A Limiting Version of a Theorem in Cohomology

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

June 27, 2004

Abstract

This paper completes the study started in [1]. Scheme-theoretic methods are used to classify linebundle-valued rank 3 quadratic bundles. The classification is done in terms of schematic specialisations of rank 4 Azumaya algebra bundles in the sense of Part A, [2]. For a quadratic form q on a rank 3 vector bundle V with values in a line bundle I over a scheme X, the degree zero subalgebra C_{0}(V, q, I) of the generalised Clifford algebra C̃(V, q, I) of the triple (V, q, I) in the sense of Bichsel-Knus [3], is seen to be such a specialised algebra by results in Part A, [2]. The Witt-invariant of (V, q, I), which may be defined as the the isomorphism class (as algebra bundle) of C_{0}(V, q, I), is shown to determine (V, q, I) upto tensoring by a twisted discriminant line bundle. Further, each specialised algebra arises in this way upto isomorphism, so that the association (V, q, I) ↦ C_{0}(V, q, I) induces a natural bijection from the set of equivalence classes of line-bundle-valued quadratic forms on rank 3 vector bundles upto tensoring by a twisted discriminant bundle and the set of isomorphism classes of schematic specialisations of rank 4 Azumaya bundles over X. This statement may be viewed as a limiting version of the natural bijection involving cohomology given by Ȟ1_{fppf}(X, O_{3})/Ȟ1_{fppf}(X, μ_{2}) ≅ Ȟ1_{étale}(X, PGL_{2}). The special, usual and the general orthogonal groups of (V, q, I) are computed and canonically determined in terms of Aut(C_{0}(V, q, I)), and it is shown that the general orthogonal group is always a semidirect product. Any element of Aut(C_{0}(V, q, I)) 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, I) are naturally isomorphic. A specialised algebra bundle A arises as C_{0}(V, q, I) with I ≅ O_{X} iff det(A) ∈ 2.Pic(X); and arises with q induced from a global bilinear form iff the line subbundle generated by 1_{A} is a direct summand of A. The use of the nice technical notion of semiregularity introduced by Kneser in [4] allows working with an arbitrary X, some (or even all) of whose points may have residue fields of characteristic two.

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

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

Contents

1 Overview of the Main Results 2

Line-bundle-valued Quadratic Forms on Rank 3 Bundles over Schemes 2

2 Notations and Preliminaries 5 3 Proof of Injectivity: Theorems 1.4 & 1.5 9

4 Proof of Surjectivity: Theorems 1.6 & 1.7 11

Acknowledgements 13

References 13

1 Overview of the Main Results

THEOREM 1.1 Over each scheme X the natural map (V, q, I) ↦ C_{0}(V, q, I) that associates to a quadratic bundle the degree zero subalgebra of its generalised Clifford algebra, induces a bijection from the set of orbits

\{ \begin{matrix}isomorphism classes (V, q, I), with \\ rank_{X}(V) = 3, rank_{X}(I) = 1 and \\ q a quadratic form on the bundle V \\ with values in the line bundle I \end{matrix} \} modulo the group \{ \begin{matrix}isomorphism classes (L, h, J), \\ with rank_{X}(L) = rank_{X}(J) = 1 \\ and h : L ⊗ L ≅ J a linear \\ isomorphism; product induced by ⊗ \end{matrix} \}

to the set of algebra-bundle-isomorphism classes of associative unital algebra structures A, on vector bundles of rank 4, that are Zariski-locally isomorphic to even Clifford algebras of rank 3 quadratic bundles. This bijection maps the subset

\{ \begin{matrix}isomorphism classes (V, q, I), with \\ q semiregular \end{matrix} \} modulo the group \{ \begin{matrix}isomorphism classes (L, h, J), \\ as above \end{matrix} \}

surjectively onto the subset of isomorphism classes of rank 4 Azumaya algebra bundles on X.

It was shown in Part A, [2] that algebra bundles such as A in the statement above are precisely the scheme-theoretic specialisations (or limits) of rank 4 Azumaya algebra bundles on X. The main result there was the smoothness of the schematic closure of Azumaya algebra structures on a fixed vector bundle of rank 4 over any scheme. Part B of [2] had applied this result to obtain desingularisations (with good specialisation properties) of certain moduli spaces over fairly general base schemes, and applications to the study of degenerations of quadratic forms on rank 3 vector bundles was initiated in [1]. Though [1] only treats forms with values in the structure sheaf, in the following we consider line-bundle-valued quadratic forms. Such a viewpoint is affordable thanks to the construction of Bichsel-Knus [3] of the generalised Clifford algebra C̃(V, q, I). This is an ℤ-graded algebra and we let C_{0}(V, q, I) denote its degree zero subalgebra. The other difference with [1] is to consider C_{0}(V, q, I) instead of the usual even Clifford algebra of a quadratic form with values in the structure sheaf. We shall see later ((b1), Theorem 1.6) that it is necessary to consider line-bundle-valued quadratic forms to obtain the surjectivity of Theorem 1.1 in those cases for which det(A) ∉ 2.Pic(X).

Remark 1.2 Assume for simplicity that X is an affine scheme. Following Sec.3, Chap.V, of the book of Knus [5], it can be seen that the association of a semiregular quadratic bundle to its even Clifford algebra induces a bijection from the set of orbits

\{ \begin{matrix}isomorphism classes (V, q, O_{X}), \\ with rank_{X}(V) = 3, and \\ q semiregular \end{matrix} \} modulo the group \{ \begin{matrix}isomorphism classes (L, h), with \\ rank_{X}(L) = 1 and h : L ⊗ L ≅ O_{X} \\ O_{X}-linear; product under ⊗ \end{matrix} \}

with the set of isomorphism classes of rank 4 Azumaya bundles on X. The group above is also called the group of discriminant bundles on X and is denoted by Disc(X). By Prop.3.2.2, Sec.3, Chap.III, [5], it is naturally isomorphic to the cohomology (abelian group) Ȟ1_{fppf}(X, μ_{2}). We may thus call the triples (L, h, J) occurring in the statement of Theorem 1.1 as twisted discriminant bundles. Further, by Lemma 3.2.1, Sec.3, Chap.IV, [5] the cohomology Ȟ1_{fppf}(X, O_{3}) classifies the set of isomorphism classes of semiregular rank 3 quadratic bundles with values in the trivial line bundle. On the other hand, the set of isomorphism classes of rank 4 Azumaya algebras on X may be interpreted as the cohomology Ȟ1_{étale}(X, PGL_{2}) (see page 145, Sec.5, Chap.III, [5]). Thus we have the bijection Ȟ1_{fppf}(X, O_{3})/Ȟ1_{fppf}(X, μ_{2}) ≅ Ȟ1_{étale}(X, PGL_{2}) and Theorem 1.1 may be considered as the limiting version of this bijection.

