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
- The questioner was satisfied with and accepted the answer, or
- The answer was evaluated as being 100% correct by the judge.
1 Attachment

- answered
- 541 views
- $25.00
Related Questions
- Element satisfying cubic equation in degree $5$ extension
- Reverse this equation/function (2d to isometric)
- Solving for two unknown angles, from two equations.
- Root of $x^2+1$ in field of positive characteristic
- How do you go about solving this question?
- Differentiate $f(x)=\int_{\tan x}^{0} \frac{\cos t}{1+e^t}dt$
- Algebra Word Problem #1
- Sinusodial graph help (electrical)