How do the $\mathrm{Spec}(\mathbb{C}\left [ X \right ])$ and $\text{m-Spec}(\mathbb{C}\left [ X \right ])$ look like?
I understand the definitions of $\mathrm{Spec}(R)$ and $\text{m-Spec}(R)$ for a commutative ring $R$. In the first case this is the set of the prime ideals and in the second the set of the maximal ideals of $R$. I also know that $\mathrm{Spec}$ contains $\text{m-Spec}$ as a subset.
Can anybody help me with a concrete example, please? Thank you very much!
Best Answer
Since $\mathbb{C}$ is algebraically closed we have a bijection between points of an affine variety and maximal ideals of the affine coordinate ring.
Hence $\mathrm{Spec}_{\text{max}}(\mathbb{C}[X])$ can be seen as the affine line over $\mathbb{C}$ with the usual Zariski topology from classical algebraic geometry (every point is closed).
Now $\mathrm{Spec}(\mathbb{C}[X])$ is the same as $\mathrm{Spec}_{\text{max}}(\mathbb{C}[X])$ but with a funny "generic point" (the zero ideal), whose closure is the whole spectrum. You can thing of this generic point as some sort of soul underlying your affine line over $\mathbb{C}$.
More generally, whenever you are working over an algebraically closed field (like $\mathbb{C}$) you can proceed in a similar way to visualize the spectra. There is a bijection between closed points of an affine variety and maximal ideals of its affine coordinate ring. Hence you can visualize the maximal spectrum as the variety with the usual Zariski topology (all points being closed). And now for the prime spectrum you will have the maximal ideals (which is the variety visualized as maximal spectrum) and then for every subvariety you will have a generic point, which you can again think of as some sort of soul underlying your subvariety, whose closure is that subvariety itself.
Even more generally, for any scheme $X$ there is a bijection between points of $X$ and irreducible closed subschemes, which sends every point to its closure (which is trivially irreducible).