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Thread: Algebraic Geometry - D&F Section 15.1, Exercose 24

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    Algebraic Geometry - D&F Section 15.1, Exercose 24

    Dummit and Foote Section 15.1, Exercise 24 reads as follows:

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    Let $\displaystyle V = \mathcal{Z} (xy - z) \subseteq \mathbb{A}^3 $.

    Prove that V is isomorphic to $\displaystyle \mathbb{A}^2 $

    and provide an explicit isomorphism $\displaystyle \phi $ and associated k-algebra isomorphism $\displaystyle \widetilde{\phi} $ from $\displaystyle k[V] $ to $\displaystyle k[ \mathbb{A}^2] $ along with their inverses.

    Is $\displaystyle V = \mathcal{Z} (xy - z^2) $ isomorphic to $\displaystyle \mathbb{A}^2 $?

    -------------------------------------------------------------------------------------------------------------

    I would appreciate some help and guidance with getting started with this exercise [I suspect I might need considerable guidance! :-( ]


    Some of the background and definitions are given in the attachment.


    Peter
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