MHB 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

 

[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?

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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

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 $$ V $$ 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 $$ V \subseteq \mathbb{A}^3 $$ to $$ W = \mathbb{A}^2 $$, so I would say we need a morphism or polynomial map $$ \phi : \ V \to W $$.

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

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

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

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

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

... ... ...

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

$$ \widetilde{\phi}: \ k[W] \to k[V] $$

$$ f \mapsto f \circ \phi $$

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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