebra. Since A is a quaternion algebra bundle by Prop.1.13, we have the associated trace map tr_{σ_{A}} : A ⟶ 𝔸_{X}^{1}. Since (ii) is equivalent to 2 ∈ Γ(X, 𝒪_{X}^{∗}), when (ii) holds, the map 2^{−1}tr_{σ_{A}} splits the canonical inclusion 𝒪_{X}.1_{A} ↪ A, showing that (2) holds. If A is Azumaya, then as we saw in the discussion following Theorem 1.14, A ≅ C_{0}(V, q) with (V, q) semiregular. So (1) follows from (b), Theorem 1.5. Q.E.D., Theorem 1.6.

Relations with Self-Duality. In the following we let X be a scheme and W a rank 4 vector bundle over X. We recall the following result: PROPOSITION 5.1 (Prop.3.3, Part A, [3])

1. Let T be an X-scheme and A an associative unital algebra structure on W_{T} := W ⊗_{X} T. Then the

subset

U(T, A) := {t ∈ T | 𝒜_{t} is an Azumaya 𝒪_{T,t} − algebra}

is an open (possibly empty) subset. When U(T, A) is nonempty, denote by the same symbol the canonical open subscheme structure. Then if f : T′ ⟶ T is an X-morphism such that the topological image intersects U(T, A), then U(T′, f^{∗}(A) = A ⊗_{T} T′) ≅ U(T, A) ×_{T} T′ as open subschemes of T′. Further U(T, A) ↪ T is an affine morphism.

2. U(T, A) is the maximal open subset restricted to which A is Azumaya.

3. Further let f : T′ ⟶ T be a morphism of X-schemes such that f^{∗}(A) is Azumaya. Then f factors through the open subscheme U(T, A) defined above.

Proof of Theorem 1.7. We first make certain observations when X is any reduced scheme and W is a rank n^{2} vector bundle over X with nowhere-vanishing global section w. It is not hard to see that the structure morphism Id-w-Azu_{W} ⟶ X is in fact a morphism of finite presentation. Hence, in view of Prop.17.5.7, EGA IV [7], assertion (3) of Theorem 4.4 implies that Id-w-Azu_{W} is reduced. Since Id-w-Sp-Azu_{W} is the schematic image of Id-w-Azu_{W}, it follows from (1), Prop.4.3 that Id-w-Sp-Azu_{W} is reduced as well. When X is integral, it is easy to see that Id-w-Azu_{W} is integral as well. Consider any affine open subscheme U = Spec(R) ↪ X such that W |U is trivial and w|U is part of a global basis. There are (w|U)-unital Azumaya algebra structures on W |U which are isomorphic to the (n × n)-matrix algebra over R. Consider the orbit morphism Stab_{w|U} ⟶ Id-(w|U)-Azu_{W |U} corresponding to one such algebra structure. Assertions (2) and (3) of Theorem 4.4 show that the topological image of this morphism is dense. Further, Stab_{w|U} is integral since it is ≅ 𝔸_{U}^{12} and since U is integral. Thus Id-(w|U)-Azu_{W |U} is integral. Recall (e.g., Prop.2.1.6 & Cor.2.1.7, Chap.0.2, EGA I [6]) that a nonempty topological space— whose set of irreducible components is locally finite—is locally irreducible iff each irreducible component is open; further it is irreducible iff it is locally irreducible and connected. Now since Id-w-Azu_{W} can be covered by irreducible open subschemes which pairwise intersect (as X is irreducible), it follows that Id-w-Azu_{W} is integral as well. Therefore (1), Prop.4.3 implies that Id-w-Sp-Azu_{W} is also integral in the case when X is integral. Using again the facts that Id-w-Azu_{W} behaves well under base change and that the schematic image of an irreducible scheme is irreducible (by (1), Prop.4.3), we may infer from the foregoing that X is irreducible iff Id-w-Azu_{W} is irreducible iff Id-w-Sp-Azu_{W} is irreducible.

We next observe that assertion (b) follows from (a) and from the following result: The vector bundle underlying a rank 4 Azumaya algebra bundle A is self-dual. To see this, first remember from Prop.1.13 that A comes with a standard involution σ = σ_{A}. The associated norm n_{σ} : x ↦ x.σ(x) is a (global) quadratic form with values in 𝒪 and its associated bilinear form b_{n_{σ}} may be given in terms of the associated trace tr_{σ} as b_{n_{σ}} (x, x′) = tr_{σ}(xσ(x′)) = xσ(x′) + x′σ(x). Now n_{σ} is nonsingular (=regular), as may be verified locally over affine open subschemes—for details see para 7.3.5, Chap.I, [2].

