Cohomological invariants of algebraic tori

Abstract

Let $G$ be an algebraic group over a field $F$. As defined by Serre, a cohomological invariant of $G$ of degree $n$ with values in $\mathbb{Q}∕\mathbb{Z}\left(j\right)$ is a functorial-in-$K$ collection of maps of sets ${Tors}_G\left(K\right)\to H^n\left(K,\mathbb{Q}∕\mathbb{Z}\left(j\right)\right)$ for all field extensions $K∕F$, where ${Tors}_G\left(K\right)$ is the set of isomorphism classes of $G$-torsors over Spec $K$. We study the group of degree $3$ invariants of an algebraic torus with values in $\mathbb{Q}∕\mathbb{Z}\left(2\right)$. In particular, we compute the group $H_{nr}^3\left(F\left(S\right),\mathbb{Q}∕\mathbb{Z}\left(2\right)\right)$ of unramified cohomology of an algebraic torus $S$.