Hello!

I'd like to show that the ring $\displaystyle K[X,Y] / (XY -1)$ is isomorphic to $\displaystyle S^{-1}k[X] $ where S is the multiplicative system of powers of X.

Thanks for any comment!

Printable View

- Nov 17th 2013, 11:36 AMLolytaismomorphism between ... (conmutative algebra)
Hello!

I'd like to show that the ring $\displaystyle K[X,Y] / (XY -1)$ is isomorphic to $\displaystyle S^{-1}k[X] $ where S is the multiplicative system of powers of X.

Thanks for any comment! - Nov 24th 2013, 02:37 AMHavenRe: ismomorphism between ... (conmutative algebra)
By modding out $\displaystyle K[X,Y]$ by $\displaystyle XY-1$ we are saying that in the new ring, $\displaystyle XY-1=0$ or rather $\displaystyle XY=1$. Therefore Y is the inverse of X.

In $\displaystyle S^{-1}k[X]$, we are adding in $\displaystyle X^{-1}$ to $\displaystyle k[X]$ and all it's powers.

From here the isomorphism should be fairly clear - Nov 24th 2013, 07:35 AMLolytaRe: ismomorphism between ... (conmutative algebra)
- Nov 27th 2013, 05:58 PMHavenRe: ismomorphism between ... (conmutative algebra)
Try the homomorphism:

$\displaystyle X \mapsto X$ and $\displaystyle Y \mapsto X^{-1}$

See if you can show it's bijective.