Timeline for Normality of certain subrings of polynomial rings in characteristic p
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Feb 11, 2019 at 16:19 | comment | added | John Zek. | Thank you again, clearly I've been making a silly mistake. | |
| Feb 11, 2019 at 12:38 | comment | added | Jason Starr | Welcome new contributor. I still do not understand your assertion. If you insist that $f$ be homogeneous, then please consider the case that $n$ equals $3$ and that $Z$ is the graded $k$-subalgebra of $k[x_1,x_2,x_3]$ generated by $f=x_1^p - x_2^{p-1}x_3$. Then $Z[x_1^p,x_2^p,x_3^p]$ equals $k[x_1^p,x_2^p,x_3^p,x_2^{p-1}x_3]$. This $k$-algebra is not normal, and it is not regular in codimension $1$. | |
| Feb 11, 2019 at 2:52 | comment | added | John Zek. | Thank you, I forgot to add that Z is graded subalgebra, I hope this at least saves the question for Z=k[f]. | |
| Feb 11, 2019 at 2:50 | history | edited | John Zek. | CC BY-SA 4.0 |
added 19 characters in body
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| Feb 11, 2019 at 2:36 | comment | added | Jason Starr | Welcome new contributor. I do not understand your assertion. Consider the case that $n$ equals $2$ and $Z$ equals the subalgebra of $k[x_1,x_2]$ generated by $f=x_1^p + x_2^{p+1}$. Then $Z[x_1^p,x_2^p]$ equals the $k$-subalgebra $k[x_1^p,x_2^p,x_2^{p+1}]$ of $k[x_1,x_2]$. This $k$-subalgebra is not normal, it is not a complete intersection scheme, and it is not regular in codimension $1$, much less in codimension $2$. | |
| Feb 11, 2019 at 2:30 | review | First posts | |||
| Feb 11, 2019 at 2:31 | |||||
| Feb 11, 2019 at 2:29 | history | asked | John Zek. | CC BY-SA 4.0 |