The following results lead to a proof of Theorem 1.1. Their proofs in turn follow from techniques of [1] adapted to include the case of line-bundle-valued forms. The proof of the injectivity essentially follows from the next couple of theorems, which describe the general, usual and special orthogonal groups of (V, q, I) in terms of the (algebra) automorphism group of C_{0}(V, q, I).

A bilinear form b, with values in a line bundle I, defined on a vector bundle V over the scheme X induces an I-valued quadratic form q_{b} given on sections by x ↦ b(x, x). Further, b also naturally defines an ℤ-graded linear isomorphism ψ_{b} : C̃(V, q_{b}, I) ≅ Λ(V) ⊗ L[I] (see (2d), Theorem 2.1). Here L[I] := O_{X} ⊕ (⨁_{n>0}(T^{n}(I) ⊕ T^{n}(I^{−1}))) is the Laurent-Rees algebra of I, where elements of V (resp. of I) are declared to be of degree one (resp. of degree two). In fact, C̃(V, 0, I) = Λ(V) ⊗ L[I]. Since in general a quadratic bundle (V, q, I) on a non-affine scheme X may not be induced from a global I-valued bilinear form, one is unable to identify the ℤ–graded vector bundle underlying its generalised Clifford algebra bundle with Λ(V) ⊗ L[I]. The following result overcomes this problem.

PROPOSITION 1.3 Every isomorphism of algebra-bundles ϕ : C_{0}(V, q, I) ≅ C_{0}(V′, q′, I′) is naturally associated to an isomorphism of bundles ϕ_{Λ^{2}} : Λ^{2}(V) ⊗ I^{−1} ≅ Λ^{2}(V′) ⊗ (I′)^{−1} which induces a map

ζ_{Λ^{2}} : Iso[C_{0}(V, q, I), C_{0}(V′, q′, I′)] ⟶ Iso[Λ^{2}(V) ⊗ I^{−1}, Λ^{2}(V′) ⊗ (I′)^{−1}] : ϕ ↦ ϕ_{Λ^{2}}

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

Denote by Sim[(V, q, I), (V′, q′, I′)] the set of isomorphisms from (V, q, I) to (V′, q′, I′). These are generalised similarities i.e., pairs (g, m) such that g : V ≅ V′ and m : I ≅ I′ are linear isomorphisms and the following diagram commutes (where q and q′ are considered as morphisms of sheaves of sets):

V ⟶_{≅}^{g} V′
q↓  ↓^{q′}
I ⟶_{m}^{≅} I′

When I = I′, since an m ∈ Aut(I) may be thought of as multiplication by a scalar l ∈ Γ(X, O_{X}^{∗}) ≡ Aut(I), we may call the isomorphism (g, m) as an I-similarity with multiplier l. In such a case we may as well denote (g, m) by the pair (g, l) and we often write g : (V, q, I) ≅_{l} (V′, q′, I). Let Iso[(V, q, I), (V′, q′, I)] be the subset of isometries (i.e., those pairs (g, m) with m = Identity or I-similarities with trivial multipliers). When V = V′, the subset of isometries with trivial determinant is denoted S-Iso[(V, q, I), (V, q′, I)]. On taking q = q′ these sets naturally become subgroups of Aut(V) × Γ(X, O_{X}^{∗}) = GL(V) × Γ(X, O_{X}^{∗}) and we define

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

Of course, O(V, q, I) and SO(V, q, I) may be thought of as subgroups of GL(V) ≡ GL(V) × {1}.

THEOREM 1.4 For I-valued quadratic forms q and q′ on a rank 3 vector bundle V over a scheme X, we have 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, I), (V, q′, I)] ⟶^{≅} S-Iso[C_{0}(V, q, I), C_{0}(V, q′, I)]
inj↓  ↓inj
Iso[(V, q, I), (V, q′, I)] \underrightarrow{onto} Iso′[C_{0}(V, q, I), C_{0}(V, q′, I)]
inj↓  ↓inj
Sim[(V, q, I), (V, q′, I)] \underrightarrow{onto} Iso[C_{0}(V, q, I), C_{0}(V, q′, I)]

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 a similarity to its multiplier, det(g, l) := det(g) for an I-similarity g with multiplier l and ζ_{Λ^{2}} is the map of Prop.1.3 above):

(a) there is a family of sections s_{2k+1} : Iso[C_{0}(V, q, I), C_{0}(V, q′, I)] ⟶ Sim[(V, q, I), (V, q′, I)] 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, I), C_{0}(V, q′, I)] ⟶ Iso[(V, q, I), (V, q′, I)] such that det^{2} ∘ s′ = det ∘ ζ_{Λ^{2}};

(c) there is a family of sections s_{2k+1}^{+} : Iso[C_{0}(V, q, I), C_{0}(V, q′, I)] ⟶ Sim[(V, q, I), (V, q′, I)] 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}, I), C_{0}(V, q_{i+1}, I)] 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 maps followed by the canonical quotient map, on taking the quotient of GL(V) × Γ(X, O_{X}^{∗}) by μ_{2}(Γ(X, O_{X})).Id_{V} × {1}, become multiplicative.

