= Characteristic-ideal bridge from local units to the main conjecture
{title2=$\operatorname{char}(E/C)\operatorname{char}(X)=\operatorname{char}(U/C)\operatorname{char}(Y)$}
In the real cyclotomic tower, the <class-field unit sequence> gives $0\to E/C\to U/C\to X\to Y\to0$, where $X$ is the p-ramified <Iwasawa module> and $Y$ the unramified one. The modules are torsion over each one-variable character component. Splitting the four-term sequence into two <short exact sequences> proves the displayed identity by multiplicativity of <characteristic ideals>. The <Iwasawa theorem on local cyclotomic units> computes $\operatorname{char}(U/C)$ analytically. Thus the main conjecture for $X$ is equivalent to $\operatorname{char}(E/C)=\operatorname{char}(Y)$; local interpolation alone does not prove this remaining global equality.
Back to article page
