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Thread: ismomorphism between ... (conmutative algebra)

  1. #1
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    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!
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  2. #2
    Member Haven's Avatar
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    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
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  3. #3
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    Re: ismomorphism between ... (conmutative algebra)

    Quote Originally Posted by Haven View Post
    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
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  4. #4
    Member Haven's Avatar
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    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.
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