THEOREM 1.5 For a rank 3 quadratic bundle (V, q, I) 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 I-(self)similarity its multiplier:

[figure: SO(V, q, I); inj; ≅; S-Aut(C_{0}(V, q, I)); inj; 1; μ_{2}(Γ(X, O_{X})); inj; O(V, q, I); inj; Aut_{′}(C_{0}(V, q, I)); inj; 1; 1; Γ(X, O_{X}^{∗}); GO(V, q, I); det^{2}×l^{−3}; Aut(C_{0}(V, q, I)); 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, I)) ⟶ GO(V, q, I) 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, I)) does not necessarily take values in O(V, q, I), but the further restriction to S-Aut(C_{0}(V, q, I)) does take values in SO(V, q, I). In particular, GO(V, q, I) is a semidirect product. The maps s_{2k+1} and s′ of Theorem 1.4 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, I) has determinant 1. Hence Aut(C_{0}(V, q, I)) = Aut′(C_{0}(V, q, I)) = S-Aut(C_{0}(V, q, I)) and O(V, q, I) is the semidirect product of μ_{2}(Γ(X, O_{X})) and SO(V, q, I).

The proofs of the above results, and of the injectivity part of Theorem 1.1, will be given in §3. As for the proof of the surjectivity part of Theorem 1.1, we have

THEOREM 1.6 Let X be a scheme and A a specialisation of rank 4 Azumaya algebra bundles on X. Let O_{X}.1_{A} ↪ A be the line sub-bundle generated by the nowhere-vanishing global section of A corresponding to the unit for algebra multiplication.

(a) There exist a rank 3 vector bundle V on X, a quadratic form q on V with values in the line bundle I := det^{−1}(A), and an isomorphism of algebra bundles A ≅ C_{0}(V, q, I). This gives the surjectivity in the statement of Theorem 1.1. Further, the following linear isomorphisms may be deduced:

(1) det(A) ⊗ Λ^{2}(V) ≅ A/O_{X}.1_{A}, from which follow:

(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) There exists a quadratic bundle (V′, q′, I′) such that C_{0}(V′, q′, I′) ≅ A and with

(1) I′ = O_{X} iff det(A) ∈ 2.Pic(X);

(2) with q′ induced from a global I′-valued bilinear form iff O_{X}.1_{A} is an O_{X}-direct summand of A; (3) with both I′ = O_{X} and with q′ induced from a global bilinear form (with values in I′) iff O_{X}.1_{A}

is an O_{X}-direct summand of A and det(A) ∈ 2.Pic(X).

The next theorem is a key ingredient in the proof of part (a) above. It describes specialisations as bilinear forms under certain conditions. As a preparation towards its statement, we recall a few definitions and results from Part A, [2]. 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. If Stab_{w} ⊂ GL_{W} is the stabiliser subgroupscheme of w, recall from Theorems 3.4 and 3.8, Part A, [2], 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,I)} be the associated rank 9 vector bundle of bilinear forms on V with values in the line bundle I. Let Bil_{(V,I)}^{sr} ↪ Bil_{(V,I)} correspond to the open subscheme of semiregular bilinear forms. We say that a bilinear form b is semiregular if there is a trivialisation {U_{i}} of I, such that over each open subscheme U_{i}, the quadratic form q_{i} with values in the trivial line bundle induced from q_{b}|U_{i} is semiregular (it may turn out that a semiregular bilinear form may be degenerate). This definition is independent of the choice of a trivialisation, since q_{i} is semiregular iff λq_{i} is semiregular for every λ ∈ Γ(U_{i}, O_{X}^{∗}). Let W := Λ^{even}(V, I) := ⨁_{n≥0} Λ^{2n}(V)⊗I^{−⊗n} and let w ∈ Γ(X, W) be the nowhere-vanishing global section corresponding to the unit for the natural multiplication in the twisted even-exterior algebra bundle W. There is an obvious natural action of GL_{V} on Bil_{(V,I)}. There is also a natural morphism of groupschemes GL_{V} ⟶ Stab_{w} given on valued points by g ↦ ⨁_{n≥0} Λ^{2n}(g) ⊗ Id 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 1.7

(1) Let V be a rank 3 vector bundle on the scheme X, W := Λ^{even}(V, I) and w ∈ Γ(X, W) correspond to 1 in the twisted even-exterior algebra bundle. There is a natural GL_{V} -equivariant morphism of X–schemes Υ′ = Υ′_{X} : Bil_{(V,I)} ⟶ 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,I)} ⟶ Id-w-Sp-Azu_{W}, then Υ is a GL_{V} -equivariant isomorphism and it maps the GL_{V} -stable open subscheme Bil_{(V,I)}^{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,I)}. 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}, resp. I_{T}) denote the pullback of V (resp. W, resp. I) 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,I)} ×_{X} T ⟶ Id-w-Assoc_{W} ×_{X} T and Υ_{X} ×_{X} T : Bil_{(V,I)} ×_{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} ,I_{T})} ⟶ Id-w_{T} -Assoc_{W_{T}} and Υ_{T} : Bil_{(V_{T} ,I_{T})} ≅ Id-w_{T} -Sp-Azu_{W_{T}} .

The proofs of Theorems 1.6 and 1.7 will be given in §4. Some notations and preliminaries are explained in §2.

2 Notations and Preliminaries

This section collects together some definitions and results. It is essentially Section 2 of [1] redone for forms with values in line bundles. We omit the proofs which follow by localisation and the corresponding results of Sec.2, [1]. For the systematic treatment of the generalised Clifford algebra and its properties, we refer the reader to the paper of Bichsel-Knus [3].

Quadratic and Bilinear Forms with Values in a Line Bundle. Let V be a vector bundle

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

0 ⟶ Alt_{(V,I)}^{2} ⟶ Bil_{(V,I)} ⟶ Quad_{(V,I)} ⟶ 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,I)}^{2}-bundle:

Alt_{(V,I)}^{2} ↪ Bil_{(V,I)} ↠ Quad_{(V,I)}.

Given a quadratic form q ∈ Γ(U, Quad_{(V,I)}), recall that the usual ‘associated’ symmetric bilinear form b_{q} ∈ Γ(U, Bil_{(V,I)}) is given on sections (over open subsets of U) by v ⊗ v′ ↦ q(v + v′) − q(v) − q(v′). 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). 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,I)}^{2}) = 0.

