Zariski Topology and Regular Functions on Algebraic Varieties in Affine Space

i) Show that the Zariski topology on $Spec(k[x,y])$ is NOT the product of the Zariski topologies on $Spec(k[x])\times Spec(k[y])$ (unlike the metric topology, in wich the product of the topologies is equivalent to the topology of the product).

ii) Show that the ring of regular functions $A(P)$ of the parabola $P$ defined by $y=x^2$ in $\mathbb{A}^2_k:=Spec[x,y]$ is isomorph to ring of polynomials in one variable over $k$.

iii) Show that the ring of regular functions $A(P)$ of the plave curve $C$ defined by $x\cdot y=2$ in $\mathbb{A}^2_k$ is NOT isomorph to ring of polynomials in one variable over k.

iv) Let $C$ be a curve in $\mathbb{A}^3_k$ given by the parameterization $\{(t,t^2,t^3)|t\in k\}$.
a) Find generators of the ideal $I(C)$.
b) Show that $A(C)$ is isomorph to ring of polynomials in one indeterminate.

Answer

Answers can only be viewed under the following conditions:
  1. The questioner was satisfied with and accepted the answer, or
  2. The answer was evaluated as being 100% correct by the judge.
View the answer

1 Attachment

The answer is accepted.
Join Matchmaticians Affiliate Marketing Program to earn up to a 50% commission on every question that your affiliated users ask or answer.