论文标题
光滑的涂鸦组，I：几何化Kummer理论
Smooth profinite groups, I: geometrizing Kummer theory
论文作者
论文摘要
在这一系列的三篇论文中，我们介绍和研究了循环组合对和光滑的涂鸦基团。它们是针对田野的Kummer理论的几何公理化，其系数为$ p $ - 统一的基础，用于主要$ p $。这些系数增强了，以$ g $ linearized的线条束，超过$ g $ $ $ p $。在第二篇论文中，该升级将进一步推向方案理论环境。 在第一篇文章中，我们介绍了循环型对，光滑的涂鸦组和$（g，s）$ - 协同学。我们证明了第一个以$ g $线性的托架为单位捆绑的抬起定理（定理A）。在第二篇文章中开发的代数几何工具的帮助下，这种形式主义是在第三篇中应用的，以证明平滑度定理，其本质如下所示。令$ g $为涂鸦集团。假设，对于每个开放子组$ h \ subset g $，对于$ n = 1 $，天然箭头$ h^n（h，\ m athbb {z}/p^2）\ to h^n（h，\ mathbb {z}/p）$是支流的。然后，对于每个这样的$ h $，也是每个$ n \ geq 2 $的汇总。适用于绝对Galois群体的平滑度定理提供了一种新的证据，证明了标准残基同构定理，这完全是与动机共同体脱节的。
In this series of three papers, we introduce and study cyclotomic pairs and smooth profinite groups. They are a geometric axiomatisation of Kummer theory for fields, with coefficients $p$-primary roots of unity, for a prime $p$. These coefficients are enhanced, to $G$-linearized line bundles in Witt vectors, over $G$-schemes of characteristic $p$. In the second paper, this upgrade is pushed even further, to the scheme-theoretic setting. In this first article, we introduce cyclotomic pairs, smooth profinite groups and $(G,S)$-cohomology. We prove a first lifting theorem for $G$-linearized torsors under line bundles (Theorem A). With the help of the algebro-geometric tools developed in the second article, this formalism is applied in the third one, to prove the Smoothness Theorem, whose essence reads as follows. Let $G$ be profinite group. Assume that, for every open subgroup $H \subset G$, and for $n=1$, the natural arrow $H^n(H,\mathbb{Z}/p^2) \to H^n(H,\mathbb{Z}/p)$ is surjective. Then, it is also surjective for every such $H$, and every $n \geq 2$. Applied to absolute Galois groups, the Smoothness Theorem provides a new proof of the Norm Residue Isomorphism Theorem, entirely disjoint from motivic cohomology.
