I am looking for a list comparing the geometric/topological properties of an affine scheme $X \simeq \text{Spec}(R)$ with the algebraic properties of the corresponding ring $R$ and vice versa.
Particular emphasis should be placed on affine $K$-varieties to make the geometric picture clearer.
An example of this would be:
- An affine $K$-variety $X$ is irreducible iff $K[X]$ is an integral domain. eg. $\mathbb{C}[x,y]/(x)$ is an ID and consists of only a single curve, whereas $\mathbb{C}[x,y]/(xy)$ is not an ID and consists of the union of two curves.
- the classical points of an affine scheme $X$ lie dense iff $R$ is Jacobson
- An affine $K$-variety $X$ has no "thick subschemes" iff $K[X]$ is reduced. Where by a "thick subscheme" I mean something like the thick $y$-axis $\mathbb{R}[x,y]/(x^2)$ as opposed to $\mathbb{R}[x,y]/(x)$
As suggested by the examples I am not looking for ultra precise statements on the geometric side of things, but rather one giving justification for either why certain algebraic properties of rings are worth studying or why certain geometric properties are worth studying.
If possible (or you feel the need) please provide an illustrative example (as for instance in my first example)
If possible (as for instance in my second example) a proof or link to a proof or resource is appreciated. In any case, please provide some explanation, or why you think of algebraic property Y giving geometric property Z.
If, in order to describe the geometric properties, you use non-standard language (as for instance in my third example) please explain.
Finally, maybe think of the question like this: You read about some new class of rings and ask yourself "Why does that definition make sense? Why should those rings be interesting?" This question should be one place to look for in seeking an answer as, in the geometric setting, the importance may become more clear.