On uniqueness in equivariant Iwasawa theory
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.