Timeline for A group generated by all the root subgroups above ideal of $\mathbb Z[\alpha]$ is of finite index in $G(\mathbb Z[\alpha])$
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Sep 7, 2019 at 3:29 | comment | added | Venkataramana | Yes. I meant to write $I\subset {\mathbb Z}[\alpha]$ contains $J=NO-k$. | |
| Sep 6, 2019 at 21:17 | comment | added | Ami | you meant $I$ contains $J=NO_k$ ? | |
| Sep 6, 2019 at 14:29 | comment | added | Venkataramana | $I\subset {\mathbb Z}[\alpha]$ has finite index in $O_k$. Hence $J$ contains $I=NO_k$ for some non-zero integer $N$ since $O_k$ is a finitely generated torsion free abelian group. Now $I$ is a nonzero ideal | |
| Sep 6, 2019 at 10:12 | comment | added | Ami | Thank you very much, can you please refer to a source regarding the ideal property you mentions? | |
| Sep 6, 2019 at 4:20 | comment | added | Venkataramana | The reason is that $G({\mathbb Z}[\alpha ])$ is commensurate to $G(O_k)$ where $k$ is the field over $\mathbb Q$ generated by $\alpha$; every non-zero ideal $I$ of ${\mathbb Z}[\alpha]$ contains a nonzero ideal $J$ of $O_k$; thus Raghunathan's theorem applied to $O_k$ and $J$ gives what you want. | |
| Sep 6, 2019 at 3:36 | comment | added | Venkataramana | If he uses the ring of S integers in the number field k, then that answers your question; in any case, the answer to your question is indeed yes; that is what is proved in Raghunathan's paper. Not just the special case of $\mathbb Z$. | |
| Sep 5, 2019 at 23:41 | history | edited | Ami | CC BY-SA 4.0 |
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| Sep 5, 2019 at 18:31 | comment | added | Ami | I believe in page 365 (Theorem 1.2). He uses in the paper the ring of S-integers in $k$, I don't think I understand it enough but from what I gather for $k=\mathbb Q$ we can get $Λ=\mathbb Z$. | |
| Sep 5, 2019 at 18:19 | comment | added | Venkataramana | where in Raghunathan's paper, do you see the statement for $G({\mathbb Z})$? | |
| Sep 5, 2019 at 18:15 | history | edited | YCor | CC BY-SA 4.0 |
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| Sep 5, 2019 at 17:42 | history | asked | Ami | CC BY-SA 4.0 |