# Structure theorem for finitely generated $\Lambda$-modules - uniqueness part

In Iwasawa theory, one of the fundamental results is the following structure theorem for finitely generated modules over the ring $$\Lambda = \mathbf{Z}_p[[T]]$$.

If $$M$$ is a finitely generated torsion $$\Lambda$$-module, then there is a pseudo-isomorphism $$M \to \left( \bigoplus_{i=1}^s \Lambda/(p^{n_i}) \right) \oplus \left( \bigoplus_{j=1}^t \Lambda/(f_i) \right)$$ where the $$f_i \in \Lambda$$ are distinguished polynomials.

My question is the following: are the numbers $$s$$ and $$t$$ in the above theorem uniquely determined by $$M$$? That is, is the number of direct summands in the above decomposition uniquely determined by $$M$$?

As to why I'm asking this, I'm in a situation where I have a finitely generated torsion $$\Lambda$$ module $$M$$ and I happen to know that $$M$$ is generated by $$n$$ elements $$x_1, \dots, x_n$$. (EDIT: I also know that the elements $$\{x_1,...,x_n\}$$ form a minimal generating set, so no strict subset of $$\{x_1,...,x_n\}$$ generates all of $$M$$.) I'd then like to use this to conclude that there are $$n$$ summands in the direct-sum composition above (i.e: that $$s+t=n$$). To do this, however, one needs to know that $$s$$ and $$t$$ are uniquely determined by $$M$$, which is why I'm asking this question.

Any tips would be appreciated. Thanks!

• Surely you cannot conclude this just from the information you've given, since I can always increase $n$ by tossing in redundant generating elements? Commented Dec 10, 2021 at 1:48
• Being generated by $n$ elements does not tell you enough. After all, $\mathbf Z$ is generated as a $\mathbf Z$-module by $8$ elements: $3, 5, 7, 9, 11, 13, 15, 17$. That does not make $\mathbf Z$ a direct sum of $8$ cyclic subgroups. Can you prove some minimality condition on your $n$ that you forgot to mention in your question? Commented Dec 10, 2021 at 1:51
• @KConrad Thanks for the comment. I've edited the question to include the fact that the generating set is minimal. Hopefully that should clarify things. Commented Dec 10, 2021 at 3:45
• Minimality in the sense of subset containment also is not enough: in the spirit of @KConrad's example, $\{2, 3\}$ is a minimal generating set for $\mathbb Z$. Commented Dec 10, 2021 at 4:18
• As Will Sawin points out below: $\Lambda/\mathfrak{m}_{\Lambda}=\mathbb{Z}/p\mathbb{Z}$ and $\Lambda/(f)$ and $\Lambda/p$ all are generated by one element, but $s+t$ is zero in the first case. Commented Dec 10, 2021 at 9:16

By Nakayam's lemma, since $$M$$ is finitely generated, $$x_1,\dots, x_n$$ generate $$M$$ if and only if they generate $$M/ (p, T)M$$ (where $$(p,T)$$ is the maximal ideal of the local ring $$\mathbb Z_p[[T]]$$.)
So $$x_1,\dots, x_n$$ are a minimal generating set if and only if they are a basis of $$M/ (p,T)M$$, and thus in this case $$n$$ is the rank of $$M/(p,T)M$$.
If your pseudo-isomorphism were an isomorphism, you'd be done at this point, as that rank would be $$s+t$$. However, for a general torsion Iwasawa module, we can have, for example $$s=t=0$$ with $$M$$ and thus its minimal generating set nonempty, as happens when $$M$$ is finite.
So you need some additional condition on $$M$$.