Let $p$ be an odd prime, and $K=\mathbb{Q}(\mu_{p})$. Consider the cyclotomic extension of $K$:
$$K\subset K_1\subset K_2\subset\cdots\subset K_\infty ,$$ where $K_n=\mathbb{Q}(\mu_{p^{n+1}})$ and $K_\infty=\mathbb{Q}(\mu_{p^{\infty}})$. The Iwasawa algebra is $$\Lambda=\varprojlim_n\mathbb{Z}_p[\text{Gal}(K_n/K)].$$ If $M$ is a $\Lambda$-module, $\text{char}(M)$ is its characteristic ideal, and if it is also a $\mathbb{Z}_p[\Delta]$-module, for $\Delta=\text{Gal}(K/\mathbb{Q})$, then $M(\chi)$ is the $\chi$-component, for $\chi$ a $p$-adic character of $\Delta$
The following Main Conjecture holds: if $\chi$ is an even $p$-adic character of $\Delta=\text{Gal}(K/\mathbb{Q})$, then $$\text{char}(C_{\infty}(\chi))=\text{char}(E_{\infty}(\chi)/V_{\infty}(\chi)),$$ where $C_{\infty}$ is the inverse limit of the $p$-parts of the ideal class groups of $K_n$, $E_{\infty}$ is the inverse limit of the closure of the units of $K_n$ in the principal units of $\mathbb{Q}_p(\mu_{p^{n+1}})$ and $V_{\infty}$ is the inverse limit of the closure of the cyclotomic units of $K_n$ in the principal units of $\mathbb{Q}_p(\mu_{p^{n+1}})$ (all these under norm maps). This makes sense since we know that if $\chi$ is even, then the modules consider are finitely generated torsion $\Lambda$-modules.
My question: is it true that $$\text{char}(C_{\infty})=\text{char}(E_{\infty}/V_{\infty})?$$ Does this make sense? Is $E_{\infty}/V_{\infty}$ again finitely generated and torsion? This is true for $C_{\infty}$. Also, we know that every module can be decomposed into the direct sum of its component with respect to the characters, but the main conjecture holds only if $\chi$ is even, therefore we cannot directly use this fact, since we lack information about $\chi$ odd.
Best Answer
The main Conjecture (MC) has at least 3 different equivalent formulations. To ease the notations in your question, since here $p\nmid [\mathbf Q(\mu_p):\mathbf Q]$, we may as well sum up the isotypical components relative to even (odd) characters and deal only with the $\pm1$ components of the modules involved.
(MC$1$) Sticking to your notations, the MC reads char(${C_{\infty}}^+)$ = char (${E_{\infty}}^+/{V_{\infty}}^+)$, and you ask whether this could be extended to the minus part. No, because the minus parts of the units at finite levels are reduced to $\pm 1$, so upstairs char(${C_{\infty}}^-)$ would be trivial, whereas downstairs the arithmetic of the minus part of the $p$-class group is definitely not (Kummer, Bernoulli numbers, etc.) On the opposite, the plus part is conjectured to be trivial : downstairs, it is Vandiver's conjecture that $p\nmid h^+$; upstairs, Greenberg's conjecture that ${C_{\infty}}^+$ is finite. To have an idea of the strength of these conjectures, note that Greenberg's (the weakest one) easily implies directly the MC. Actually (MC$1$), which avoids the appeal to $p$-adic $L$-functions, is rather specific of the proofs of the MC based on Euler systems.
(MC$2$) In view of the comments above concerning (MC$1$), it appears that - at the time being - our true knowledge of the arithmetic of the $p$-class groups is rather one-handed. To stress the most striking feature, the link with $p$-adic $L$-functions, we must come back to characters, since the trivial character must be excluded because of the pole at $s=1$. The character-wise formulation (MC$2$) on the minus size then reads : Let $\omega$ denote the Teichmüller character. For any non trivial even character $\chi$ of Gal($\mathbf Q(\mu_p)/\mathbf Q)$, the characteristic series $f_{\chi}$ of the $\omega \chi^{-1}$-part of ${C_{\infty}}$ satisfies $f_{\chi}((1+p)^s -1)=L_p(\chi, s)$ for all $s\in \mathbf Z_p$ .
(MC$3$) To come back to the plus size we must change modules, introducing ${B_{\infty}}$= the Galois group over $\mathbf Q(\mu_{p^\infty})$ of the maximal pro-p-abelian extension of $\mathbf Q(\mu_{p^\infty})$ which is $p$-ramified, i.e. unramified outside $p$. Then (MC$3$) reads : For any non trivial even character $\chi$ of Gal($\mathbf Q(\mu_p)/\mathbf Q)$, the characteristic series $g_{\chi}$ of the $ \chi$-part of ${B_{\infty}}$ satisfies $f_{\chi}((1+p)^{1-s} -1)=L_p(\chi, s)$ for all $s\in \mathbf Z_p$ . Comparing with (MC$2$), note the shift between the pair ($s,\omega \chi^{-1}$) and the pair ($1-s, \chi^{-1}$), which is due to a combination of class-field theoretic iso. and Kummer duality, usually called Spiegelung ./.