In classification problems over the real field ℝ first Galois cohomology sets play an important role, as they often make it possible to classify the orbits of a real Lie group. In this talk we outline an algorithm to compute the first Galois cohomology set H1(G,ℝ) of a complex reductive algebraic group G defined over the real field ℝ. The algorithm is in a large part based on computations in the Lie algebra of G. This is joint work with Mikhail Borovoi.
We consider Galois cohomology groups over function fields F of curves that are defined over a complete discretely valued field. Motivated by work of Kato and others for n=3, we show that local-global principles hold for Hn(F,ℤ/mℤ(n−1)) for all n>1. In the case n=1, a local-global principle need not hold. Instead, we will see that the obstruction to a local-global principle for H1(F,G), a Tate-Shafarevich set, can be described explicitly for many (not necessarily abelian) linear algebraic groups G. Concrete applications of the results include central simple algebras and Albert algebras.
You must be logged in to post a comment.