We know that there is an equivalence of categories between the two following categories:
$1)$ Classical varieties over $k$, where $k$ is an algebraically closed field. (Informally I mean locally ringed spaces formed by patching affine irriducible algebraic sets over $k$. This is the definition of algebraic variety present for example in Perrin's book.)
$2)$ Schemes over $k$ which are integral and of finite type over $k$
Now I need an example, if it does exist, of two non isomorphic (as classical varieties) algebraic sets that are non isomorphic as $k$-schemes (this is obvious by the previous functor) but isomorphic in the category of schemes.
To be more specific I need two non isomorphic algebraic sets such that when I use the above functor $1)\longrightarrow 2)$ and then I forget the structure morphism on Spec $k$,
I finally obtain two isomorphic schemes.
Thanks in advance
Best Answer
Let's say $k=\mathbb{C}$ (although something like this should work over any algebraically closed field). Let $V_1=\mathbb{P}^1-\{0,1,\infty,\pi\}$ and $V_2 = \mathbb{P}^1-\{0,1,\infty,e\}$. One can see that $V_1$ and $V_2$ are not isomorphic as varieties/schemes over $\mathbb{C}$: such an isomorphism would extend to a map $\mathbb{P}^1\to\mathbb{P}^1$ taking $\{0,1,\infty,\pi\}$ to $\{0,1,\infty,e\}$, which cannot exist because the cross ratio of any permutation of $\{0,1,\infty,\pi\}$ does not equal the cross ratio of $\{0,1,\infty,e\}$. However, there exists an abstract field automorphism $\sigma$ of $\mathbb{C}$ taking $\pi$ to $e$, so there will be an isomorphism of schemes $V_1\to V_2$ which is $\sigma$-semilinear.