The Generalised Clifford Algebra of Bichsel-Knus. LetR be a commutative ring (with 1),

I an invertible R-module and V a projective R-module. Let L[I] := R ⊕ (⨁_{n>0}(T^{n}(I) ⊕ T^{n}(I^{−1}))) be the Laurent-Rees algebra of I, and define the ℤ-gradation on the tensor product of algebras T V ⊗ L[I] by requiring elements of V (resp. of I) to be of degree one (resp. of degree two). Let q : V ⟶ I be an Ivalued quadratic form on V. Following the definition of Bichsel-Knus [3], let J(q, I) be the two-sided ideal of T V ⊗L[I] generated by the set {(x⊗_{T V} x)⊗1_{L[I]} −1_{T V} ⊗q(x) | x ∈ V } and let the generalised Clifford algebra of q be defined by C̃(V, q, I) := T V ⊗ L[I]/J(q, I). This is an ℤ-graded algebra by definition. Let C_{n} be the submodule of elements of degree n. Then C_{0} is a subalgebra, playing the role of the even Clifford algebra in the classical situation (i.e., I = R) and C_{1} is a C_{0}-bimodule. Bichsel and Knus baptize C_{0} and C_{1} respectively as the even Clifford algebra and the Clifford module associated to the triple (V, q, I). The generalised Clifford algebra satisfies an appropriate universal property which ensures it behaves well functorially. Since V is projective, the canonical maps V ⟶ C̃(V, q, I) and L[I] ⟶ C̃(V, q, I) are injective. For proofs of these facts, see Sec.3 of [3]. If (V, q, I) is an I-valued quadratic form on the vector bundle V over a scheme X, with I a line bundle, then the above construction may be carried out to define the generalised Clifford algebra bundle C̃(V, q, I) which is an ℤ-graded algebra bundle on X.

Bourbaki’s Tensor Operations with Values in a Line Bundle. Let R be a commutative

ring and L[I] := R ⊕ (⨁_{n>0}(T^{n}(I) ⊕ T^{n}(I^{−1}))) as above. We denote by ⊗_{T} (resp. by ⊗_{L}) the tensor product and by 1_{T} (resp. 1_{L}) the unit element in the algebra T V (resp. in L[I]).

THEOREM 2.1 (with the above notations)

(1) Let q : V ⟶ I be an I-valued quadratic form on V and f ∈ Hom_{R}(V, I). Then there exists an R-linear endomorphism t_{f} of the algebra T V ⊗ L[I] which is unique with respect to the first three of the following properties it satisfies:

(a) t_{f} (1_{T} ⊗ λ) = 0 ∀ λ ∈ L[I];

(b) t_{f} ((x ⊗_{T} y) ⊗ λ) = y ⊗ (f(x) ⊗_{L} λ) − (x ⊗_{T} 1_{T})t_{f} (y ⊗ λ) for any x ∈ V, y ∈ T V, and λ ∈ L[I];

(c) following the definition of Bichsel-Knus [3], let J(q, I) be the two-sided ideal of T V ⊗ L[I] generated by the set {(x ⊗_{T} x) ⊗ 1_{L} − 1_{T} ⊗ q(x) | x ∈ V }; then t_{f} (J(q, I)) ⊂ J(q, I);

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

(e) recall from the definition of Bichsel-Knus [3] that the generalised Clifford algebra of q is C̃(V, q, I) := T V ⊗ L[I]/J(q, I); by (c) above, t_{f} induces a ℤ-graded endomorphism of degree +1 denoted by d_{f}^{q} : C̃(V, q, I) ⟶ C̃(V, q, I);

(f) t_{f} ∘ t_{f} = 0;

(g) if g ∈ Hom_{R}(V, I), then t_{f} ∘ t_{g} + t_{g} ∘ t_{f} = 0; (h) if α ∈ End_{R}(V), then t_{f} ∘ (T (α) ⊗ Id_{L[I]}) = (T (α) ⊗ Id_{L[I]}) ∘ t_{α^{∗}f} where α^{∗}f ∈ Hom_{R}(V, I) is

defined by x ↦ f(α(x));

(i) t_{f} ≡ 0 on the R-subalgebra of T V ⊗ L[I] generated by kernel(f) ⊗ L[I]. In fact, atleast when V is a projective R-module, the smallest R-subalgebra of T V ⊗ L[I] containing kernel(f) ⊗ R.1_{L} is Image(T (kernel(f)) ⊗ R.1_{L} and t_{f} vanishes on this R-subalgebra.

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

(a) Ψ_{b}(1_{T} ⊗ λ) = (1_{T} ⊗ λ) ∀λ ∈ L[I];

(b) Ψ_{b}((x ⊗_{T} y) ⊗ λ) = (x ⊗ 1_{L}).Ψ_{b}(y ⊗ λ) + t_{b_{x}}(Ψ_{b}(y ⊗ λ)) for any x ∈ V, y ∈ T V and λ ∈ L[I];

(c) Ψ_{b}(J(q′, I)) ⊂ J(q, I);

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

ψ_{b} : C̃(V, q′, I) ≅ C̃(V, q, I);

in particular, given a quadratic form q_{1} : V ⟶ I, since there always exists an I-valued bilinear form b_{1} that induces q_{1} (i.e., such that q_{1} = q_{b_{1}}), setting q′ = q_{1}, q = 0 and b = b_{1} in the above gives an ℤ-graded linear isomorphism ψ_{b_{1}} : C̃(V, q_{1}, I) ≅ C̃(V, 0, I) = Λ(V) ⊗ L[I];

(e) Ψ_{b}(T^{2n}V ⊗ L[I]) ⊂ ⊕_{(i≤n)}(T^{2i}V ⊗ L[I]), Ψ_{b}(T^{2n+1}V ⊗ L[I]) ⊂ ⊕_{(odd i≤2n+1)}(T^{i}V ⊗ L[I]), Ψ_{b}(T^{2n}V ⊗ I^{−n}) ⊂ ⊕_{(i≤n)}(T^{2i}V ⊗ I^{−i}) and Ψ_{b}(T^{2n+1}V ⊗ I^{−n}) ⊂ ⊕_{(odd i≤2n+1)}(T^{i}V ⊗ I^{\frac{1−i}{2}} );

