Venkata Balaji, Thiruvalloor Eesanaipaadi: A Limiting Version of a Theorem in Cohomology --- Dedicated to Professor Martin Kneser

Submission: 2004, Jul 4

This paper completes the study started in [1]. 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, [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 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 [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 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 [4] 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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