Let $j:\mathbb{A}^{n}\rightarrow\mathbb{A}^{n}$ an open immersion over a field $k$. Is it an isomorphism?



open immersion between affine spaces
If $U \cong \mathbb A^n$ is the image of $j$, then the long exact sequence of a pair shows that $\mathbb A^n \setminus U$ has no cohomology in any degree, which is only possible if $U = \mathbb A^n$.


More generally (using heavier machinery), no variety $X$ is isomorphic to a Zariski open subset $U \subsetneq X$. The same long exact sequence shows that the Hodge-Deligne polynomial of $X \setminus U$ would need to vanish, which is only possible for $X \setminus U = \varnothing$.


