ismomorphism between ... (conmutative algebra)

• Nov 17th 2013, 11:36 AM
Lolyta
ismomorphism 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 AM
Haven
Re: 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 AM
Lolyta
Re: ismomorphism between ... (conmutative algebra)
Quote:

Originally Posted by Haven
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

I understand everything you say but I can't see the ismorphism... :S
• Nov 27th 2013, 05:58 PM
Haven
Re: 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.