Cohomological invariants of quadratic forms
Abstract. This talk is about joint work with T. Chinburg, B. Morin
and M.J. Taylor. Our aim is to establish comparison formulas
between the Hasse-Witt invariants
of a symmetric bundle over a scheme and
the invariants of some of its twists. In particular we will consider a
special kind of twist, which has been first
studied by A. Fröhlich for quadratic forms on fields.
This arises from twisting the form
by a cocycle obtained from an orthogonal representation of a group scheme.
A simple important example of this twisting procedure is
the trace form of an étale algebra, which is obtained
by twisting the standard/sum of squares form by the
orthogonal representation attached to the algebra.
We will start by recalling the classical results of Fröhlich and Serre
for quadratic forms on fields. Then we will define the Hasse-Witt
invariants one can associate to a symmetric bundle on a scheme and
we will prove a general comparison formula from which almost all the
previous formulas can be deduced. Finally we will indicate some
applications of our results to embeddings problems.