Degree 4 cohomological invariants of algebraic tori
Cyril Demarche, Hanqing Long
TL;DR
The paper addresses the problem of determining degree‑4 cohomological invariants of algebraic tori and their unramified cohomology. It develops a motivic framework, using slice filtration and the purity of the motive of the classifying space $BT$ to compute low‑degree étale motivic cohomology and derive exact sequences that describe $\mathrm{Inv}^4(T,\mathbb{Q/Z}(3))_{norm}$ and $H^4_{nr}(F(T),\mathbb{Q/Z}(3))$ in terms of $A^2(BT,K^M_3)$, CH groups, and Milnor K‑theory. The main contributions include a proof that $M(BT)$ is a pure Tate motive, explicit exact sequences relating degree‑4 invariants to Chow groups and Milnor K‑theory, and an explicit example of a torus with a nontrivial degree‑4 invariant not arising from degree‑3 invariants, illustrating the richness of the higher invariants. These results advance understanding of rationality questions and obstruction theories for tori by providing concrete formulas, diagrams, and examples for higher‑degree invariants and their unramified counterparts.
Abstract
In this paper, we determine the motive of the classifying torsor of an algebraic torus. As a result, we give an exact sequence describing the degree 4 cohomological invariants of algebraic tori. Using results by Blinstein and Merkurjev, this provides a formula for the degree 4 unramified cohomology group of an algebraic torus, via a flasque resolution.