Mixed Characteristic Cyclic Matters

Abstract

The Artin-Schreier polynomial $Z^p - Z - a$ 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 $\theta$ is a root then $\sigma(\theta) = \theta + 1$ 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 $\sigma(\theta) = \rho\theta + 1$, where $\rho$ is a primitive $p$ root of one. The Azumaya algebra analog is generated by $x,y$ subject to the relations $xy - \rho{y}x = 1$. 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 $p^n$ cyclic extensions and exponent $p^n$ 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.