Venkata Balaji, Thiruvalloor Eesanaipaadi: Degenerations of Rank 3 Quadratic Bundles and Rank 4 Azumaya Bundles over Schemes --- Dedicated to Professor Martin Kneser

Submission: 2004, May 4, corrected version from 2004, July 8

This paper uses scheme-theoretic methods to extend 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 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, C0(V,q), is shown to determine (V,q) uniquely 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 the automorphism group of C0(V,q), and it is shown that the general orthogonal group is always a semidirect product. Any automorphism of C0(V,q) can be lifted to a self-similarity, and in fact to an isometry provided the determinant of the automorphism is a square. The special orthogonal group and the group of determinant 1 automorphisms of C0(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 C0(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, 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 may be seen as limiting versions of the natural bijection (involving first cohomology groups) from H1(X-fppf, O(3)) mod Disc(X) to H1(X-etale, PGL(2)). 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.

2000 Mathematics Subject Classification: 11Exx, 11E12, 11E20, 11E88, 11R52, 14L15, 15A63, 16S, 16H05.

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

Full text: dvi.gz 97 k, dvi 266 k, ps.gz 1172 k, pdf.gz 457 k, pdf 519 k.

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