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
- 527 views
- $25.00
Related Questions
- Calculating Speed and Velocity
- Use Rouche’s Theorem to show that all roots of $z ^6 + (1 + i)z + 1 = 0$ lines inside the annulus $ \frac{1}{2} \leq |z| \leq \frac{5}{4}$
- Mechanical principle help (maths)
- Fields and Galois theory
- Prove that $A - B=A\cap B^c$
- Please help me with this math question
- Find $x$, if $\sqrt{x} + 2y^2 = 15$ and $\sqrt{4x} − 4y^2 = 6$.
- Algebra Word Problem #1