2024/10/06 by Mikhail Borovoi, Borovoi, Mikhail
Mathematics · #11E72 #20G10 #20G25 #20G30 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Group Theory (math.GR) #Number Theory (math.NT) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2410.04474
openalex publication_date 2024/10/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let K be a local or global field. For a connected reductive group G over K, in another preprint [5] we defined a power operation (ξ,n)↦ ξ\Diamond n \colon H1(K,G)× \mathbb Z→ H1(K,G) of raising to power n in the Galois cohomology pointed set H1(K,G). In this paper, for a cohomology class ξ in H1(K,G), we compare the period \rm per(ξ) defined to be the least integer n≥ 1 such that ξ\Diamond n=1, and the index \rm ind(ξ) defined to be the greatest common divisor of the degrees [L:K] of finite separable extensions L/K splitting ξ. These period and index generalize the period and index a central simple algebra over K. For an arbitrary reductive K-group G, we proved in [5] that \rm per(ξ) divides \rm ind(ξ). In this paper we show that the index may be strictly greater than the period. In [5] we proved that for any K, G, and ξ∈ H1(K,G) as above, the index \rm ind(ξ) divides \rm per(ξ)d for some positive integer d, and we gave upper bounds for d in the local case and in the case of a number field. Here we give a characteristic-free proof of the fact that \rm ind(ξ) divides \rm per(ξ)d for some positive integer d in the global field case, and our proof gives an upper bound for d that is valid also in the case of a function field.