As for assertion (a), we begin with a general remark: if X′ ⟶ X is a surjective k-morphism, k being the algebraically closed base field, and if (a) holds for the pair (X′, A′ := A ⊗_{X} X′), then it also holds for the pair (X, A) where A is an associative unital algebra structure on W. For, by (3), Prop.5.1, U(X′, A′) = X′ ⟹ U(X, A) = X. Since the canonical morphism X_{RED} ↪ X is a surjective closed immersion, we may therefore assume that X is reduced. Let x ∈ U(X, A) ∩ X_{α} where X_{α} is an irreducible component of X given the canonical reduced induced closed subscheme structure. If (a) holds for (X_{α}, A ⊗_{X} X_{α}), then by (1), Prop.5.1, we have that A ⊗_{X} X_{α} is Azumaya, and hence by (3) of the same Proposition, the canonical closed immersion X_{α} ↪ X factors through U(X, A). Now as X is connected, we observe that (a) holds for X if it holds for each irreducible component given the reduced induced closed subscheme structure (which is also a proper scheme of finite type over k); thus we may as well assume that X is irreducible. By Chow’s Lemma (Theorem 5.6.1, EGA II [8]) there exists an integral projective scheme X′ of finite type over k alongwith a surjective projective k-morphism X′ ⟶ X. Therefore we may further assume that X is projective. Finally, we may also assume that X is normal, for if X̃ is the normalisation of X, then the canonical morphism X̃ ⟶ X is finite and surjective, hence also projective. So we have reduced to the case when X is an integral normal projective scheme over k. However, the following proof works with the hypothesis of X being regular in codimension 1, which is satisfied when X is normal.

Now given an associative w-unital algebra structure A on W such that the open subset U(X, A) of Prop.5.1 is nonempty, the restriction to the dense open subset U(X, A) of the section over X corresponding to A factors through Id-w-Azu_{W} as a morphism (and not just topologically); further, since both X and Id-w-Sp-Azu_{W} are reduced, this section factors through Id-w-Sp-Azu_{W} itself. Thus any A such that U(X, A) ≠ ∅ corresponds to a specialised algebra structure. Let there be given an isomorphism of vector bundles ϕ : W ≅ W^{∨} and an A such that U(X, A) ≠ ∅. To prove (a), we must show that U(X, A) = X. We proceed by contradiction: suppose D(X, A) := X\U(X, A) is nonempty. Consider the composition of the following morphisms of vector bundles:

W ⊗_{X} W ≡ A ⊗_{X} A^{op} ⟶^{(a⊗b)↦(x↦axb)} End_{Linear} (A) ≡ W^{∨} ⊗_{X} W ⟶^{ϕ×Id} W ⊗_{X} W

The above composite gives an endomorphism of the vector bundle W ⊗ W which is an isomorphism precisely at the local rings of the points of U(X, A). Therefore, the induced element s(ϕ) ∈ Γ(X, 𝒪_{X}) ≡ H^{0}(X, End(det(W ⊗ W))) goes into the maximal ideal of the local ring at each point of D(X, A) and to a unit of the local ring at each point of U(X, A). It follows that if D(X, A) is the irredundant union of divisors {D_{i}|1 ≤ i ≤ n}, then the divisor defined by (s(ϕ), 𝒪_{X}) namely Σ_{i}n_{i}D_{i} is principal. Under the present hypotheses on X, every principal divisor has degree zero (see for e.g., (d), Ex.6.2, Chap.II, [9]), so that we get 0 = Σ_{i}n_{i}.(degree(D_{i})). This is a contradiction, since n_{i} ≥ 1 as it is the order of vanishing of s(ϕ) on D_{i} for each i, and since the degree of each D_{i} is positive (see for e.g., (a), Prop.7.6, Chap.I, [9]). Q.E.D., Theorem 1.7.

