Timeline for Flatness of certain subrings
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 11, 2023 at 12:31 | comment | added | user237522 | Thank you very much. I have posted a similar question on MSE math.stackexchange.com/questions/4716672/… | |
| Jun 11, 2023 at 12:24 | comment | added | R. van Dobben de Bruyn | The inclusion $\mathbf C[f(t)] \subseteq \mathbf C[t]$ is the same thing as the map $\mathbf C[x] \to \mathbf C[t]$ given by $x \mapsto f(t)$, so it is étale if and only if $f'(t)$ is nowhere vanishing, which on polynomial rings means $f(t)$ is linear. The question for $\mathbf C[f(t),g(t)] \subseteq \mathbf C[t]$ is a lot more subtle, as it also depends on the relations between $f$ and $g$. As in my answer, a necessary criterion is that the parametrised curve $t \mapsto (f(t),g(t))$ is smooth. Of course when $f(t)$ and $g(t)$ are both linear, this is true. | |
| Jun 11, 2023 at 11:20 | comment | added | user237522 | Please, what if $s,k \in \mathbb{C}[t]$, $R_1=\mathbb{C}[s+k]$ and $R_2=\mathbb{C}[s-k]$ and then $R=\mathbb{C}[s,k]$, where $s$ is an even (symmetric) polynomial and $k$ is an odd (skew-symmetric) polynomial (w.r.t. the involution $t \mapsto -t$. However, I am not sure for which $s,k$, we have $\mathbb{C}[s+k] \subseteq \mathbb{C}[t]$ and $\mathbb{C}[s-k] \subseteq \mathbb{C}[t]$ flat and separable. | |
| Jun 11, 2023 at 10:02 | comment | added | user237522 | Thank you very much again for your help! Please, are there 'non-trivial' special cases where the above $R \subseteq S$ is flat? Perhaps a condition involving the generators of $R_1$ and $R_2$ will guarantee flatness? | |
| Jun 11, 2023 at 9:31 | vote | accept | user237522 | ||
| Jun 11, 2023 at 1:55 | history | edited | R. van Dobben de Bruyn | CC BY-SA 4.0 |
Improved notation.
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| Jun 11, 2023 at 1:32 | history | edited | R. van Dobben de Bruyn | CC BY-SA 4.0 |
Improved readability.
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| Jun 11, 2023 at 1:25 | history | edited | R. van Dobben de Bruyn | CC BY-SA 4.0 |
Added more details.
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| Jun 11, 2023 at 1:20 | history | answered | R. van Dobben de Bruyn | CC BY-SA 4.0 |