Algebraic Geometry - D&F Section 15.1, Exercise 24

[Math Amateur]

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

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

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 $\mathbb{A}^2$?

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

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