(f) in particular, for x, x′ ∈ V, Ψ_{b}((x ⊗_{T} x′) ⊗ 1_{L}) = (x ⊗_{T} x′) ⊗ 1_{L} + 1_{T} ⊗ b(x, x′) so that for ψ_{b} : C_{0}(V, q_{b}, I) ≅ C_{0}(V, 0, I) = ⊕_{n≥0}(Λ^{2n}(V)⊗I^{−n}) we have ψ_{b}(((x⊗_{T} x′)⊗ζ) mod J(q_{b}, I)) = (x ∧ x′) ⊗ ζ + ζ(b(x, x′)).1 for any x, x′ ∈ V and ζ ∈ I^{−1} ≡ Hom_{R}(I, R);

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

(h) for I-valued bilinear forms b_{i} on V, Ψ_{b_{1}+b_{2}} = Ψ_{b_{1}} ∘ Ψ_{b_{2}} and Ψ_{0} = Identity on T V ⊗ L[I];

(i) for any α ∈ End_{R}(V), Ψ_{b} ∘ (T (α) ⊗ Id_{L[I]}) = (T (α) ⊗ Id_{L[I]}) ∘ 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, I), +) ⟶ (Aut_{R}(T V ⊗ L[I]), ∘) : b ↦ Ψ_{b}; the associative unital monoid (End_{R}(V), ∘) acts on Bil_{R}(V, I) on the right by b′ ⇝ b′.α and acts on the left (resp. on the right) of End_{R}(T V ⊗L[I]) by α.Φ := (T (α)⊗Id_{L[I]})∘Φ (resp. by Φ.α := Φ ∘ (T (α) ⊗ Id_{L[I]}), 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, I) by g.b : (x, x′) ↦ b(g^{−1}(x), g^{−1}(x′)) and on the left of Aut_{R}(T V ⊗ L[I]) by conjugation via the natural group homomorphism GL_{R}(V) ⟶ Aut_{R}(T V ⊗ L[I]) : g ↦ T (g) ⊗ Id_{L[I]} i.e., g.Φ := (T (g) ⊗ Id_{L[I]}) ∘ Φ ∘ (T (g^{−1}) ⊗ Id_{L[I]}), 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}) ⟶ (I ⊗_{R} S =: I_{S}) be the I_{S}-valued quadratic forms induced from the quadratic forms q, q′ of (2) above and (b ⊗_{R} S) ∈ Bil_{S}(V_{S}, I_{S}) the I_{S}-valued S-bilinear 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-graded S-linear isomorphism (ψ_{b} ⊗_{R} S) : C̃(V_{S}, (q′ ⊗_{R} S), I_{S}) ≅ C̃(V_{S}, (q ⊗_{R} S), I_{S}) induced from ψ_{b} of (2d) above may be canonically identified with ψ_{(b⊗_{R}S)}.

Remark 2.2 As mentioned in [3], F. van Oystaeyen has observed that L[I] is a faithfully-flat splitting for I, and the generalised Clifford algebra C̃(V, q, I) is nothing but the “classical” Clifford algebra of the triple (V ⊗_{R}L[I], q⊗_{R} L[I], I ⊗_{R}L[I] ≡ L[I]) over L[I]. In the same vein, I-valued forms (both multilinear and quadratic) on an R-module V can be treated as the usual (L[I]-valued) forms on V ⊗_{R} L[I]. With this in mind, the proof of the above proposition follows from the usual Bourbaki tensor operations with respect to V ⊗_{R} L[I] on L[I]. (See §9, Chap.9, [6] or para.1.7, Chap.IV, [5], or Theorem 2.1 [1] for the “classical” Bourbaki operations). However one needs to remember that the ℤ-gradation on T V ⊗_{R} L[I] as defined above is different from the usual ℤ_{≥0}-gradation on T (V ⊗_{R} L[I]).

Tensoring by Symmetric Bilinear Bundles and Twisted Discriminant Bundles. Let

V, M be vector bundles on a scheme X and let I, J be line bundles on X. Let q be a quadratic form on V with values in I and let b be a symmetric bilinear form on M with values in J. PROPOSITION 2.3 (with the above notations)

(1) The tensor product of (V, q, I) with (M, b, J) gives a unique quadratic bundle (V ⊗ M, q ⊗ b, I ⊗ J). 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 M is a line bundle, (M, b, J) is regular (=nonsingular) iff (M, q_{b}, J) is semiregular iff b : M ⊗ M ≅ J is an isomorphism (i.e., in other words iff (M, b, J) is a twisted discriminant bundle).

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

PROPOSITION 2.4 Let V and V′ be vector bundles of the same rank on the scheme X, (L, h, J) a twisted discriminant line bundle on X and α : V′ ≅ V ⊗ L an isomorphism of bundles.

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

V′ ⊗ V′ ⊗ J^{−1} ⟶^{α⊗α⊗Id (≅)} V ⊗ L ⊗ V ⊗ L ⊗ J^{−1} ⟶^{SWAP(2,3) (≡)} V ⊗ V ⊗ L^{2} ⊗ J^{−1} ⟶^{Id⊗h⊗Id (≅)}

⟶^{Id⊗h⊗Id (≅)} V ⊗ V ⊗ J ⊗ J^{−1} ⟶^{CANON (≡)} V ⊗ V.

Then the association b′ ↦ b induces linear isomorphisms shown by vertical upward arrows in the following diagram of associated locally-free sheaves (with exact rows) making it commutative:

0 ⟶ Alt_{(V,I⊗J^{−1})}^{2} ⟶ Bil_{(V,I⊗J^{−1})} ⟶ Quad_{(V,I⊗J^{−1})} ⟶ 0
≅↑  ≅↑  ≅↑
0 ⟶ Alt_{(V′,I)}^{2} ⟶ Bil_{(V′,I)} ⟶ Quad_{(V′,I)} ⟶ 0.