Proof of assertions (a)—(d) of Theorem 1.8. The proofs follow essentially from the properties of Id-w-Sp-Azu_{W} ⟶ X and Id-w-Azu_{W} ⟶ X as mentioned in Theorems 4.5 and 4.4. In this regard see also the first paragraph of the proof of Theorem 1.7. We indicate proofs for the not-so-obvious assertions, especially (c) and for the implication (X irreducible ⇒ D_{X} irreducible) of (a). The converse implication would follow from the fact that D_{X} ⟶ X is topologically surjective. Since the structure morphisms Id-w-Sp-Azu_{W} ⟶ X and Id-w-Azu_{W} ⟶ X are smooth, they are faithfully flat (hence surjective). Therefore in view of (b)—which is actually the result of good base-change properties of Id-w-Sp-Azu_{W} and Id-w-Azu_{W} relative to X, the irreducible components of D_{X} and of Id-w-Sp-Azu_{W} are induced from those of X by pullback, provided we check the particular case when X is irreducible and reduced.

So let X be integral and first assume that W is trivial and w is part of a global basis. Without loss of generality, we may take (W, w) = (Λ^{even}(V), 1) for V a rank 3 trivial vector bundle over X. After fixing a suitable basis for V, we may define the morphism Θ, which by Theorem 1.11 is an isomorphism that maps the closed subset (Quad_{V} ×_{X} L_{w})\(Quad_{V}^{sr} ×_{X} L_{w}) onto D_{0} := Id-w-Sp-Azu_{W} \Id-w-Azu_{W} . Therefore the irreducibility of D_{0} is equivalent to that of the closed subset Quad_{V} \Quad_{V}^{sr}. Recall from the discussion on semiregular forms (page 24, Section 4) that the open subscheme Quad_{V}^{sr} corresponds to localisation by the polynomial P_{3}. This polynomial is irreducible as an element of R[ζ_{i}, ζ_{ij}] ≅ R[Quad_{V}] when X = Spec(R) is affine, though it is not clear if it is a prime element (unless we assume something more e.g., R a UFD). The closed subset Quad_{V} \Quad_{V}^{sr} may be given the canonical closed subscheme structure Z(P_{3}) corresponding to the vanishing of P_{3}. Let q^{(2)} be the global quadratic form on V given by (x_{1}, x_{2}, x_{3}) ↦ x_{1}x_{2}. It can be checked that q^{(2)} is not semiregular, but that its restriction to the rank two (direct summand) vector subbundle generated by {e_{1}, e_{2}} is regular. Therefore the X-valued point corresponding to q^{(2)} lands topologically inside the closed subset underlying Z(P_{3}). Consider the orbit morphism O(q^{(2)}) : GL_{V} ⟶ Quad_{V} corresponding to this X-valued point which also lands topologically inside Z(P_{3}). It will follow from assertion (2), Theorem 1.17, that the topological image of O(q^{(2)}) is dense in Z(P_{3}). On the other hand this topological image is irreducible, since GL_{V} ≅ 𝔸_{X}^{9}. It follows that Quad_{V} \Quad_{V}^{sr}, and hence D_{0}, is irreducible in the case when W is trivial and w is part of a global basis over X. Since the reduced closed subscheme structure on Z(P_{3}) is given by the radical of the ideal (P_{3}), it follows that \sqrt{(P_{3})} is the minimal prime divisor of (P_{3}) and by Krull’s Hauptidealsatz this prime has height 1. Therefore, the codimension of D_{0} is also 1 in the present case.

Now consider the case of a general (W, w), choose an affine open covering {U_{i} = Spec(R_{i})}_{i∈I} of X such that W_{i} := W |U_{i} is trivial and w_{i} := w|U_{i} is part of a global basis ∀ i. The subset D_{i} := Id-w_{i}-Sp-Azu_{W_{i}}\Id-w_{i}-Azu_{W_{i}} is irreducible for each i by the preceding paragraph. On the other hand, by Theorems 4.4 and 4.5, the subsets D_{i} form an open cover of D := Id-w-Sp-Azu_{W} \Id-w-Azu_{W} . Since X is irreducible, we have D_{i} ∩ D_{j} ≠ ∅ when i ≠ j. Thus D is locally irreducible and connected, and hence irreducible (for e.g., by Cor.2.1.7, Chap.0, EGA I [6]). Since X is integral and noetherian, assertion (2) of Theorem 4.5 implies that Id-w-Sp-Azu_{W} is integral and noetherian as well. Therefore the codimension of D in Id-w-Sp-Azu_{W} is atleast 1. On the other hand (for e.g., by Prop.14.2.3, Chap.0, EGA IV, [7]) this codimension is bounded above by 1 = Codim (D_{i}, Id-w_{i}-Sp-Azu_{W_{i}}) for any i, since Id-w_{i}-Sp-Azu_{W_{i}} ↪ Id-w-Sp-Azu_{W} is an open subset whose intersection with D is precisely D_{i}.

