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A Limiting Version of a Theorem in Cohomology

Scheme-theoretic methods are used to classify ternary quadratic forms with values in line bundles over arbitrary schemes and to canonically determine the isomorphisms between them. The association of a quadratic bundle to its even Clifford algebra induces a natural bijection from the set of equivalence classes of line-bundle-valued quadratic forms on rank 3 vector bundles upto tensoring by twisted discriminant bundles and the set of isomorphism classes of schematic specialisations of rank 4 Azumaya bundles over any fixed scheme X. This statement is a limiting version of the following statement: the set of orbits of Disc(X) in the 1-cohomology of X in the fppf topology with values in O(3) is in bijection with the 1-cohomology with values in PGL(2). The various orthogonal groups of a quadratic bundle are canonically determined in terms of the automorphisms of its even Clifford algebra. Any automorphism of the latter arises from a similarity, and in fact from an orthogonal transformation if its determinant is a square. The special orthogonal group is thus identified with the subgroup of automorphisms with trivial determinant. A specialised algebra arises from a honest quadratic form iff its determinant has a square root and arises from a bilinear form iff the line subbundle generated by 1 is a direct summand.

1 Overview of the Main Results THEOREM 1.1 Over each scheme X the natural map (V, q, I) → C0(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 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 isomorphism classes (V, q, I), with q semiregular modulo the group isomorphism classes (L, h, J), as above 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 C(V, q, I). This is an Z-graded algebra and we let C0(V, q, I) denote its degree zero subalgebra. The other difference with [1] is to consider C0(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 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)Ȟ 1 fppf (X, µ2). 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 cohomologyȞ 1 fppf (X, O3) 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 cohomologyȞ 1 etale (X, PGL2) (see page 145, Sec.5, Chap.III, [5]). Thus we have the bijectionȞ 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 C0(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 ) 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, C(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.
] 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 and those for which det(φ Λ 2 ) = 1 by the smaller subset S-Iso[C0(V, q, I), C0(V, q ′ , I)]. Taking q = q ′ in these sets and replacing " Iso" by "Aut" in their notations respectively defines the groups Aut(C0(V, q, I)) ⊃ Aut ′ (C0(V, q, I)) ⊃ S-Aut(C0(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): 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 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[C0(V, q, I), C0(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 ( (c) there is a family of sections s + 2k+1 : Iso[C0(V, q, I), C0(V, q ′ , I)] −→ Sim[(V, q, I), (V, q ′ , I)] indexed by the integers which is multiplicative when followed by the natural inclusions into The maps s 2k+1 and s ′ above may not be multiplicative but are mutliplicative upto µ2(Γ(X, OX )) i.e., these maps followed by the canonical quotient map, on taking the quotient of GL(V ) × Γ(X, O * X ) by µ2(Γ(X, OX )).IdV × {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: Further, we have: (a) There are splitting homomorphisms s + 2k+1 : Aut(C0(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 ′ (C0(V, q, I)) does not necessarily take values in O(V, q, I), but the further restriction to S-Aut(C0(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 µ2(Γ(X, OX )).
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 OX.1A ֒→ 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 ∼ = C0(V, q, I). This gives the surjectivity in the statement of Theorem 1.1. Further, the following linear isomorphisms may be deduced: (3) with both I ′ = OX and with q ′ induced from a global bilinear form (with values in 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-AzuW 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-AssocW 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 Stabw ⊂ GLW is the stabiliser subgroupscheme of w, recall from Theorems 3.4 and 3.8, Part A, [2], that there exists a canonical action of Stabw on Id-w-Sp-Azu W such that the natural inclusions 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 qi is semiregular iff λqi is semiregular for every λ ∈ Γ(Ui, 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 GLV on Bil (V,I) . There is also a natural morphism of groupschemes GLV −→ Stabw given on valued points by g → n≥0 Λ 2n (g) ⊗ Id and therefore the natural inclusions marked by (♣) above are GLV -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-AzuW invariant. (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 GLV -equivariant morphism of X-schemes Υ ′ = Υ ′ X : Bil (V,I) −→ Id-w-AssocW whose schematic image is precisely the scheme of specialisations Id-w-Sp-Azu W . Further if Υ ′ factors canonically through Υ = ΥX : (3) For an X-scheme T , let VT (resp. WT , resp. IT ) denote the pullback of V (resp. W , resp. I) to T , and let wT be the global section of WT induced by w. Then the base-changes of Υ ′ X and ΥX to T , namely Υ ′ X ×X T : Bil (V,I) ×X T −→ Id-w-AssocW ×X T and ΥX ×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 The proofs of Theorems 1.6 and 1.7 will be given in §4. Some notations and preliminaries are explained in §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 2 (V,I) , resp. of Quad (V,I) ), or equivalently, an element of . of Γ(U, Quad (V,I) ) ), where the sheaf Quad (V,I) -the (coherent locally-free) sheaf of OX -modules corresponding to the bundle Quad (V,I) of I-valued quadratic forms on V -is defined by the exactness of the following sequence: 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 2 (V,I) -bundle: Given a quadratic form q ∈ Γ(U, Quad (V,I) ), recall that the usual 'associated' symmetric bilinear form bq ∈ Γ(U, Bil (V,I) ) is given on sections (over open subsets of Given a (not-necessarily symmetric!) bilinear form b, we also have the induced quadratic form q b given on sections by . 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 2 (V,I) ) = 0. . This is an Z-graded algebra by definition. Let Cn be the submodule of elements of degree n. Then C0 is a subalgebra, playing the role of the even Clifford algebra in the classical situation (i.e., I = R) and C1 is a C0-bimodule. Bichsel and Knus baptize C0 and C1 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 −→ C(V, q, I) and L[I] −→ C(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 C(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 1T (resp. 1L) the unit element in the algebra T V (resp. in L[I]).