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

Alt_{(V,I⊗J^{−1})}^{2} closed ; ⟶ ; immersion Bil_{(V,I⊗J^{−1})} locally ; ⟶ ; trivial Quad_{(V,I⊗J^{−1})}
≅↑  ≅↑  ≅↑
Alt_{(V′,I)}^{2}  closed ; ⟶ ; immersion Bil_{(V′,I)}  locally ; ⟶ ; trivial Quad_{(V′,I)}

(2) Let b′ ∈ Γ (X, Bil_{(V′,I)}) be a global bilinear form and let it induce b ∈ Γ (X, Bil_{(V,I⊗J^{−1})}) via α and h as defined in (1) above. Let Ψ_{b′} ∈ Aut_{O_{X}} (T V′ ⊗ L[I]) (resp. Ψ_{b} ∈ Aut_{O_{X}} (T V ⊗ L[I ⊗ J^{−1}])) be the ℤ-graded linear isomorphism induced by b′ (resp. by b) defined locally (and hence globally) as in (2), Theorem 2.1 above. Let Z_{(α,h)} : ⊕_{n≥0}(T_{O_{X}}^{2n} (V′) ⊗ I^{−n}) ≅ ⊕_{n≥0}(T_{O_{X}}^{2n} (V) ⊗ I^{−n} ⊗ J^{n}) be the O_{X}-algebra isomorphism induced via the isomorphism ζ_{(α,h)} defined in (1) above. Then, taking into account (2e), Theorem 2.1, the following diagram commutes:

⊕_{n≥0}(T_{O_{X}}^{2n} (V′) ⊗ I^{−n}) ⟶_{≅}^{Z_{(α,h)}} ⊕_{n≥0}(T_{O_{X}}^{2n} (V) ⊗ I^{−n} ⊗ J^{n})
Ψ_{b′} ↓≅  ≅↓^{Ψ_{b}}
⊕_{n≥0}(T_{O_{X}}^{2n} (V′) ⊗ I^{−n}) ⟶_{Z_{(α,h)}}^{≅} ⊕_{n≥0}(T_{O_{X}}^{2n} (V) ⊗ I^{−n} ⊗ J^{n})

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

C_{0}(V′, q_{b′}, I)  ⟶_{≅}^{via Z_{(α,h)}}  C_{0}(V, q_{b}, I ⊗ J^{−1})
ψ_{b′} ↓≅  ≅↓^{ψ_{b}}
⊕_{n≥0}(Λ_{O_{X}}^{2n} (V′) ⊗ I^{−n}) ⟶_{via Z_{(α,h)}}^{≅} ⊕_{n≥0}(Λ_{O_{X}}^{2n} (V) ⊗ I^{−n} ⊗ J^{n})

(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′, I) ≅ (V, b, I ⊗ J^{−1}) ⊗ (L, h, J) and also an isometry of the induced quadratic bundles α : (V′, q_{b′}, I) ≅ (V, q_{b}, I ⊗ J^{−1}) ⊗ (L, h, J). Moreover, if we are just given a global I ⊗ J^{−1}-valued quadratic form q on V (resp. an I-valued q′ on V′), then we may define the global I-valued quadratic form q′ on V′ (resp. I ⊗J^{−1}-valued q on V) via q′ := (q ⊗h)∘α (resp. via q := (q′ ∘α^{−1})⊗(h^{∨})^{−1})) and again α : (V′, q′, I) ≅ (V, q, I ⊗ J^{−1}) ⊗ (L, h, J) becomes an isometry of quadratic bundles.

PROPOSITION 2.5 Let g : (V, q, I) ≅_{l} (V′, q′, I) be an I-similarity with multiplier l ∈ Γ(X, O_{X}^{∗}).

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

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

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

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

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

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

A local computation shows that tensoring by a twisted discriminant bundle amounts to (locally) applying a similarity. In this case also one gets a global isomorphism of even Clifford algebras:

PROPOSITION 2.6 Let (V, q, I) be a quadratic bundle on a scheme X and (L, h, J) be a twisted discriminant bundle. There exists a unique isomorphism of algebra bundles

γ_{(L,h,J)} : C_{0} ((V, q, I) ⊗ (L, h, J)) ≅ C_{0}(V, q, I)

given by γ_{(L,h,J)} ((v ⊗ λ).(v′ ⊗ λ′).(s ⊗ t)) = t(h(λ ⊗ λ′))v.v′.s for any sections v, v′ of V, λ, λ′ of L, s of I^{−1} ≡ I^{∨} and t of J^{−1} ≡ J^{∨}.

3 Proof of Injectivity: Theorems 1.4 & 1.5

Proof of Prop.1.3: Start with an isomorphism of algebra-bundles ϕ : C_{0}(V, q, I) ≅ C_{0}(V′, q′, I′). Let {U_{i}}_{i∈I} be an affine open covering of X (which may also be chosen so as to trivialise some or any of the involved bundles if needed). Choose bilinear forms b_{i} ∈ Γ(U_{i}, Bil_{(V,I)}) and b′_{i} ∈ Γ(U_{i}, Bil_{(V′,I′)}) 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, q, I)|U_{i}  ⟶_{≅}^{ϕ|U_{i}}  C_{0}(V′, q′, I′)|U_{i}
ψ_{b_{i}} ↓≅  ≅↓^{ψ_{b′_{i}}}
(O_{X} ⊕ Λ^{2}(V) ⊗ I^{−1})|U_{i} ⟶_{ϕ_{Λ{ev \atop i}}}^{≅} (O_{X} ⊕ Λ^{2}(V′) ⊗ (I′)^{−1})|U_{i}

The linear isomorphism ϕ_{Λ_{i}^{ev}} preserves 1 and therefore it induces a linear isomorphism from (Λ^{2}(V) ⊗ I^{−1})|U_{i} to (Λ^{2}(V′) ⊗ (I′)^{−1})|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)⊗I^{−1})|U_{i} (resp. onto (Λ^{2}(V′)⊗(I′)^{−1})|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) ⊗ I^{−1} ≅ Λ^{2}(V′) ⊗ (I′)^{−1}. Q.E.D, Prop.1.3.