Proof of assertion (e) of Theorem 1.8. Let us remind the reader that the pullbacks W ⊗_{X} Id-w-Sp-Azu_{W} and W ⊗_{X} Id-w-Azu_{W} are naturally endowed with (w ⊗_{X} Id-w-Sp-Azu_{W})-unital associative 𝒪_{Id-w-Sp-Azu_{W}} -algebra structures with which they become respectively the universal specialisation and universal Azumaya algebra relative to the pair (W, w)—for details see the proof of Theorem 3.4, Part A, [3]. By (c) and (d), the natural map Pic(Id-w-Sp-Azu_{W}) ⟶ Pic(Id-w-Azu_{W}) is surjective and its kernel is generated by the image of ℤ.(D_{X}). Given an isomorphism ϕ : W ≅ W^{∨}, we consider its pullback to Id-w-Sp-Azu_{W}, and proceeding along the lines of the proof of Theorem 1.7, with the triple (Id-w-Sp-Azu_{W} , W ⊗_{X} Id-w-Sp-Azu_{W} , ϕ ⊗_{X} Id-w-Sp-Azu_{W}) replacing the triple (X, A, ϕ) there, we infer that n.(D_{X}) is principal for some n ≥ 1. Now W ⊗_{X} Id-w-Azu_{W} is an Azumaya algebra bundle, and as seen in the discussion following Theorem 1.14, it is isomorphic to the even Clifford algebra of a canonically obtained rank 3 quadratic bundle on Id-w-Azu_{W} . Therefore it follows from (b), Theorem 1.5 that det(W ⊗_{X} Id-w-Sp-Azu_{W}) maps to an element of 2.Pic(Id-w-Azu_{W}). The remaining assertions in (e) are now consequences of Theorem 1.6. Q.E.D., Theorem 1.8.

6 Stratification of the Variety of Specialisations

In this section, we prove Prop.1.15 and Theorem 1.17.

Proof of Prop.1.15. Fix an S-basis {e_{1}, e_{2}, e_{3}} for V, and with respect to this basis, let q^{1} denote the quadratic form given by (x_{1}e_{1} + x_{2}e_{2} + x_{3}e_{3}) ⟼ x_{1}x_{2} + x_{3}^{2}. It is easy to see that this quadratic form is semiregular. We show that any semiregular quadratic form q can be moved to q^{1} i.e., that ∃ g ∈ GL(V) such that g · q = q^{1}.

By Prop.3.17, Chap.IV, [2], there exists a basis {e′_{1}, e′_{2}, e′_{3}} for V such that q restricted to the submodule generated by e′_{1} and e′_{2} is regular and further such that q(e′_{3}) ∈ S^{∗}, b_{q}(e′_{1}, e′_{2}) = 1 and b_{q}(e′_{1}, e′_{3}) = 0 = b_{q}(e′_{2}, e′_{3}). Let g′ ∈ GL(V) be the automorphism that maps e′_{i} onto the e_{i} for each i and consider the quadratic form q′ := g′ · q. Then by definition of the GL(V)-action on the set Quad(V) of quadratic S-forms on V we have q′(e_{3}) ∈ S^{∗}, b_{q′}(e_{1}, e_{2}) = 1 and b_{q′}(e_{1}, e_{3}) = 0 = b_{q′}(e_{2}, e_{3}). So if we assume that q′(e_{i}) = λ_{i} (⇔ q(e′_{i}) = λ_{i}), then q′(x_{1}e_{1} + x_{2}e_{2} + x_{3}e_{3}) = λ_{1}x_{1}^{2} + λ_{2}x_{2}^{2} + λ_{3}x_{3}^{2} + x_{1}x_{2} ∀ x_{i} ∈ S. Thus it is enough to show that q′ can be moved to q^{1}. We look for an invertible matrix g′′ = (u_{ij}) ∈ GL(V) such that g′′ · q′ = q^{1}. Writing this condition equivalently as q′ = (g′′)^{−1} · q^{1} and comparing the polynomials in the x_{i} gives 6 equations in terms of the u_{ij} and the λ_{i} which are to be satisfied. We choose the u_{ij} as follows. First set u_{11} = u_{22} = 0 and let u_{12} ∈ S^{∗} be a free parameter. Since every element of S has square roots in S, it makes sense to choose u_{31} = ±\sqrt{λ_{1}} and u_{32} = ±\sqrt{λ_{2}}. We let α = 1 + 2u_{31}u_{32}, β = 1 − 2u_{31}u_{32} and u_{21} = β/u_{12}. Since q′ = g · q and since q is semiregular, q′ is also semiregular. Its half-discriminant relative to the present basis of V is (remembering that λ_{3} ∈ S^{∗}) d_{q′}(e_{1}, e_{2}, e_{3}) = λ_{3}.(4λ_{1}λ_{2} − 1) ∈ S^{∗}. This implies that αβ = 1 − 4λ_{1}λ_{2} ∈ S^{∗} ⟹ α, β ∈ S^{∗}. Therefore it makes sense to define u_{33} = ±\sqrt{(βλ_{3})/α}, u_{13} = −2u_{31}u_{33}u_{12}/β and u_{23} = −2u_{32}u_{33}/u_{12}. Note that u_{33} ∈ S^{∗}. A computation shows that the determinant of the matrix g′′ = (u_{ij}) defined above is −u_{33}α ∈ S^{∗} and hence g′′ is invertible. It is also easily checked that g′′ · q′ = q^{1}. Q.E.D., Prop.1.15.

