Iwasawa module structure theorem

A finitely generated module over $\Lambda ={\mathbb{Z}}_p[[T]]$ is pseudo-isomorphic towhere the $f_j$ are irreducible distinguished polynomials. Height-one localizations are discrete valuation rings, which give the elementary divisors; the remaining errors are supported only at the maximal ideal and are finite. This is a finite-error classification, not necessarily an actual direct-sum decomposition.

For a finitely generated torsion Iwasawa module, its characteristic ideal is the product of the height-one prime ideals raised to the lengths of the corresponding localizations. In the elementary-divisor description it is generated by $p^{\sum a_i}\prod f_j^{b_j}$. It is unchanged by finite modules and multiplicative in short exact sequences of torsion modules.

The invariants $\mu ,\lambda$ of a finitely generated torsion Iwasawa module measure its $p$-power elementary factors and distinguished-polynomial factors. Vanishing of $\mu$ means that the module is finitely generated over ${\mathbb{Z}}_p$, up to finite error. It does not mean that the module is finite: a factor $\Lambda /(T)$ has $\mu =0$ and is ${\mathbb{Z}}_p$.

For a finite abelian extension of $\mathbb{Q}$, the unramified Iwasawa module of its cyclotomic Zp-extension has $\mu =0$. This removes $p$-power elementary factors from its characteristic ideal. The theorem does not assert $\mu =0$ for arbitrary noncyclotomic Zp-extensions.