Seminars & Colloquia Calendar

Many important algebraic objects can be viewed as \(G\)-torsors over a field \(F\), where \(G\) is an algebraic group over \(F\). For example, there is a natural bijection between F-isomorphism classes of central simple \(F\)-algebras of degree \(n\) and \(PGL_n(F)\)-torsors over Spec\((F)\). Much as one may study principal bundles on a manifold via characteristic classes, one may likewise study \(G\)-torsors over a field via certain associated Galois cohomology classes. This principle is made precise by the notion of a cohomological invariant, which was first introduced by Serre.

In this talk, we will determine the cohomological invariants for algebraic groups of multiplicative type with values in \(H^1(-, Q/Z(1))\). Our main technical analysis will center around a careful examination of \(mu_n\)-torsors over a smooth, connected, reductive algebraic group. Along the way, we will compute a related group of invariants for smooth, connected, reductive groups