Proof of assertion (2) of Theorem 1.17. (We shall not prove assertion (1) since it is wellknown). Recall from the discussion on semiregular forms (page 24, Section 4) that the open subscheme Quad_{V}^{sr} corresponds to localisation by the polynomial P_{3} and that this polynomial is prime as an element of k[ζ_{i}, ζ_{ij}] ≅ k[Quad_{V}]. Here a quadratic form q corresponding to the point (λ_{i}, λ_{ij}) ∈ 𝔸_{k}^{6} is given by (x_{1}, x_{2}, x_{3}) ↦ Σ_{i}λ_{i}x_{i}^{2} + Σ_{i<j}λ_{ij}x_{i}x_{j}. For ease of readability (and typesetting !) let us denote the closure \overline{T} of a subset T (given the reduced closed subscheme structure) by ⟨T ⟩ in what follows. Since Quad_{V}^{(1)} is the same as the variety underlying the open subscheme Quad_{V}^{sr} of semiregular quadratic forms, that ⟨Quad_{V}^{(1)}⟩ = Quad_{V} follows from the fact that Quad_{V} is irreducible.

By assertion (1) of the present Theorem, Quad_{V} is the disjoint union of the Quad_{V}^{(i)}; therefore ⟨Quad_{V}^{(1)}⟩\Quad_{V}^{(1)} is the disjoint union of {Quad_{V}^{(i)}|2 ≤ i ≤ 4} and also equals the closed subset Z(P_{3}) defined by the vanishing of P_{3}. An explicit computation shows that the dimension of the stabilizer of q^{(2)} in GL(V) is 4. Since Quad_{V}^{(2)} is an open dense subvariety of ⟨Quad_{V}^{(2)}⟩ ⊂ V (P_{3}), its closure is thus 5-dimensional. But since P_{3} is an irreducible polynomial, Z(P_{3}) is also an irreducible 5-dimensional subvariety. It follows that ⟨Quad_{V}^{(2)}⟩ = ⟨Quad_{V}^{(1)}⟩\Quad_{V}^{(1)}.

Since Quad_{V}^{(2)} is smooth in its closure (= Z(P_{3}) as seen above), the singularities of its closure are contained in Quad_{V}^{(3)} ∪ Quad_{V}^{(4)} which consists of quadratic forms that are perfect squares i.e., squares of linear forms. These singularities may be identified with points (λ_{i}, λ_{ij}) ∈ 𝔸_{k}^{6} ≅ Quad_{V} at which all the partial derivatives of P_{3} vanish. A simple computation shows that this set is the symmetric determinantal variety given by the vanishing of the (2×2)-minors of the matrix of the symmetric bilinear form associated to the generic quadratic form given by (x_{1}, x_{2}, x_{3}) ↦ Σ_{i}ζ_{i}x_{i}^{2} +Σ_{i<j}ζ_{ij}x_{i}x_{j} ; but it can also be shown that this set precisely corresponds to the perfect squares. Therefore Sing (⟨Quad_{V}^{(2)}⟩) = ⟨Quad_{V}^{(2)}⟩\Quad_{V}^{(2)}.