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) ⊗ 1L − 1T ⊗ 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 C(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 q f : C(V, q, I) −→ C(V, q, I); 1L 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 ∈ BilR(V, Further, for any x ∈ V denote by bx the element of HomR(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: ).1 for any x, x ′ ∈ V and ζ ∈ I −1 ≡ HomR(I, R); (g) if f ∈ HomR(V, I), and t f is given by (1)      (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 : is the linear isomorphism given by the composition of the following natural morphisms: 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: Therefore one also has the following commutative diagram of vector bundle morphisms with the vertical upward arrows being isomorphisms: (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) 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) : the OX -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: thereby inducing by (2d), Theorem 2.1 the following commutative diagram of OX -linear isomorphisms (3) Let b and b ′ be as in (2)   for any section v of V and any section c of C0(V, q). Thus C1(g, l, I) is C0(g, l, I)-semilinear.
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:  Let {Ui} 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 bi ∈ Γ(Ui, Bil (V,I) ) and b ′ i ∈ Γ(Ui, Bil (V ′ ,I ′ ) ) such that q|Ui = q b i and q ′ |Ui = 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 ev i so as to make the following diagram commute: The linear isomorphism φ Λ ev i preserves 1 and therefore it induces a linear isomorphism from (Λ 2 (V ) ⊗ I −1 )|Ui to (Λ 2 (V ′ ) ⊗ (I ′ ) −1 )|Ui, which we denote by (φ Λ 2 ) i . Observe that (φ Λ 2 ) i is independent of the choice of the bilinear forms bi and b ′ i . For, replacing these respectively by bi and b ′ i , it follows from (2f), Theorem 2.1, that 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 Ui ∩ Uj. Therefore they glue to give a global isomorphism of vector bundles Firstly we deduce a linear isomorphism det( Since V and V ′ are of rank 3, there are canonical isomorphisms η : Therefore the composition of the following sequence of isomorphisms of algebra bundles on X C0(V ′ , q ′′ , I ′ ) is an element of Iso[C0(V ′ , q ′′ , I ′ ), C0(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 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.
Proof of Theorem 1.5. (The case I = OX 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(C0(g, l, I)) = det ((C0(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 C0(g, l, I) is the identity on C0(V, q, I), then g = l −1 det(g).IdV , and further if g ∈ O(V, q, I) then g = det(g).IdV with det 2 (g) = 1. This gives exactness at GO(V, q, I) and at O(V, q, I).
to give A as well. Now consider the similarity g φ ij l ij = s + −1 (φij) : (O ⊕3 X |Uij , qi|Uij ) ∼ =l ij (O ⊕3 X |Uij , qj |Uij ) with multiplier lij := det(φij) −1 given by (c), Theorem 1.4. Since s + −1 is multiplicative, and since φij satisfy the cocycle condition, it follows that s + −1 (φij) also satisfy the cocycle condition and therefore glue the O ⊕3 X |Ui along the Uij to give a rank 3 vector bundle V on X. While the qi do not glue to give an OX-valued quadratic form on V , the facts that the multipliers {lij } 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 qi glue to give an I-valued quadratic form q on V and that C0(V, q, I) ∼ = A. It was verified in page 28, [1] that (φij ) Λ 2 = det(φij).Λ 2 (g φ ij l 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.