Timeline for When is the module of Kahler differentials free?
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9 events
| when toggle format | what | by | license | comment | |
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| May 23, 2020 at 18:26 | comment | added | Plank | If $R$ is an $S$-algebra and $\Omega_{R/S}$ is projective of rank $n$, then it is free of rank $n$ if, and only if, it can be generated by $n$ elements. A reference is pg 556 of "Lombardi, Henri, and Claude Quitté. "Commutative algebra: Constructive methods." Traduction anglaise, révisée et augmentée, de l’édition française (Algebre commutative. Méthodes constructives. Calvage et Mounet, 2011). Springer, Berlin 1.2 (2015): 3." | |
| May 23, 2020 at 7:59 | comment | added | abx | To give one more precise reference (for $\Omega ^1_{X/k}$ free of rank $\dim X$ $\Longleftrightarrow \ X$ smooth over $k$): Kunz Kähler differentials, Theorem 8.1. | |
| May 23, 2020 at 2:53 | answer | added | Mohan | timeline score: 4 | |
| May 23, 2020 at 2:13 | comment | added | Plank | @DevlinMallory Ah yes, I see what you mean that these modules are closely related to smoothness. I mean this is the essence of smooth varities and when the sheaf of differentials is finite locally free, there is a lot of interaction. Thanks for the reference, that covers part of the question nicely! | |
| May 23, 2020 at 2:11 | comment | added | Plank | @Mohan Could you explain this a bit further please? Perhaps add some details? This would answer a huge problem I have been facing with these modules! | |
| May 23, 2020 at 1:58 | comment | added | Mohan | For $S$ as above, $\Omega^1_X$ is free of rank $=\dim X$ (X=\operatorname{Spec} S$ implies $S=k[x_0,\ldots,x_n, x_{n+1},\ldots x_m]/(g_1,\ldots, g_{l+m})$ where the $g_i$ form a regular sequence, that is $S$ is a complete intersection possibly after adding more variables. | |
| May 23, 2020 at 1:15 | comment | added | Devlin Mallory | Freeness of $\Omega_{S/k}$ is very closely related to smoothness of $S$ as a $k$-algebra (one has to assume also that it has the correct rank, to avoid issues with characteristic $p$); this should be in most commutative algebra resources. If $S$ is not smooth, then it seems a bit subtle exactly when $\Omega_{S/k}$ is reflexive, but in general one expects to have both torsion and cotorsion (see, e.g., arxiv.org/pdf/1012.5940.pdf for the failure of reflexivity for some fairly mild singularities). | |
| May 22, 2020 at 23:13 | history | edited | Plank | CC BY-SA 4.0 |
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| May 22, 2020 at 23:01 | history | asked | Plank | CC BY-SA 4.0 |