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.
![Sarah M](https://matchmaticians.com/storage/user/100072/thumb/matchmaticians-hd8sie-file-1-avatar-512.jpg)
Answer
- The questioner was satisfied with and accepted the answer, or
- The answer was evaluated as being 100% correct by the judge.
1 Attachment
![Poincare](https://matchmaticians.com/storage/user/100023/thumb/matchmaticians-xhd1nv-file-2-avatar-512.jpg)
- answered
- 277 views
- $25.00
Related Questions
- Get area of rotated polygon knowing all coordinates and angle.
- Module isomorphism and length of tensor product.
- Algebra Word Problem 2
- Evaluate $\int \ln(\sqrt{x+1}+\sqrt{x}) dx$
- If both $n$ and $\sqrt{n^2+204n}$ are positive integers, find the maximum value of $𝑛$.
- Center of algebra of functions
- Let $R$ be an integral domain and $M$ a finitely generated $R$-module. Show that $rank(M/Tor(M))$=$rank(M)$
- Five times the larger of two consecutive odd integers is equal to one more than eight times the smaller. Find the integers.