2017/04/10 by Jason Starr, Starr, Jason Michael · 1 citation
Computer Science · Mathematics · #12E30 #14G05 #14G27 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1704.02932
openalex publication_date 2017/04/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The cohomological dimension of a field is the largest degree with non-vanishing Galois cohomology. Serre's "Conjecture II" predicts that for every perfect field of cohomological dimension 2, every torsor over the field for a semisimple, simply connected algebraic group is trivial. A field is perfect and "pseudo algebraically closed" (PAC) if every geometrically irreducible curve over the field has a rational point. These have cohomological dimension 1. Every transcendence degree 1 extension of such a field has cohomological degree 2. We prove Serre's "Conjecture II" for such fields of cohomological degree 2 provided either the field is of characteristic 0 or the field contains primitive roots of unity for all orders n prime to the characteristic. The method uses "rational simple connectedness" in an essential way. With the same method, we prove that such fields are C2-fields, and we prove that "Period equals Index" for the Brauer groups of such fields. Finally, we use a similar method to reprove and extend a theorem of Fried-Jarden: every perfect PAC field of positive characteristic is C2