Timeline for Lifting flat modules over ring quotients
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 6, 2018 at 18:04 | comment | added | Fred.Fred | @MahdiMajidi-Zolbanin If the directed system is not well-ordered, lifting the maps between the finitely generated free $\bar{R}$-modules of the system arbitrarily may not yield a directed system of $R$-modules, or at least, I do not see a way how to ensure that. | |
| Jan 5, 2018 at 23:05 | comment | added | Mahdi Majidi-Zolbanin | Where does the argument using Lazard's Theorem break down for a general directed system? | |
| Jan 3, 2018 at 14:52 | comment | added | Jeremy Rickard | @Mohan An example of a projective that doesn’t lift to a projective is $R=\mathbb{Z}$, $I=6\mathbb{Z}$, $P=\mathbb{Z}/2\mathbb{Z}$. But applying Fred.Fred’s construction, $P$ does lift to a flat $\mathbb{Z}$-module, namely $\mathbb{Z}[\frac{1}{3}]$. Is this example enough to defuse your skepticism? | |
| Jan 3, 2018 at 14:42 | comment | added | Mohan | There are projective modules (=finitely generated flat modules) over $\overline{R}$ which do not lift to projective modules over $R$, so I am skeptical of your argument using Lazard's theorem. | |
| Jan 3, 2018 at 9:16 | history | asked | Fred.Fred | CC BY-SA 3.0 |