Cohomological invariants of line bundle-valued symmetric bilinear forms / Asher Auel.

Auel, Asher
x, 132 p. ; 29 cm.
Local subjects:
Penn dissertations -- Mathematics.
Mathematics -- Penn dissertations.
The object of this dissertation is to construct cohomological invariants for symmetric bilinear forms with values in a line bundle L on a scheme X. These generalize the classical Hasse-Witt (or Stiefel-Whitney) invariants when L is the trivial line bundle. In this case, Jardine computes the etale cohomology ring of the classifying scheme of the orthogonal group to define universal invariants. There is no comparable theory when L is not trivial. Our approach is to utilize coboundary maps on nonabelian cohomology sets arising from covers of the orthogonal similitude group scheme. A new feature of this construction is a four-fold cover of the orthogonal similitude group by the Clifford group which "interpolates" between the Kummer double cover of the multiplicative group and the classical spin cover of the orthogonal group. This four-fold cover allows us to define an analogue of the 2nd Hasse-Witt invariant for L -valued forms.
As for calculating the new invariants, we provide explicit formulas in the cases of odd rank forms and L -valued metabolic forms. We also relate the invariants to parametrizations of L -valued forms arising from exceptional isomorphisms of algebraic groups. One interesting case concerns forms of rank 6 with trivial Arf invariant. These arise from the reduced pfaffian construction of Knus, Parimala, and Sridharan, applied to 2-torsion Azumaya algebras of degree 4. We relate the new invariant of a reduced pfaffian form to the class of the corresponding Azumaya algebra in a refined involutive Brauer group defined by Parimala and Srinivas.
Adviser: Tony Pantev.
Thesis (Ph.D. in Mathematics) -- University of Pennsylvania, 2009.
Includes bibliographical references.
Local notes:
University Microfilms order no.: 3363244.
Pantev, Tony, advisor.
University of Pennsylvania.
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