Algebraic Geometry - D&F Section 15.1, Exercise 24

[Math Amateur]

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

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

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

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

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

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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

 

[R136a1]

So you you need to find an isomorphism between $\mathbb{A}^2$ and the surface $z=xy$. A suitable isomorphism should be $\varphi(x,y) = (x,y,xy)$. Does that help?

 

[Math Amateur]

Thanks for the help, R136a1

Will now reflect on your guidance

Peter