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Let $\Lambda$ be a commutative integral ring with an automorphism $\sigma$ (I have in mind $\mathbb Z_p[[t]]$ and $\sigma(t) = (1+t)^\alpha - 1$ with $\alpha \in \Lambda^\times$) and $R = \Lambda\{F\}$ with $F\lambda = \sigma(\lambda)F$ for $\lambda \in \Lambda$.

Is there a classification of finitely generated modules over $R$ that are free and finite as modules over over $\Lambda$? I allow faithfully flat base changes of $\Lambda$ so that we can assume it's fraction field is algebraically closed (among other things).

Ultimately, I am only interested in the eigenvalues of F, if that makes sense.

When $\Lambda$ is a field, there is a classification similar to the standard one over PID's in chapter three of "The theory of rings" by Nathan Jacobson.

What about the general case or at least my specific example? Or even when $\Lambda$ is a PID? Ideally, I would want any finitely generated module to be isomorphic to a direct sum of modules generated by one element, perhaps up to finite kernel and cokernel.

Asvin
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  • Do you mean that $R$ is a skew-polynomial ring in the sense of https://en.wikipedia.org/wiki/Polynomial_ring#Differential_and_skew-polynomial_rings, so that typical elements are of the form $\sum_{i=0}^n \lambda_i F^i$ with $\lambda_i\in \Lambda$? I think you must but wanted to check. – Simon Wadsley Apr 12 '20 at 13:35
  • Yes, that's exactly right. I didn't know the name for it. – Asvin Apr 12 '20 at 13:36
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    I don't know about your specific example, but for PIDs in general: for $\Lambda=\mathbb{Z}$ and $\sigma=\text{id}$, so you're looking at $\mathbb{Z}[t]$- modules, I believe it's known that the classification of these (even if you restrict to those that are finitely generated and free over $\mathbb{Z}$) is a wild problem. – Jeremy Rickard Apr 12 '20 at 14:49
  • Really? At least for similar examples to that, I thought I had an atgument where you tensor with Q, use the classification over PIDs and finally use the freeness over Z to classify such modules. I must have made a mistake somewhere. Do you have a reference for modules over Z[t] that are free as Z modules? – Asvin Apr 12 '20 at 15:02
  • @Jeremy Rickard I am not sure why it is wild. If you restrict to finitely-generated free it seems to be covered by Latimer-MacDuffee: https://en.wikipedia.org/wiki/Latimer%E2%80%93MacDuffee_theorem – Bugs Bunny Apr 12 '20 at 15:24
  • I'm not sure right now of a reference for wildness, but there's a pair of papers of Heller and Reiner from the early 60s in Annals Representations of cyclic groups in rings of integers, I and II where they at least show that there are finitely many $\mathbb{Z}$-free indecomposable $\mathbb{Z}C_{p^k}$-modules iff $k<3$ (and such modules are the same as those $\mathbb{Z}$-free $\mathbb{Z}[t]$ where $t$ acts invertibly with order dividing $p^k$. – Jeremy Rickard Apr 12 '20 at 15:27
  • @BugsBunny I think that is just reducing it to another wild classification problem. – Jeremy Rickard Apr 12 '20 at 15:28
  • No. Take a matrix by which $t$ acts, compute its characteristic polynomial. Then Latimer-MacDuffee tells you that there are finitely many possibilities with this characteristic polynomial. It does not look wild to me (and it looks like the wild-tame-finite trichotomy is not applicable) – Bugs Bunny Apr 12 '20 at 15:31
  • Yes, I see that now. That's a wonderful theorem – Asvin Apr 12 '20 at 15:32
  • @BugsBunny But you have to classify the finitely many possibilities for all the infinitely many possible characteristic polynomials simultaneously to get a classification. – Jeremy Rickard Apr 12 '20 at 15:32
  • @Asvin If you are happy to assume, assume. Otherwise, you cannot classify finitely-generated $\Lambda$-modules for an arbitrary $\Lambda$ already. – Bugs Bunny Apr 12 '20 at 15:33
  • On the other hand, I am okay with extending scalars and in that case, it does seem like it would decompose into a sum of cyclic modules. – Asvin Apr 12 '20 at 15:33
  • @Jeremy Rickard Yes, but this does not make it wild. No relation to pairs of matrices. It sound cotame, hence, the trichotomy may not be applicable. – Bugs Bunny Apr 12 '20 at 15:35
  • Okay, I modified the problem to make it (hopefully, much) simpler in response to the comments. – Asvin Apr 12 '20 at 15:39
  • @BugsBunny There's a reference in my answer to this question to a paper of Nagornyı where he claims to prove that a classification of $4n\times4n$ matrices over $\mathbb{Z}/p^2\mathbb{Z}$ up to conjugacy would imply a classification of pairs of $n\times n$ matrices over $\mathbb{Z}/p\mathbb{Z}$ up to simultaneous conjugacy. So even classifying similarity classes of integer matrices mod $p^2$ seems to be a wild problem. – Jeremy Rickard Apr 12 '20 at 15:59

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I haven't thought about base changes, but the original problem for $\alpha=1$ (so $\sigma$ is the identity map and $R=\mathbb{Z}_p[[t]][F]$ is just a polynomial ring over $\mathbb{Z}_p[[t]]$, and classifying $R$-modules that are finitely generated and free over $\mathbb{Z}_p[[t]]$ up to isomorphism is equivalent to classifying square matrices over $\mathbb{Z}_p[[t]]$ up to conjugacy) is a "wild" problem (i.e., if you could classify these then you could classify pairs of matrices over some field up to simultaneous conjugacy), and so is probably intractable.

In fact, Theorem 2 of

Gudivok, P. M.; Oros, V. M.; Rojter, A. V., Representations of finite $p$-groups over the ring of formal power series with integral $p$-adic coefficients, Ukr. Math. J. 44, No. 6, 678-689 (1992); translation from Ukr. Mat. Zh. 44, No. 6, 753-765 (1992). ZBL0787.20006.

shows that the classification of representations of the cyclic group $C_{p^2}$ over $\mathbb{Z}_p[[t]]$ is a wild problem, and this is the subproblem of classifying those matrices whose $p^2$th power is the identity.

For $\alpha\neq1$, I think it should still be a wild problem, as the problem of classifying $R$-modules that are finitely generated and free over $\mathbb{Z}_p[[t]]$ should be at least as hard as classifying representations of finite cyclic groups over $\mathbb{Z}_p$, and this is a wild problem for $G=C_{p^3}$ ($p$ odd) and $C_{16}$ ($p=2$) (see the main theorem of

Dieterich, Ernst, Group rings of wild representation type, Math. Ann. 266, 1-22 (1983). ZBL0506.16021.)

Jeremy Rickard
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