Timeline for $\mathbb Z_n[x_1^\pm,\dots,x_D^\pm]$-modules extended from $\mathbb Z_n$
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Oct 15 at 19:03 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Jun 17 at 19:01 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Feb 18 at 19:00 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| S Jan 9 at 16:06 | history | bounty ended | CommunityBot | ||
| S Jan 9 at 16:06 | history | notice removed | CommunityBot | ||
| Jan 1 at 19:47 | answer | added | Andrea Marino | timeline score: 0 | |
| Jan 1 at 17:33 | comment | added | Andrea Marino | (false proof that every module is quasi-free: consider $M_0 = M \otimes_R \mathbb{Z}_n$. Then $M_0 \otimes_{\mathbb{Z}_n} R \simeq (M \otimes_R \mathbb{Z}_n ) \otimes_{\mathbb{Z}_n} R \simeq M \otimes_R (\mathbb{Z}_n \otimes_{\mathbb{Z}_n} R ) \simeq M \otimes_R R \simeq M$ ) | |
| Jan 1 at 16:40 | comment | added | Andrea Marino | I think $M$ quasi-free and fgen can be characterized quite explicitly. Note that $M \otimes_R \mathbb{Z}_n \simeq M_0$ must be fgen too. By the classification theorem, $M_0 \simeq \bigoplus_{\lambda \in \Lambda} \mathbb{Z}_{q_{\lambda}}$ with $\Lambda$ finite and $q_{\lambda} \mid n$ prime powers. This implies $M \simeq \bigoplus R/q_{\lambda} R$, that is a finite sum of quotients for primes coming from $R \to \mathbb{Z}_n$. | |
| S Jan 1 at 14:09 | history | bounty started | Blazej | ||
| S Jan 1 at 14:09 | history | notice added | Blazej | Draw attention | |
| Jun 7, 2022 at 10:27 | comment | added | Blazej | In the first version of this question I mentioned that quasi-free modules $M$ satisfy $\mathrm{Ext}^i_R(M,N)=0$ for every $i>0$ and every $N$ free over $\mathbb Z_n$ (equivalently, $N$ of finite projective dimension) and asked if the converse is true. Then I realized that module $M = \mathrm{im} \begin{pmatrix} 2 & 0 \\ 1-x & 2 \end{pmatrix}$ with $n=4$ and $D=1$ provides a counterexample to this converse. | |
| Jun 7, 2022 at 10:23 | history | edited | Blazej | CC BY-SA 4.0 |
I realized that one point in my question is not true.
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| Jun 7, 2022 at 8:20 | history | asked | Blazej | CC BY-SA 4.0 |