Let $\Lambda$ denote Iwasawa algebra $\mathbb{Z}_p[[\Gamma]]$, where $\Gamma$ is a group isomorphic to $\mathbb{Z}_p$(ring of $p$-adic integers) . The structure theorem of the Iwasawa module tells; If M is a finitely generated $\Lambda$-module, then
$$M \sim \Lambda^r \oplus \bigoplus_{i=1}^t \Lambda/p^{n_i} \oplus \bigoplus_{j=1}^s \Lambda/f_j^{m_j}$$ is a pseudo isomorphism (maps with finite kernel and cokernel). If $M$ is $\Lambda$-torsion (i.e $r=0$), then in most of the articles they define
The characteristic ideal of $M$ is $\displaystyle \prod_j^s f_j\Lambda$.
The $\mu$-invariant of $M$ is $\displaystyle \sum_{i=1}^n n_i $.
Clearly, the definitions of both invariants are independent on $r$(rank). My question is:
- why do we need the torsion condition to define them?
I understand that most of the objects that appear in theory are torsion, but what are the drawbacks if we define them for a non-torsion f.g module?
Note: There is a similar thread Pseudo-isomorphism in Iwasawa Theory for Characteristic ideal, but the answer is not convincing to me. Is there something I'm missing in that post?
There is in fact a general definition of the $\mu$-invariant of a finitely generated $\Lambda$-module which is not necessarily torsion. The definition also works for more general groups, e.g., when considering multiple $\mathbb{Z}_p$-extensions. Bascially, you look at the torsion submodule. This is used in the recent literature, e.g. to describe Iwasawsa modules associated to Selmer and fine Selmer groups.
Let $G$ be a uniform pro-$p$ group of dimension $l$, i.e. there exists a filtration $G=G_0 \supseteq G_1 \supseteq \dots$ such that each $G_{i+1}$ is normal in $G_{i}$ and $G_i / G_{i+1} \cong (\mathbb{Z}/p\mathbb{Z})^l$. For example, $G=\Gamma\cong \mathbb{Z}_p$ or $G=\mathbb{Z}_p^l$. Let $\Lambda=\mathbb{Z}_p[[G]]$ and let $M$ be a finitely generated $\Lambda$-module. Let $M(p)$ be the $p$-power torsion submodule of $M$. Then define
$$ \mu (M) = \sum_{i \geq 0} \text{rank}_{\mathbb{F}_p[[G]]} (p^i M(p) / p^{i+1} M(p) ) $$ (see Howson, S.: Euler characteristics as invariants of Iwasawa modules. Proc. Lond. Math. Soc. 85(3), 634–658 (2002)). Note that only finitely many terms in the above sum are nonzero. It is easy to see that this agrees with the classical definition of $\mu$ for $\mathbb{Z}_p[[\Gamma]]$-torsion modules.
Regarding exact sequences $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$, one has $$ \mu(B) \leq \mu(A) + \mu(C)$$ with equality if the modules are torsion.