2004/07/06 by Venkata Balaji Thiruvalloor Eesanaipaadi, Eesanaipaadi, Venkata Balaji Thiruvalloor
Mathematics · #11E12 #11E20 #11E88 #11Exx #11R52 #14L15 #15A63 #16H05 #16S #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Rings and Algebras (math.RA) #math.AG #math.RA #msc:11E12 #msc:11E20 #msc:11E88 #msc:11Exx #msc:11R52 #msc:14L15 #msc:15A63 #msc:16H05 #msc:16S
paper · pdf · doi:10.48550/arxiv.math/0407079
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
arxiv created 2004/07/06 · openalex publication_date 2004/07/06 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.