I am interested in the analog between etale and usual topology. The following question is one example. Any reference would be appreciate.


Let $k$ be an algebraically closed field and $X\_1$ and $X\_2$ be $k$-varieties. My question is as follows: For a given etale neighbourhoods $U,u\to X\_1\times\_k X\_2,(a\_1,a\_2)$, are there etale neighbourhood $U\_1,u\_1\to X\_1,a\_1$ and $U\_2,u\_2\to X\_2,a\_2$ such that the induced etale morphism $U\_1\times\_k U\_2, (u\_1,u\_2) \to X\_1\times\_k X\_2$ can be factored through $U,u$?



etale neighbourhood of a product
In fact, the example I gave in the comments shows that such $U\_i$ do not exist in general:


**Example.** Let $X\_1 = X\_2 = \mathbb A^1$, with base points $a\_1 = a\_2 = 0$. Let $U = D(xy-1)$ be the complement of the hyperbola $xy = 1$.


Suppose there exist étale neighbourhoods $U\_1$ and $U\_2$ of $0$ such that $U\_1 \times U\_2 \to \mathbb A^2$ factors through $U$. Since étale maps are open, the (set-theoretic) images of $U\_1$ and $U\_2$ in $\mathbb A^1$ are opens $W\_1, W\_2 \subseteq \mathbb A^1$ containing $0$. Then the image of $U\_1 \times U\_2 \to \mathbb A^2$ is $W\_1 \times W\_2$.


But no such set can miss $V(xy-1)$: there are only finitely many $x$ not in $W\_1$, and only finitely many $y$ not in $W\_2$. Thus, for almost all $x \neq 0$, we have $x \in W\_1$ and $\tfrac{1}{x} \in W\_2$, providing a point
$$\left(x,\tfrac{1}{x}\right) \in V(xy-1) \cap W\_1 \times W\_2.$$
We conclude that such $U\_i$ cannot exist. $\square$


**Remark.** Basically, the Zariski topology (!) has too few opens.


