Cohomological invariants of algebraic tori. (English) Zbl 1368.11034

Summary: 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 \(\mathbb Q/\mathbb Z(j)\) is a functorial-in-\(K\) collection of maps of sets \(\mathrm{Tors}_G(K)\to H^n(K,\mathbb Q/\mathbb Z(j))\) for all field extensions \(K/F\), where \(\mathrm{Tors}_G(K)\) is the set of isomorphism classes of \(G\)-torsors over \(\mathrm{Spec}\, K\). We study the group of degree 3 invariants of an algebraic torus with values in \(\mathbb Q/\mathbb Z(2)\). In particular, we compute the group \(H_{nr}^3(F(S),\mathbb Q/\mathbb Z(2))\) of unramified cohomology of an algebraic torus \(S\).


11E72 Galois cohomology of linear algebraic groups
12G05 Galois cohomology
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