Abstract:

In terms of non-abelian Galois cohomology, quadratic forms of rank n are classified by torsors
under the orthogonal group O_n, so that Gauss diagonalization of quadratic forms implies that
those torsors actually have a very nice shape called loop torsors. We will review what is known
about loop torsors in non-abelian Galois cohomology of algebraic groups and will pursue with other
nice rings, for example the localization of a regular complete ring with respect to a system of parameters.