Continuous map example

Question

I am trying to understand the distinction between continuous maps between varieties and morphisms between varieties, and I believe a concrete example illustrating the distinction will help. What is an example of a continuous map $\pi:A\rightarrow B$ where $A,B$ are varieties and $\pi$ is not a morphism?

Angina Seng · Accepted Answer · 2018-12-13T07:09:16.383

0

How about $A=B$ being the affine line over $\Bbb C$, and $\pi:z\mapsto\overline z$?

edited Dec 13 '18 at 07:09

answered Dec 13 '18 at 06:48

Angina Seng

158,341

It's not given by a polynomial @topoSpace – Angina Seng Dec 13 '18 at 18:35

score 0 · Answer 2 · answered Dec 13 '18 at 06:49

0

One easy source of examples is $\Bbb A^1_k$. Mapping the generic point to the generic point and permuting the closed points will produce a continuous map, and most examples of this fail to be morphisms. For instance, you can swap $0$ and $1$ and leave all other points invariant, which cannot be the result of a morphism.

answered Dec 13 '18 at 06:49

KReiser

65,137

Two maps $f,g:X\to Y$ with $X$ reduced and $Y$ separated which agree on a dense set must be equal. Since this agrees with the identity morphism on a dense set but is not equal to it, it can't be a map of varieties. – KReiser Dec 13 '18 at 20:47

Continuous map example

2 Answers2