Timeline for Flatness from constancy of dimension of fibers
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 14, 2017 at 10:27 | vote | accept | gregorsamsa | ||
| Jan 14, 2017 at 10:22 | comment | added | Francesco Polizzi | Ah, sorry, I slightly misread the question. You are asking about flatness of a (finitely presented) module over the ring $R$, so in geometric terms you want to know about the flatness of a coherent sheaf over an affine scheme (and not about the flatness of a morphism of schemes). Then the correct answer is given by abx in his comment: in the case where the fibres are of constant dimension, flatness occurs whenever $R$ is without nilpotents. | |
| Jan 14, 2017 at 10:08 | answer | added | abx | timeline score: 14 | |
| Jan 14, 2017 at 9:59 | comment | added | Francesco Polizzi | In general equi-dimensionality of the fibres is necessary, but not sufficient in order to guarantee flatness. However, it is also sufficient when $R$ is regular of dimension $1$, or when $R$ is regular of arbitrary dimension and $M$ is Cohen-Macaulay. See S. Kovacs' answer to the following MO question: mathoverflow.net/questions/75317/… | |
| Jan 14, 2017 at 9:53 | review | First posts | |||
| Jan 14, 2017 at 10:16 | |||||
| Jan 14, 2017 at 9:50 | history | asked | gregorsamsa | CC BY-SA 3.0 |