AI 中文总结
本文作为等变岩泽理论主猜想相关研究的续篇，通过可除代数分类、交叉乘积序的Π滤过及换位子乘积构造，研究岩泽代数全分式环的SK₁群零化问题，推进了等变岩泽理论的唯一性研究。
AI 中文摘要
本文是我们《论等变岩泽理论的“主猜想”》一文的续篇，研究了$SK_1(QG)$的零化可能性，其中$QG$是岩泽代数$ΛG=Z_p[[G]]$的全分式环，$G$为前述文献中伽罗瓦扩张$K/k$的伽罗瓦群。该零化等价于对$QG$的韦德伯恩分量中所有可除代数$D$都有$SK_1(D)=0$，因此可通过这些$D$的分类$D_{r,s,F}$展开研究，该分类给出了$D$中的一个交叉乘积序$Δ$以及取值环为$Δ_\bullet$的赋值$v_\bullet$。$Δ$包含一个特殊元素$Π$，它生成$D$在其中心上的一个极大子域，满足$V_\bullet(Π)=1$且$ΠΔΠ^{-1}=Δ$。由$Π$诱导的$Δ$上的滤过使得我们能够基于对$d\inΔ$的$nr(d)$模$ΠΔ$的同余式，研究既约范数。给定$d\inΔ^\times$满足$nr(d)=1$，且$n\ge 1$是使得$d\in 1+Π^nΔ$的最大整数（称为$d$的水平），核心问题是找到合适的换位子乘积$c$满足$c\equiv d$模$Π^{n+1}Δ$。此时$c^{-1}d\equiv 1$模$Π^{n+1}Δ$，因此令$d'=c^{-1}d$，则有$nr(d')=1$且水平$n'>n$。当水平足够大时，重复这一过程最终会落在$[Δ^\times,Δ^\times]$中。
英文摘要
This sequel to our paper `On the ``main conjecture'' of equivariant Iwasawa theory' studies the possibility of the vanishing of $SK_1(QG)$, when $QG$ is the total ring of fractions of the Iwasawa algebra $ΛG=Z_p[[G]]$, with $G$ the Galois group of the Galois extension $K/k$ of loc. cit. The vanishing is equivalent to $SK_1(D)=0$ for all division algebras $D$ in the Wedderburn components of $QG$, so can be studied via the classification $D_{r,s,F}$ of these $D$'s, which provides a crossed product order $Δ$ in $D$ and the valuation $v_\bullet$ with valuation ring $Δ_\bullet$. $Δ$ contains a special element $Π$, which generates a maximal subfield of $D$ over its centre with $V_\bullet(Π)=1$ and $ΠΔΠ^{-1}=Δ$. The induced $Π$-filtration on $Δ$ enables a study of the reduced norm built on a congruence for $nr(d)$ mod $ΠΔ$ for $d\inΔ$. Given $d\inΔ^\times$ with $nr(d)=1$, and $n\ge 1$ maximal with $d\in 1+Π^nΔ$ (called the level of $d$), the main problem is to find a suitable commutator product $c\equiv d$ mod $Π^{n+1}Δ$. Then $c^{-1}d\equiv 1$ mod $Π^{n+1}Δ$ hence, setting $d'=c^{-1}d$, has $nr(d')=1$ and level $n'>n$. Repetition ends in $[Δ^\times,Δ^\times]$ when the level gets sufficiently large.
Comments108 pages
