A Limiting Version of a Theorem in Cohomology
Loading...
Date
Journal Title
Journal ISSN
Volume Title
Publisher
Abstract
Description
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.
Dedicated to Professor Martin Kneser (13p.) Keywords: semiregular form, quadratic bundle, Azumaya bundle, Witt-invariant, line-bundle-valued form, Clifford algebra, discriminant bundle, orthogonal group, similarity, similitude
Dedicated to Professor Martin Kneser (13p.) Keywords: semiregular form, quadratic bundle, Azumaya bundle, Witt-invariant, line-bundle-valued form, Clifford algebra, discriminant bundle, orthogonal group, similarity, similitude