That ⟨Quad_{V}^{(i+1)}⟩ = ⟨Quad_{V}^{(i)}⟩\Quad_{V}^{(i)} for i = 2 follows from the above and the obvious fact that any quadratic form can be specialised to the zero quadratic form. The case i = 3 is trivial.

To see that ⟨Quad_{V}^{(3)}⟩ is smooth if Char(k)=2, we first note from the above and assertion (1) of the present Theorem that the closure of Quad_{V}^{(3)} consists of perfect squares. Since char(k)=2, under the identification Quad_{V} ≅ 𝔸_{k}^{6}, the perfect squares are seen to correspond to the copy of 𝔸_{k}^{3} in 𝔸_{k}^{6} given by the vanishing of the last three coordinates λ_{ij}, 1 ≤ i < j ≤ 3.

To see that the zero quadratic form is a singularity of ⟨Quad_{V}^{(3)}⟩ if char(k) ≠ 2, we note from the above that ⟨Quad_{V}^{(3)}⟩ is defined by the same equations that define the singularities of Z(P_{3}) and is a certain symmetric determinantal variety. It is a well-known fact—an application of Standard Monomial Theory (for e.g., see [10] or [11])—that the ideal defining these equations is itself reduced, i.e., it is the ideal of the variety. Checking the Jacobian criterion now shows that the zero quadratic form is indeed a singular point.

Proof of assertion (3) of Theorem 1.17. The proof will follow from a series of observations. We remind the reader of the X-morphism θ : Quad_{V} ⟶ Id-w-Sp-Azu_{W} which was used to define the isomorphism Θ (page 15, Section 3).

Claim 1: There are exactly 4 H-orbits in SpAzu.

By Theorem 1.11, any point of SpAzu is of the form \underline{t} · θ(q). Its H-orbit is H · θ(q). There exists a unique i, (1 ≤ i ≤ 4), and some g ∈ GL(V) such that q = g · q^{(i)}. Consider the algebra isomorphism C_{0}(g, 1) of (1), Prop.2.4 and the induced isomorphism h(g, 1, q^{(i)}, q) of Prop.3.5. By definition, θ(q) = θ(g · q^{(i)}) = h(g, 1, q^{(i)}, q) · θ(q^{(i)}) ⇒ H · θ(q) = H · θ(q^{(i)}) so that there are atmost 4 orbits. To see that there are atleast 4, we use the following result.

Claim 2. For q ∈ Quad_{V}, we have L_{w} · Θ((GL(V) · q) × {I_{4}}) = Θ((GL(V) · q) × L_{w}) = H · θ(q).

On the one hand, we have

Θ((GL(V) · q) × L_{w}) = \{\underline{t} · θ(g · q) | \underline{t} ∈ L_{w} and g ∈ GL(V)\}
= \{\underline{t} · (h(g, 1, q, g · q) · θ(q)) | \underline{t} ∈ L_{w} and g ∈ GL(V)\}
⊂ H · θ(q).

Conversely take any h · θ(q) ∈ H · θ(q). By Theorem 1.11, there exists a unique \underline{t′} ∈ L_{w} and a unique q′ ∈ Quad_{V} such that h · θ(q) = \underline{t′} · θ(q′). Therefore ((\underline{t′})^{−1}.h) · θ(q) = θ(q′). Since k is algebraicallyand hence quadratically closed, by (b), Theorem 1.3, there exists g ∈ GL(V) such that q′ = g · q and (\underline{t′})^{−1}.h = h(g, 1, q, q′). Hence h · θ(q) = \underline{t′} · θ(g · q) = Θ(g · q, \underline{t′}) ⇒ H · θ(q) ⊂ Θ((GL(V) · q) × L_{w}). This settles Claim 2. As for Claim 1, if q, q′ ∈ Quad_{V} are such that their GL(V)-orbits are distinct, then because Θ is an isomorphism, Claim 2 shows that H · θ(q) is distinct from H · θ(q′).

Claim 3. For each i, with 1 ≤ i ≤ 4, ⟨Quad_{V}^{(i)} × L_{w}⟩ = ⟨Quad_{V}^{(i)}⟩ × L_{w}.