Reduction of Proof of Injectivity of Theorem 1.1 to Theorem 1.4. We start with an

isomorphism of algebra-bundles ϕ : C_{0}(V, q, I) ≅ C_{0}(V′, q′, I′), construct the isomorphism of vector bundles ϕ_{Λ^{2}} : Λ^{2}(V) ⊗ I^{−1} ≅ Λ^{2}(V′) ⊗ (I′)^{−1} and keep the notations introduced in the proof of Prop.1.3. Firstly we deduce a linear isomorphism det((ϕ_{Λ^{2}})^{∨})^{−1} : det((Λ^{2}(V) ⊗ I^{−1})^{∨}) ≅ det((Λ^{2}(V′) ⊗ (I′)^{−1})^{∨}). 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} and J := I′⊗I^{−1} then we get a twisted discriminant line bundle (L⊗J^{−1}, h, J) and a vector bundle isomorphism α : V′ ≅ V ⊗(L⊗J^{−1}).

Now for each i ∈ I, the bilinear form b_{i} ∈ Γ(U_{i}, Bil_{(V,I)}) induces, via α|U_{i} and (LJ^{−1}, h, J)|U_{i} and (1), Prop.2.4, a bilinear form b′′_{i} ∈ Γ(U_{i}, Bil_{(V′,IJ)}). 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}, IJ|U_{i}) ≅ (V |U_{i}, b_{i}, I|U_{i})⊗(LJ^{−1}, h, J)|U_{i} and also an isometry of quadratic bundles α|U_{i} : (V′|U_{i}, q_{b′′_{i}}, IJ|U_{i}) ≅ (V |U_{i}, q_{b_{i}} = q|U_{i}, I|U_{i}) ⊗ (LJ^{−1}, h, J)|U_{i}.

On the other hand, by an assertion in (3), Prop.2.4, we could also define the global quadratic bundle (V′, q′′, IJ) using (V, q, I), α and (LJ^{−1}, h, J), so that we have an isometry of quadratic bundles α : (V′, q′′, IJ) ≅ (V, q, I) ⊗ (LJ^{−1}, h, J). 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.2.5, there exists a unique isomorphism of algebra bundles

C_{0}(α, 1, IJ) : C_{0}(V′, q′′, IJ) ≅ C_{0} ((V, q, I) ⊗ (LJ^{−1}, h, J))

and by Prop.2.6 we have a unique isomorphism of algebra bundles

γ_{(LJ^{−1},h,J)} : C_{0} ((V, q, I) ⊗ (LJ^{−1}, h, J)) ≅ C_{0}(V, q, I).

Therefore the composition of the following sequence of isomorphisms of algebra bundles on X

\begin{matrix}C_{0}(V′, q′′, I′) ≅^{using I′≅IJ} C_{0}(V′, q′′, IJ) ⟶^{C_{0}(α,1,IJ) (≅)} C_{0} ((V, q, I) ⊗ (LJ^{−1}, h, J)) ⟶^{γ_{(LJ^{−1},h,J)} (≅)} \\ ⟶^{γ_{(LJ^{−1},h,J)} (≅)} C_{0}(V, q, I) ⟶^{ϕ (≅)} C_{0}(V′, q′, I′) \end{matrix}

is an element of Iso[C_{0}(V′, q′′, I′), C_{0}(V′, q′, I′)], which, granting Theorem 1.4, is induced by a similarity in (g, l) ∈ Sim[(V′, q′′, I′), (V′, q′, I′)]. Hence we would have

g : (V′, q′′, I′) ≅ (V′, q′, I′) ⊗ (O_{X}, mult.by l^{−1}, O_{X})

where l ∈ Γ(X, O_{X}^{∗}). This combined with the fact that (V, q, I) and (V′, q′′, I′) ≅ (V′, q′′, IJ) are isomorphic upto the twisted discriminant bundle (LJ^{−1}, h, J) by the construction above would imply that (V, q, I) and (V′, q′, I′) also differ by a twisted discriminant bundle. Therefore the proof of the injectivity asserted in Theorem 1.1 reduces to the proof of Theorem 1.4.

Reduction of Theorem 1.4 to the Case when I is free: For a similarity g with multiplier l,

we have C_{0}(g, l, I) given by (1), Prop.2.5, so that we may define the map

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

The equality det(C_{0}(g, l, I)) = det ((C_{0}(g, l, I))_{Λ^{2}}) = l^{−3}det^{2}(g) was shown to hold (locally, hence globally) (1), Lemma 3.11, [1]. Thus Iso[(V, q, I), (V, q′, I)] and S-Iso[(V, q, I), (V, q′, I)] are respectively mapped into Iso′[C_{0}(V, q, I), C_{0}(V, q′, I)] and S-Iso[C_{0}(V, q, I), C_{0}(V, q′, I)] as claimed. We start with an isomorphism of algebra-bundles ϕ : C_{0}(V, q, I) ≅ C_{0}(V, q′, I), which by Prop.1.3 leads to the automorphism of vector bundles ϕ_{Λ^{2}} ∈ Aut(Λ^{2}(V) ⊗ I^{−1}). Firstly, define the global bundle automorphism g′ ∈ GL (V ⊗ (det(V))^{−1} ⊗ I) so that the following diagram commutes:

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

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} ⊗ I) be the image of g′ i.e., the image of g′ ⊗ det(V) ⊗ I^{−1} under the canonical identification GL(V ⊗ (det(V))^{−1} ⊗ I ⊗ det(V) ⊗ I^{−1}) ≡ 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. Suppose we show the following locally for the Zariski toplogy on X (more precisely, for each open subscheme of X over which V and I are free):

(1) that g_{l}^{ϕ} is an I-similarity from (V, q, I) to (V, q′, I) with multiplier l;

(2) that g_{l}^{ϕ} induces ϕ i.e., with the notations of (1), Prop.2.5, that C_{0}(g_{l}^{ϕ}, l, I) = ϕ;

(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 and

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

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 1.4. But these maps are not necessarily multiplicative since a computation reveals that if ϕ_{i} ∈ Iso[C_{0}(V, q_{i}, I), C_{0}(V, q_{i+1}, I)] is associated to g_{l_{i}}^{ϕ_{i}} ∈ Sim[(V, q_{i}, I), (V, q_{i+1}, I)], 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, I), C_{0}(V, q′, I)], 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 1.4 to the case when the rank 3 vector bundle V and the line bundle I are free. More generally, even for the case V not necessarily free but I = O_{X}, this was treated in Theorem 1.3 of [1]. Q.E.D., Theorem 1.4 & injectivity of Theorem 1.1.

