Zariski topologies are defined via closed-set definition; with the closed sets being algebraic varieties $V(I)$ of ideals $I\subseteq R$ of commutative rings $R$... But what are these varieties, exactly?
I can understand easily the concept of an algebraic variety in the context of polynomial rings, since its just a set of solutions. Moreover, i can easily spot the "main" underlying sets: ideals $I \subseteq K[x_1,x_2,...,x_n]$ in the polynomial ring are subsets of $K[x_1,x_2,...,x_n]$ ($K$ a field), and algebraic varieties $V(I)\subseteq K^n$ are subsets of the affine space $K^n$. For obvious reasons, every element in an affine space is called a "point"... but the definition of variety for a commutative ring is another beast. The most compact and least convoluted one i could find was this one: if $R$ is a ring and $I\subseteq R$ and ideal, then: $$V(I)=\{P\in \mathrm{Spec}(R)|P\supseteq I\}$$ But this implies that $V(I)\subseteq \mathcal{P}(R)$, with $\mathcal{P}$ denoting power set. I have some questions:
- If each $V(I)\subseteq \mathcal{P}(R)$ is just a subset of $R$, how can they have geometrical properties as a whole?
- Does this mean that $\mathcal{P}(R)$ has an inherent geometrical meaning (this is, distinct from the geometrical properties of $R$ itself)?
- Why are the varieties of a polynomial ring and of a commutative ring named after the same concept if they are seemingly different? (I.e, algebraic varieties are not subsets of $K[x_1,x_2,...,x_n]$)
I suspect that this is somehow related to the reason why elements in $\mathrm{Spec}(R)$ are called "closed points" but i don't understand that either. Any help will be deeply appreciated!