论文标题
混合特征循环问题
Mixed Characteristic Cyclic Matters
论文作者
论文摘要
Artin -Schreier多项式$ z^p -z- a $众所周知。这种类型的多项式描述了所有特征性$ p $的$ p $ $ p $（循环）galois扩展。同样有吸引力的是相关的Galois动作。如果$θ$是根，则$σ（θ）=θ+ 1 $生成Galois组。所谓的“差异交叉产品” azumaya代数由$ x，y $属于关系$ xy -yx = 1 $。在特征$ p $中，这些代数始终是Azumaya，这种代数为任何交换戒指的Brauer组的$ P $ Torsion子组（特征性$ P $）。 不可能在混合特征$ 0，p $中进行描述，但我们可以接近。在Galois理论中，我们定义了$ p $ p $ galois扩展名$σ（θ）=ρθ+ 1 $，其中$ρ$是一个原始$ p $ root。 Azumaya代数模拟由$ x，y $生成，约束$ xy-ρ{y} x = 1 $。 上述结构的强度可以通过提升结果来编码。在相当普遍的情况下，我们获得了特征$ 0 $的特征$ p $ p $ pug eunptivity $ p $ galois扩展名和指数$ p $ brauer group elements。显然，我们希望获得类似的结果$ p^n $循环扩展和指数$ p^n $ brauer组元素，尽管$ p = 2 $是一种特殊情况，但大多数情况下，我们还是完成了这一点。我们还提供结果，而没有假设约为$ p $的根。
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 $θ$ 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 $xy - ρ{y}x = 1$. 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.
