2022/12/07 by David J. Saltman, Saltman, David J.
Mathematics · #12G05 #13B05 (Primary) #14F22 (Secondary) #16K50 #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2212.03713
openalex publication_date 2022/12/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Artin-Schreier polynomial Zp - Z - a is very well known. Polynomials of this type describe all degree p (cyclic) Galois extensions over any commutative ring of characteristic p. Equally attractive is the associated Galois action. If θ is a root then σ(θ) = θ+ 1 generates the Galois group. Less well known, but equally general, is the so called "differential crossed product" Azumaya algebra generated by x,y subject to the relation xy - yx = 1. In characteristic p these algebras are always Azumaya and algebras of this sort generate the p torsion subgroup of the Brauer group of any commutative ring (of characteristic p). It is not possible for there to be descriptions this general in mixed characteristic 0,p but we can come close. In Galois theory we define degree p Galois extensions with Galois action σ(θ) = ρθ+ 1, where ρ is a primitive p root of one. The Azumaya algebra analog is generated by x,y subject to the relations xy - ρyx = 1. The strength of the above constructions can be codified by lifting results. We get characteristic 0 to characteristic p surjectivity for degree p Galois extensions and exponent p Brauer group elements in quite general circumstances. Obviously we want to get similar results for degree pn cyclic extensions and exponent pn Brauer group elements, and mostly we accomplish this though p = 2 is a special case. We also give results without assumptions about p roots of one.