Proof of Theorem 1.5. (The case I = O_{X} was treated in Theorem 1.4 of [1]). Taking q′ = q in Theorem 1.4 gives the commutative diagram of groups and homomorphisms as asserted in the statement of the theorem. We continue with the notations above. For g ∈ GO(V, q, I) with multiplier l, that the equality det(C_{0}(g, l, I)) = det ((C_{0}(g, l, I))_{Λ^{2}}) = l^{−3}det^{2}(g) holds (locally, hence globally) was shown in (1), Lemma 3.11, [1]. Assertion (2) of the same Lemma shows the following (locally, and hence globally): if C_{0}(g, l, I) is the identity on C_{0}(V, q, I), then g = l^{−1}det(g).Id_{V}, and further if g ∈ O(V, q, I) then g = det(g).Id_{V} with det^{2}(g) = 1. This gives exactness at GO(V, q, I) and at O(V, q, I).

We proceed to prove assertion (b). Let ϕ ∈ Aut(C_{0}(V, q, I)), 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 and I are trivial over X. Fix a global basis {e_{1}, e_{2}, e_{3}} for V 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, [5]—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. In general, even if V and I are not necessarily trivial, since this equality holds over a covering of X which trivialises both V and I, it also holds over all of X. Q.E.D., Theorem 1.5.

4 Proof of Surjectivity: Theorems 1.6 & 1.7

Proof of Theorem 1.7. We adopt the notations introduced just before Theorem 1.7. Let T be an X-scheme. Given a bilinear form b ∈ Bil_{(V,I)}(T) = Γ(T, Bil_{(V_{T} ,I_{T})}), consider the linear isomorphism ψ_{b} : C_{0}(V_{T} , q_{b}, I_{T}) ≅ O_{T} ⊕ Λ^{2}(V_{T}) ⊗ (I_{T})^{−1} = 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}, I_{T}). By definition, A_{b} ∈ Id-w-Assoc_{W} (T) and we get a map of T -valued points

Υ′(T) : Bil_{(V,I)}(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,I)} ⟶ Id-w-Assoc_{W} . The morphism Υ′ is GL_{V} -equivariant due to (2j), Theorem 2.1. Notice that the schemes Bil_{(V,I)}, Bil_{(V,I)}^{sr} and Id-w-Assoc_{W} are well-behaved relative to X with respect to base-change. In fact, so are Id-w-Azu_{W} and Id-w-Sp-Azu_{W}, as may be recalled from Theorems 3.4 and 3.8, Part A, [2]. In view of these observations, by taking a trivialisation for I over X, we may reduce to the case when I is trivial. But this case was treated in Theorem 1.9 of [1]. Q.E.D., Theorem 1.7.

Proofs of assertions in (a), Theorem 1.6 and Surjectivity part of Theorem 1.1. 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). Recall that A_{i} is also a specialised algebra structure by Theorem 3.8, Part A, [2]. Hence by Theorem 1.7 applied to X = U_{i}, V = O_{U_{i}}^{⊕3} and I = O_{U_{i}}, we can also find an O_{U_{i}}-valued 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 1.4. 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. While the q_{i} do not glue to give an O_{X}-valued quadratic form on V, the facts that the multipliers {l_{ij}} form a cocycle for I := det^{−1}(A) and that s_{(−1)}^{+} is a section together imply, taking into account the uniqueness in (1), Prop.2.5, that actually the q_{i} glue to give an I-valued quadratic form q on V and that C_{0}(V, q, I) ≅ A. It was verified in page 28, [1] 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 1.6, 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.

Proofs of assertions in (b), Theorem 1.6. We first prove (b1). Let A be a given specialisation, and let A ≅ C_{0}(V, q, I) as in part (a) of Theorem 1.6 with I = det^{−1}(A). By the injectivity part of Theorem 1.1, we have C_{0}(V, q, I) ≅ A ≅ C_{0}(V′, q′, O_{X}) iff there exists a twisted discriminant bundle (L, h, J) and an isomorphism (V, q, I) ≅ (V′, q′, O_{X}) ⊗ (L, h, J). The latter implies that I ≅ J ≅ L^{2} and hence det(A) ∈ 2.Pic(X). On the other hand, if this last condition holds, we could take for L a square root of J := I^{−1}, alongwith an isomorphism h : L^{2} ≅ J and we would have by Prop.2.6 an algebra isomorphism γ_{(L,h,J)} : C_{0}(V ⊗ L, q ⊗ h, O_{X}) ≡ C_{0} ((V, q, I) ⊗ (L, h, J)) ≅ C_{0}(V, q, I) ≅ A.

For the proof of (b2), suppose that the line subbundle O_{X}.1_{A} ↪ A is a direct summand of A. We may choose a splitting A ≅ O_{X}.1_{A} ⊕ (A/O_{X}.1_{A}). Using assertion (1) of (a), Theorem 1.6, 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} ⊕ (Λ^{2}(V) ⊗ I^{−1}) ≅ O_{X}.1 ⊕ Λ^{2}(V) ⊗ I^{−1} =: W where I := det^{−1}(A) and the last isomorphism is chosen so as to map O_{X}.1_{A} isomorphically onto O_{X}.1. Therefore if (W, w) := (O_{X}.1⊕Λ^{2}(V)⊗I^{−1}, 1), then by the above identification A induces an element of Id-w-Sp-Azu_{W} (X), and since Υ : Bil_{(V,I)} ≅ Id-w-Sp-Azu_{W} is an X-isomorphism by (1), Theorem 1.7, it follows that there exists an I-valued global quadratic form q = q_{b} induced from an I-valued global bilinear form b on V such that the algebra structure Υ(b) ≅ A. (We recall that Υ(b) is the algebra structure induced from the linear isomorphism ψ_{b} : C_{0}(V, q = q_{b}, I) ≅ O_{X}.1 ⊕ Λ^{2}(V) ⊗ I^{−1} = W of (2d), Theorem 2.1, which preserves 1 by (2a) of the same Theorem). The proof of (b3) follows from a combination of those of (b1) and (b2). Q.E.D., Theorem 1.6 and surjectivity part of Theorem 1.1.

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.

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