3

In my course Algebraic Geometry I always find it hard to come up with examples or counterexamples. For instance in the following question:

Give an example of two affine varieties $X,Y$ and a morphism $\phi:X \rightarrow Y$ such that the image of $\phi$ is not locally closed in $Y$.

We defined locally closed here as: A subset $Z$ of a topological space is called locally closed if $Z$ is the intersection of an open subset and a closed subset of $X$.

TheBeiram
  • 435
  • 5
    Try $\mathbb{C}^2\to\mathbb{C}^2$, given by $(x,y)\mapsto (x,xy)$. – Mohan May 25 '16 at 15:56
  • I am not sure how to show this works. I can see that under this map $\phi$ we get that $\phi = \mathbb{C}^2 \setminus Z(x) \cup Z(x,y)$, where $Z(x)$ is the zero-set of $x$. But this is the union of an open and closed set, but how do I know that we cannot write it as a locally closed set in $\mathbb{C}^2$? – TheBeiram May 29 '16 at 15:01
  • What are the closed subsets containing the image? If the image is locally closed, what can you say about the image, if you answer this question? – Mohan May 29 '16 at 16:23
  • So do you mean that $\mathbb{C}^2$ is the only closed subset containing the image? – TheBeiram May 29 '16 at 19:27
  • Please check if you can find a non-zero polynomial vanishing on the image. – Mohan May 29 '16 at 21:22
  • Exercise: A set is locally closed iff it is open in its closure. Using this is easy to see that the image is not locally closed. – nowhere dense May 07 '20 at 10:25

0 Answers0