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 [itex] V = \mathcal{Z} (xy - z) \subseteq \mathbb{A}^3 [/itex].

Prove that [itex] V [/itex] is isomorphic to [itex] \mathbb{A}^2 [/itex]

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

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

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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