Cohomological invariants of algebraic tori

Blinstein, Sam; Merkurjev, Alexander

doi:10.2140/ant.2013.7.1643

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Cohomological invariants of algebraic tori

Sam Blinstein and Alexander Merkurjev

Vol. 7 (2013), No. 7, 1643–1684

DOI: 10.2140/ant.2013.7.1643

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 $ℚ ∕ ℤ (j)$ is a functorial-in- $K$ collection of maps of sets ${Tors}_{G} (K) \to H^{n} (K, ℚ ∕ ℤ (j))$ for all field extensions $K ∕ F$ , where ${Tors}_{G} (K)$ 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 $ℚ ∕ ℤ (2)$ . In particular, we compute the group $H_{nr}^{3} (F (S), ℚ ∕ ℤ (2))$ of unramified cohomology of an algebraic torus $S$ .

Keywords

algebraic tori, cohomological invariants, Galois cohomology

Mathematical Subject Classification 2010

Primary: 11E72

Secondary: 12G05

Milestones

Received: 23 April 2012

Revised: 8 October 2012

Accepted: 9 November 2012

Published: 12 October 2013

Authors

Sam Blinstein
Department of Mathematics University of California, Los Angeles Los Angeles CA 90095-1555 United States

Alexander Merkurjev
Department of Mathematics University of California, Los Angeles Los Angeles CA 90095-1555 United States

Vol. 7, No. 7, 2013