This is part of an exercise from Eisenbud:
$k$ is a field, describe as explicitly as possible
a) $k[x]/(x^n) \otimes_{k[x]} k[y]/(y^m)$
b) $k[x] \otimes_{k} k[y]$
Any hint ?
Asked
Active
Viewed 261 times
1
-
1Well start by changing the name of the variable in one side from $x$ to $y$ so it becomes $k[x]\otimes k[y]$ – omar Apr 12 '14 at 06:55
-
@WLOG, what are you tensoring over? $k[x]$or $k$? – Apr 12 '14 at 09:57
-
@user114539: I have edited – WLOG Apr 12 '14 at 11:16
1 Answers
1
For $a)$ you can use the fact that $R/I\otimes_R R/J\cong R/(I+J)$, so it is $k[x]/((x^n)+(x^m))$. What does that become if $n\mid m$? I believe that the second one has been asked on this website, $k[x]\otimes_k k[y]\cong k[x,y]$. I know this is not an answer yet, if you need further help, let me know and i will write more.
Later: You can find it here
-
-
$n | m$ is not required, for what you are trying to imply $n \le m$ suffices – Brozovic Aug 31 '19 at 15:54