Timeline for The structure of the module of Kähler differentials of R[[x]] over R
Current License: CC BY-SA 2.5
10 events
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| Apr 13, 2010 at 14:44 | comment | added | BCnrd | It is a fun exercise to prove that if $X$ is complex-analytic or rigid-analytic space (allow any non-arch. ground field, including char. > 0) and $x \in X$ is point then completed stalk at $x$ for the sheaf of 1-forms is naturally isomorphic to the module of continuous Kahler differentials for the completed local ring at $x$. One likewise proves a relative version for relative 1-forms with respect to analytic map $X \rightarrow Y$. [Gindi's assertion above would be correct more generally if he used module of cont. Kahler differentials for the max-adic topology, denoted with \widehat{\Omega}. | |
| Apr 13, 2010 at 14:11 | comment | added | Harry Gindi | @Brian: This is a special case where $\widehat{\Omega}_{R/k}=\Omega_{R/k}$, unless I'm mistaken. | |
| Apr 13, 2010 at 14:07 | comment | added | Harry Gindi | If $k$ is perfect, any derivation kills $R_0:=k[[x_1^p,...,x_n^p]]$. But (using our previous notation), $R$ is module-finite over $R_0$, so it is obviously finitely generated over $R_0$. These are the only nontrivial generators of $\Omega_{R/k}$, so $\Omega_{R/K}$ is module-finite. | |
| Apr 13, 2010 at 13:42 | comment | added | Martin Brandenburg | sure? I think the module is never finitely generated. | |
| Apr 13, 2010 at 12:54 | answer | added | the L | timeline score: 1 | |
| Apr 13, 2010 at 12:34 | comment | added | Harry Gindi | If $R=k[[x_1,...,x_n]]$ where $k$ has characteristic $p>0$ and is perfect, $\Omega_{R/k}$ is module-finite. | |
| Apr 13, 2010 at 12:31 | history | edited | Harry Gindi | CC BY-SA 2.5 |
added 4 characters in body; edited title
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| Apr 13, 2010 at 12:02 | answer | added | Martin Brandenburg | timeline score: 2 | |
| Apr 13, 2010 at 11:37 | history | edited | Martin Brandenburg | CC BY-SA 2.5 |
some latex corrections
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| Apr 13, 2010 at 10:26 | history | asked | zcqc | CC BY-SA 2.5 |