Submission: 2004, Jul 4
This paper completes the study started in . Scheme-theoretic methods are used to classify line-bundle-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, . 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 C0(V,q,I) of the generalised Clifford algebra C(V,q,I) of the triple (V,q,I) in the sense of Bichsel-Knus , is seen to be such a specialised algebra by results in Part A, . The Witt-invariant of (V,q,I), which may be defined as the the isomorphism class (as algebra bundle) of C0(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) |---> C0(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 1-cohomology from the set of orbits H1(X-fppf, O(3)) mod H1(X-fppf, \mu_2) to H1(X-etale, PGL(2)). The special, usual and the general orthogonal groups of (V,q,I) are computed and canonically determined in terms of Aut(C0(V,q,I)), and it is shown that the general orthogonal group is always a semidirect product. Any element of Aut(C0(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 C0(V,q,I) are naturally isomorphic. A specialised algebra bundle A arises as C0(V,q,I) with I=structure sheaf iff det(A) is in 2.Pic(X); and arises with q induced from a global I-valued bilinear form iff the line subbundle generated by 1 in A is a direct summand of A. The use of the nice technical notion of semiregularity introduced by Kneser in  allows working with an arbitrary scheme X, some (or even all) of whose points may have residue fields of characteristic two.
2000 Mathematics Subject Classification: 11Exx, 11E12, 11E20, 11E88, 11R52, 14L15, 15A63, 16S, 16H05.
Keywords and Phrases: line-bundle-valued form, semiregular form, quadratic bundle, Azumaya bundle, Witt-invariant, Clifford algebra, discriminant bundle, orthogonal group, similarity, similitude
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