If f : X ⟶ Y is a smooth morphism and U ↪ Y is an open subset, then f^{−1}(⟨U⟩) = ⟨f^{−1}(U)⟩. Since L_{w} ⟶ Spec(k) is smooth, so is the induced morphism ⟨Quad_{V}^{(i)}⟩ × L_{w} ⟶ ⟨Quad_{V}^{(i)}⟩. Taking f to be this morphism and U = Quad_{V}^{(i)} gives Claim 3.

Claim 4. The GL(V)-stratification of Quad_{V} induces a GL(V)-stratification of Quad_{V} × L_{w} (the GL(V)-action on L_{w} being taken to be trivial) with strata given by (Quad_{V} × L_{w})^{(i)} := Quad_{V}^{(i)} × L_{w}, (1 ≤ i ≤ 4).

To prove Claim 4, the only thing that needs to be checked is that

⟨(Quad_{V} × L_{w})^{(i+1)}⟩ = ⟨(Quad_{V} × L_{w})^{(i)}⟩\(Quad_{V} × L_{w})^{(i)}.

This follows by applying Claim 3 twice:

⟨(Quad_{V} × L_{w})^{(i+1)}⟩ = ⟨Quad_{V}^{(i+1)} × L_{w}⟩ = ⟨Quad_{V}^{(i+1)}⟩ × L_{w} = (⟨Quad_{V}^{(i)}⟩\Quad_{V}^{(i)}) × L_{w}
= ⟨Quad_{V}^{(i)}⟩ × L_{w}\Quad_{V}^{(i)} × L_{w} = ⟨(Quad_{V} × L_{w})^{(i)}⟩\(Quad_{V} × L_{w})^{(i)}.

Now according to Claim 2, we have SpAzu^{(i)} = Θ(GL(V) · q^{(i)} × L_{w}) = Θ((Quad_{V} × L_{w})^{(i)}). This combined with Claim 4 and the fact that Θ is an isomorphism completes the proof of assertion (3) of Theorem 1.17. Q.E.D., Theorem 1.17.

Acknowledgements

The author gratefully acknowledges the Postdoctoral Fellowship (September 2003–August 2004) of the Graduiertenkolleg Gruppen und Geometrie under support from the Deutschen Forschungsgemeinschaft and the State of Niedersachsen at the Mathematisches Institut Göttingen where this paper was written. The author also thanks the National Board for Higher Mathematics, Department of Atomic Energy, Government of India, for its Postdoctoral Fellowship (August 2002–August 2003) at the Chennai Mathematical Institute, Chennai, India, during the latter half of which certain special cases of the results of this work were obtained.

References

[1] Kneser M, Quadratische Formen, Ausarbeitung einer Vorlesung, Gottingen: Mathematisches Institut, 1974.

[2] Max-Albert Knus, Quadratic and Hermitian Forms over Rings, Grundlehren der mathematischen Wissenschaften 294 (1991) (Springer-Verlag)

[3] Venkata Balaji T E, Limits of rank 4 Azumaya algebras and applications to desingularisation, Proc. Indian Acad. Sci. (Math. Sci.), Vol.112, No.4, November 2002, pp.485–537

[4] Bourbaki N, Éléments de Mathématique XXIV Livre 2, (1959) (Hermann, Paris)

[5] Venkata Balaji T E, On a Theorem of Seshadri on Limits of (2 × 2)-Matrix Algebras with Applications to Desingularisation and Quadratic Modules, Ph. D. Thesis, University of Madras, June 2000

[6] Grothendieck A, Dieudonné J A, Eléments de Géométrie Algébrique I, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Band 166 (1971) (Springer-Verlag)

[7] Grothendieck A, Dieudonné J A, Eléments de Géométrie Algébrique IV, IHES Pub.Math.Nos.20, 24, 28, 32 (1964–1967)

[8] Grothendieck A, Dieudonné J A, Eléments de Géométrie Algébrique II, IHES Pub.Math.No.8 (1961)

[9] Robin Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, GTM 52 (Springer-Verlag, New York)

[10] V. Lakshmibai and C. S. Seshadri, Geometry of G/P – II. Proc. Ind. Acad. Sci. 87 A, 1978, pp. 1–54.

[11] C. DeConcini and C. Procesi, A Characteristic-free approach to invariant theory. Advances in Math. 21, 1976, pp. 330–354.