# Finitely generated module over $A[[x]]$

I have a question about module over power series ring.

Let $$A$$ is a local ring with maximal ideal $$\mathfrak m$$, I'm more interested in the case that $$A=\mathbb Z_p/p^n\mathbb Z_p$$

If $$M\neq0$$ is a finitely generated torsion module over $$A[[x]]$$, then is $$M$$ finitely generated over $$A$$?

I think it suffices to assume that $$M$$ is generated by one element $$m$$ and $$m$$ is killed by some series $$f(x)=\sum_{i=0}a_ix^i$$. But if $$a_0\in\mathfrak m$$, I can't deduce anything.

Thanks.

• Dear @reuns , in my definition, a module $M$ over ring $A$ is called torsion if any $m\in M$, there exists a non-zero $a$ such that $a\cdot m=0$.
– user832207
Mar 17, 2021 at 16:41
• If so then $pA[[x]]$ is not finitely generated $A$-module for $n=2$. Some people require that $a$ is not a zero divisor which makes more sense. Mar 17, 2021 at 16:41
• @reuns You are right. Thanks.
– user832207
Mar 17, 2021 at 16:46
• @reuns By the way, do you know if there is a structure theorem for finitely generated module over $(\mathbb Z_p/p^n\mathbb Z_p)[[x]]$?
– user832207
Mar 17, 2021 at 16:48
• @Sate: You might be interested in the structure theory of finitely generated modules over Iwasawa algebras (en.wikipedia.org/wiki/Iwasawa_algebra); a prototypical example of an Iwasawa algebra is $\mathbb Z_p[[x]]$. By taking appropriate quotients, your case will follow from that. Mar 17, 2021 at 17:33

$$p$$ is nilpotent. For any $$f\in pA[[x]], g\in x^m A[[x]]^\times$$,

$$f-g$$ is not a zero-divisor and any non-zero divisor is of this form.

let $$h=\prod_{m=0}^{n-1}(f^{2^m}+g^{2^m})$$

we have that $$(f-g) (h)=(f^{2^n}-g^{2^n})=(g^{2^n})=(g)^{2^n}=(x^m)^{2^n}=(x^{m 2^n})$$

When including non-zero divisor in the definition of torsion module you get that $$M$$ is a finitely generated $$A[[x]]$$-module killed by such a non-zero divisor $$f-g\in A[[x]]$$, whence $$M$$ is killed by some $$x^{m2^n}$$, it is a finitely generated $$A[[x]]/(x^{m 2^n})$$ module.

Therefore it is also a finitely generated $$A$$-module.

• Note that the statement is not true in the more general setting of $A$ just being any local ring: Already in the case $A= \mathbb Z_p$, e.g. the module $A[[x]]/(p)$ is torsion ($p$ is not a zero-divisor any more) and f.g. over $A[[x]]$, but not finitely generated as $A$-module. More generally, a f.g. torsion module of the Iwasawa algebra (en.wikipedia.org/wiki/Iwasawa_algebra) $\mathbb Z_p[[x]]$ is f.g. over $\mathbb Z_p$ if and only if its $\mu$-invariant vanishes (which does not necessarily happen). Mar 17, 2021 at 17:43

You say you know the structure theory for f.g. torsion modules over the Iwasawa algebra $$\Lambda := \mathbb Z_p[[x]]$$. But then the result follows easily.

Namely, the ring $$(\mathbb Z_p/p^n \mathbb Z_p) [[x]]$$ you look at is just $$\Lambda / (p^n)$$, and like over any ring, a $$\Lambda / (p^n)$$-module is the same as a $$\Lambda$$-module on which $$p^n$$ operates as zero. Further, such a module $$M$$ being torsion and f.g. as $$\Lambda / (p^n)$$-module implies that it is f.g. and torsion as $$\Lambda$$-module. So we can apply said structure theory and now there is a homomorphism with finite kernel and cokernel

$$M \rightarrow \bigoplus_i\mathbb{Z}_p[\![x]\!]/(p^{\mu_i})\oplus\bigoplus_j\mathbf{Z}_p[\![x]\!]/(f_j^{m_j})$$

Since being f.g. over $$\mathbb Z_p$$ will not be changed by those finite kernel and cokernel, w.l.o.g. we can assume this is an isomorphism; then, to be torsion as $$\Lambda/(p^n)$$-module means that the first sum must vanish (because it's not torsion). But to have $$p^n$$ operate as $$0$$, the second sum must vanish as well. Meaning that up to something finite, $$M =0$$; meaning that $$M$$ is finite; meaning that it's finitely generated as module over whatever ring you want.