- #1
ay46
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This question is typically seen in the beginning of a commutative algebra course or algebraic geometry course.
Let $V = \mathcal{Z}(xy-z) \subset \mathbb{A}^3$. Here $\mathcal{Z}$ is the zero locus. Prove that $V$ is isomorphic to $\mathbb{A}^2$ as algebraic sets and provide an explicit isomorphism $\phi$ and associated $k$-algebra isomorphism $\tilde{\phi}$ from $k[V]$ to $k[\mathbb{A}^2]$ along with their inverses. Is $V = \mathcal{Z}(xy-z^2)$ isomorphic to $\mathbb{A}^2$?
Here is what I have so far: let $V = \mathcal{Z}(xy-z) \subset \mathbb{A}^3$. Consider the map: $\pi(x,y,z) = z$ where $\pi$ is a family of varieties i.e. a surjective morphism. This map give the hyperbola family: $\{\mathcal{Z}(xy-z) \subset \mathbb{A}^2\}_{z \in \mathbb{A}^1}$ and is injective. Does this provide an explicit isomorphism $\phi$? I am not sure how to proceed for the coordinate rings and how to define the inverses.
Thank you!
Let $V = \mathcal{Z}(xy-z) \subset \mathbb{A}^3$. Here $\mathcal{Z}$ is the zero locus. Prove that $V$ is isomorphic to $\mathbb{A}^2$ as algebraic sets and provide an explicit isomorphism $\phi$ and associated $k$-algebra isomorphism $\tilde{\phi}$ from $k[V]$ to $k[\mathbb{A}^2]$ along with their inverses. Is $V = \mathcal{Z}(xy-z^2)$ isomorphic to $\mathbb{A}^2$?
Here is what I have so far: let $V = \mathcal{Z}(xy-z) \subset \mathbb{A}^3$. Consider the map: $\pi(x,y,z) = z$ where $\pi$ is a family of varieties i.e. a surjective morphism. This map give the hyperbola family: $\{\mathcal{Z}(xy-z) \subset \mathbb{A}^2\}_{z \in \mathbb{A}^1}$ and is injective. Does this provide an explicit isomorphism $\phi$? I am not sure how to proceed for the coordinate rings and how to define the inverses.
Thank you!