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 \(\displaystyle V \) 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

 

[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 \(\displaystyle V \) 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

I am still working on this problem. Here are some more thoughts I've had ... however I am struggling to make a great deal of progress and still need substantial help ...

We have to show that \(\displaystyle V \) is isomorphic to $$\mathbb{A}^2$$

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

... ... ... well, we must look for a mapping from \(\displaystyle V \subseteq \mathbb{A}^3 \) to \(\displaystyle W = \mathbb{A}^2 \), so I would say we need a morphism or polynomial map \(\displaystyle \phi : \ V \to W \).

D&F (Section 15.1, page 662) define a morphism as follows:

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

Definition. A map \(\displaystyle \phi : \ V \to W \) is called a morphism or polynomial map or regular map of algebraic sets if there are polynomials \(\displaystyle \phi_1, ... \ ... \phi_m \in k[x_1, x_2, ... \ ... x_n] \) such that

\(\displaystyle \phi ((a_1, a_2, ... \ ... , a_n)) = ( \phi_1(a_1, a_2, ... \ ... , a_n) ... \ ... \phi_m(a_1, a_2, ... \ ... , a_n)) \) for all \(\displaystyle (a_1, a_2, ... \ ... , a_n) \in V \)

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

D&F (seemingly importantly for our problem) go on to say:

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

The map \(\displaystyle \phi : \ V \to W \) is an isomorphism of algebraic sets if there is a morphism \(\displaystyle \psi : \ W \to V \) with \(\displaystyle \phi \circ \psi = 1_W \) and \(\displaystyle \psi \circ \phi = 1_V \).

... ... ...

... \(\displaystyle \phi \) indices a well defined map from the quotient ring \(\displaystyle k[x_1, ... \ ... , x_m]/ \mathcal{I}(W) \) to the quotient ring \(\displaystyle k[x_1, ... \ ... , x_m]/ \mathcal{I}(V) \):

\(\displaystyle \widetilde{\phi}: \ k[W] \to k[V] \)

\(\displaystyle f \mapsto f \circ \phi \)

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

So the above are the concepts I now believe need to be applied, but I lack the skills and knowledge to apply them in this specific case

I would really appreciate some help.

Peter