AI 中文总结
本文通过研究混合特征局部域上ℓ进平展上同调的权及p进类似情况，肯定了亚p进域是拟高度Kummer忠实域，还扩展基域类并得出相关等价条件，证明了两类忠实性等价。
AI 中文摘要
在阿贝尔几何中，Kummer忠实域被期望作为合适的基域。近年来，为了用伽罗瓦表示刻画Kummer忠实性，Ozeki和Taguchi定义了（拟）高度Kummer忠实域的概念作为变体，并提出了一个自然问题：亚p进域是否是拟高度Kummer忠实域？本文通过研究混合特征局部域上的ℓ进平展上同调的权并讨论p进类似情况，对该问题给出了肯定回答。此外，我们将基域的类从混合特征局部域扩展到剩余域为某个有限域代数扩张的完备离散赋值域，并给出ℓ进与p进平展上同调的余不变量消失的等价条件。结果表明，对于剩余域为某个有限域代数扩张的混合特征完备离散赋值域，Kummer忠实性与拟高度Kummer忠实性等价。
英文摘要
In anabelian geometry, Kummer-faithful fields are expected to be suitable as base fields. In recent years, to characterize Kummer-faithfulness in terms of Galois representations, Ozeki and Taguchi defined a notion of (quasi-)highly Kummer-faithful fields as a variant, and posed the following natural question: Are sub-$p$-adic fields quasi-highly Kummer-faithful? In this paper, by studying the weights of $\ell$-adic étale cohomology over mixed-characteristic local fields and discussing the $p$-adic analogue, we give an affirmative answer to this question. Furthermore, we extend the class of base fields from mixed-characteristic local fields to complete discrete valuation fields whose residue fields are algebraic extensions of some finite field, and give equivalent conditions for the vanishing of the coinvariants of $\ell$-adic and of $p$-adic étale cohomology. As a result, we show that, for mixed-characteristic complete discrete valuation fields whose residue fields are algebraic extensions of some finite field, Kummer-faithfulness and quasi-high Kummer-faithfulness are equivalent.
Comments24 pages
