Timeline for Description of prime ideals in $\operatorname{Spec}(\mathbb{Z}[x_1, \dots, x_n])$
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 28 at 5:53 | history | edited | YCor | CC BY-SA 4.0 |
changed to more useful title
|
| Jun 27 at 15:47 | history | edited | Daniel Asimov | CC BY-SA 4.0 |
are —> and
|
| Jun 27 at 11:19 | comment | added | YCor | @Z.M ah, you're right. | |
| Jun 27 at 11:18 | comment | added | Z. M | @YCor But this is in general not a prime ideal (I guess that it is prime only when $n=1$). | |
| Jun 26 at 23:27 | comment | added | YCor | In $R[x,y]$, $R$ any nonzero commutative ring, the ideal $J_n$ generated by $\{x^ay^b:a+b=n\}$ is not generated by less than $n+1$ elements (because $J_n/J_{n+1}$ is isomorphic to $R^{n+1}$ as $R[x,y]$-module, where $R$ is identified to $R[x,y]/(x,y)$. | |
| Jun 26 at 21:14 | answer | added | Dave Benson | timeline score: 4 | |
| Jun 26 at 21:12 | history | edited | diracdeltafunk | CC BY-SA 4.0 |
added 14 characters in body
|
| Jun 26 at 21:11 | comment | added | diracdeltafunk | I see, thanks for your help! | |
| Jun 26 at 21:10 | comment | added | Will Sawin | Prime ideals in $\mathbb C[x_1,x_2,x_3]$ can require arbitrarily many generators math.stackexchange.com/a/2030114/84942 and the same should extend to $\mathbb Z[x_1,x_2,x_3]$, and possibly even $\mathbb Z[x_1,x_2]$. | |
| Jun 26 at 21:08 | comment | added | Will Sawin | I don't think so - this leads to a subtler issue where if $\mathbb Z[x_1,\dots, x_n]/(f_0,\dots, f_n)$ is not a UFD, then there is no reason to think there is a unique optimal way to take a polynomial in $k(\mathfrak p)[x_{n+1}]$ and clear denominators to obtain a polynomial in $\mathbb Z[x_1,\dots,x_{n+1}]$. So the ideal could have multiple generators involving $x_{n+1}$. On the level of fields it's correct that each generator $x_i$ is algebraic or transcendental over the field generated by the previous ones but for rings more can happen. | |
| Jun 26 at 21:06 | history | edited | diracdeltafunk | CC BY-SA 4.0 |
Edited to correct a mistake pointed out by Will Sawin
|
| Jun 26 at 21:05 | comment | added | diracdeltafunk | Oh good point, thank you! I am happy to remove the word monic everywhere, in which case things should be correct, yes? | |
| Jun 26 at 21:03 | comment | added | Will Sawin | Your statement is not quite right. For example for $n=1$ the ideal generated by $2 x_1-1$ should be generated by a monic polynomial in $x_1$, but of course $2 x_1-1$ is not monic, and no monic alternative exists. The issue is that while every ideal of $k(\mathfrak p)[x_{n+1}]$ is generated by a monic polynomial, that polynomial need not lie in $\mathbb Z[x_1,\dots, x_{n+1}]$ and putting it into $\mathbb Z[x_1,\dots, x_{n+1}]$ may make it nonmonic. | |
| Jun 26 at 20:59 | history | edited | diracdeltafunk | CC BY-SA 4.0 |
typos
|
| Jun 26 at 20:53 | history | asked | diracdeltafunk | CC BY-SA 4.0 |