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Shizhuo Zhang
Shizhuo Zhang
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In fact, it is a simple problem. I just want to know whether there are some interesting proof.

$Z[x_1, x_2, ......, x_{n^2-1}]$ and $Z[y_{11}, ......, y_{1n}, y_{21}, ......, y_{nn}]/(det(y_{ij})-1))$, where $Z$ is integer.

One way to prove is select a prime number,say $p=2$,then localize these two rings, one can count the number of elements in both rings and they are NOT equal.

Question: Is there any other geometric way to "see" they are obviously not isomorphic to each other?

Any related comments are welcome. Thanks

The reason I want to ask is some one argued that the proof I gave above is not natural. He thought this is not a "Grothendieck style proof"

In fact, it is a simple problem. I just want to know whether there are some interesting proof.

$Z[x_1, x_2, ......, x_{n^2-1}]$ and $Z[y_{11}, ......, y_{1n}, y_{21}, ......, y_{nn}]/(det(y_{ij})-1))$, where $Z$ is integer.

One way to prove is select a prime number,say $p=2$,then localize these two rings, one can count the number of elements in both rings and they are NOT equal.

Question: Is there any other geometric way to "see" they are obviously not isomorphic to each other?

Any related comments are welcome. Thanks

In fact, it is a simple problem. I just want to know whether there are some interesting proof.

$Z[x_1, x_2, ......, x_{n^2-1}]$ and $Z[y_{11}, ......, y_{1n}, y_{21}, ......, y_{nn}]/(det(y_{ij})-1))$, where $Z$ is integer.

One way to prove is select a prime number,say $p=2$,then localize these two rings, one can count the number of elements in both rings and they are NOT equal.

Question: Is there any other geometric way to "see" they are obviously not isomorphic to each other?

Any related comments are welcome. Thanks

The reason I want to ask is some one argued that the proof I gave above is not natural. He thought this is not a "Grothendieck style proof"

Source Link
Shizhuo Zhang
Shizhuo Zhang
  • 5.5k
  • 2
  • 40
  • 59

How to prove these two rings are not isomorphic

In fact, it is a simple problem. I just want to know whether there are some interesting proof.

$Z[x_1, x_2, ......, x_{n^2-1}]$ and $Z[y_{11}, ......, y_{1n}, y_{21}, ......, y_{nn}]/(det(y_{ij})-1))$, where $Z$ is integer.

One way to prove is select a prime number,say $p=2$,then localize these two rings, one can count the number of elements in both rings and they are NOT equal.

Question: Is there any other geometric way to "see" they are obviously not isomorphic to each other?

Any related comments are welcome. Thanks