2024/03/12 by Mikhail Borovoi, Zinovy Reichstein, Borovoi, Mikhail +2
Mathematics · #11E72 #20G10 #20G20 #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.2403.07659
openalex publication_date 2024/03/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For a connected reductive group G over a local or global field K, we define a *diamond* (or *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 (this operation is new when K is a number field). We show that this power operation has many good properties. When G is a torus, the set H1(K,G) has a natural group structure, and ξ\Diamond n then coincides with the n-th power of ξ in this group. On the other hand, we show that a power operation on H1(K,G), functorial in G, which we define over local and global fields, cannot be defined for an arbitrary field K. Our proof of this assertion relies on the results of Appendix B written by Philippe Gille. Using this power operation, for a cohomology class ξ in H1(K,G) over local or global field, we define the period \rm per(ξ) to be the least integer n≥ 1 such that ξ\Diamond n=1. We define the index \rm ind(ξ) to be the greatest common divisor of the degrees [L:K] of finite extensions L/K splitting ξ. The period and index of a cohomology class generalize the period and index a central simple algebra over K. For any connected reductive group G defined over a local or global field K, we show that \rm per(ξ) divides \rm ind(ξ) and that \rm ind(ξ) may be strictly greater than \rm per(ξ), but they always have the same prime factors.