Degree three cohomology of function fields of surfaces

2010/12/24 by R. Parimala, Parimala, R., V. Suresh +1
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1012.5367

openalex publication_date 2010/12/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let F be a finite field and l a prime not equal to the characteristic of F. Let K be the function field of a surface over F. Assume that K contains a primitive lth root of unity. In the paper we prove a certain local-global principle for elements of H³(K, μ_l) in terms of symbols in H²(K, μ_l) with respect to discrete valuations of K. We also show that this local global principle is equivalent to the vanishing of certain unramified cohomology groups of 3-folds over finite fields. Using this local-global principle we show that every element in H³(F, μ_l) is a symbol. The vanishing of the unramified cohomology groups has consequences in the study of integral Tate conjecture and Brauer-Manin obstruction for existence of zero-cycles.

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