Timeline for Classification of finitely generated modules over non-commutative rings
Current License: CC BY-SA 4.0
20 events
| when toggle format | what | by | license | comment | |
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| Apr 13, 2020 at 14:53 | answer | added | Jeremy Rickard | timeline score: 4 | |
| Apr 12, 2020 at 15:59 | comment | added | Jeremy Rickard | @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. | |
| Apr 12, 2020 at 15:39 | comment | added | Asvin | Okay, I modified the problem to make it (hopefully, much) simpler in response to the comments. | |
| Apr 12, 2020 at 15:38 | history | edited | Asvin | CC BY-SA 4.0 |
simplified problem; added 21 characters in body
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| Apr 12, 2020 at 15:35 | comment | added | Bugs Bunny | @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. | |
| Apr 12, 2020 at 15:33 | comment | added | Asvin | 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. | |
| Apr 12, 2020 at 15:33 | comment | added | Bugs Bunny | @Asvin If you are happy to assume, assume. Otherwise, you cannot classify finitely-generated $\Lambda$-modules for an arbitrary $\Lambda$ already. | |
| Apr 12, 2020 at 15:32 | comment | added | Jeremy Rickard | @BugsBunny But you have to classify the finitely many possibilities for all the infinitely many possible characteristic polynomials simultaneously to get a classification. | |
| Apr 12, 2020 at 15:32 | comment | added | Asvin | Yes, I see that now. That's a wonderful theorem | |
| Apr 12, 2020 at 15:31 | comment | added | Bugs Bunny | 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) | |
| Apr 12, 2020 at 15:28 | comment | added | Jeremy Rickard | @BugsBunny I think that is just reducing it to another wild classification problem. | |
| Apr 12, 2020 at 15:27 | comment | added | Jeremy Rickard | 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 $p^k$. | |
| Apr 12, 2020 at 15:24 | comment | added | Bugs Bunny | @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: en.wikipedia.org/wiki/Latimer%E2%80%93MacDuffee_theorem | |
| Apr 12, 2020 at 15:02 | comment | added | Asvin | 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? | |
| Apr 12, 2020 at 14:49 | comment | added | Jeremy Rickard | 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. | |
| Apr 12, 2020 at 14:13 | history | edited | YCor | CC BY-SA 4.0 |
added ac tag (it's close to commutative so the tag would be useful)
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| Apr 12, 2020 at 14:10 | history | edited | Asvin | CC BY-SA 4.0 |
added 5 characters in body
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| Apr 12, 2020 at 13:36 | comment | added | Asvin | Yes, that's exactly right. I didn't know the name for it. | |
| Apr 12, 2020 at 13:35 | comment | added | Simon Wadsley | Do you mean that $R$ is a skew-polynomial ring in the sense of en.wikipedia.org/wiki/…, 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. | |
| Apr 12, 2020 at 13:00 | history | asked | Asvin | CC BY-SA 4.0 |