Let k be algebraic closed field. then $\mathbb{A}_n(k)$ as $\operatorname{Hom}(k_n,k)$ and $V(\alpha)$ as $\operatorname{Hom}(k_n/\alpha,k)$ which is true by using noether normalization theorem. so every point is like morphism over k. at this point of view, Hilbert nullstellensatz give a classify of $\operatorname{Hom}(k_n/I,k) \rightarrow \operatorname{Hom}(k_n,k)$ which classify by $\operatorname{Hom}(k_n/\sqrt{I},k) \rightarrow \operatorname{Hom}(k_n,k)$.

1. But how can we say some things of this [ n generated k-algebra / k] this slice category by Hilbert nullstellensatz? Furthermore, Can I see all of finitely generated k-algebra together, and we can get what?
2. Zariski topology is saying $\operatorname{Hom}(k_n/\sqrt{I},k) \rightarrow \operatorname{Hom}(k_n,k)$ closed under pullback. and intersection like what?
3. From this point of view, can I dream of étale topology ( because this is like site over $\operatorname{Hom}(k_n,k)$ )?