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- Jun 22, 2012

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

Prove that \(\displaystyle V \) is isomorphic to [TEX] \mathbb{A}^2 [/TEX]

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

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

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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