I'm trying to understand how to visualize schemes and so I'm starting with visualizing $\operatorname{Spec}$ of different rings. I've marked my questions with question numbers as I go:
I understand that $\operatorname{Spec}\mathbb{C}[x,y]/(x)$ is the set of prime ideals in $\mathbb{C}[x,y]$ containing $(x)$ which is just $(x)$ and maximal ideals $(x,y-a)$, so this is the affine line (corresponding to the obvious fact that $\mathbb{C}[x,y]/(x) \cong \mathbb{C}[y]$)
What about the localizations $\mathbb{C}[x,y]_x$ and $\mathbb{C}[x,y]_{(x)}$? I understand that these correspond to prime ideals of $\mathbb{C}[x,y]$ not containing $x$ and contained in $x$ respectively. For the former, $\operatorname{Spec}\mathbb{C}[x,y]_x$ is the picture of the entire affine plane that's 'away' from the y axis. That is every prime ideal that doesn't 'pass through' the y axis, which is every prime ideal not equal to $(x)$ or $(x, y-a)$. (Maybe 'pass through' isn't the best phrase here, since the parabola $(y-x^2)$ is perfectly fine, even though it intersects the y-axis?)
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It seems like it's just the compliment of the quotient example at the start. Is there some algebraic way of saying this geometric intuition aside from just noting the ideals contain and resp. don't contain x? Or is that the main point?
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The second thing I'm confused about is the second localization $\mathbb{C}[x,y]_{(x)}$. What geometric picture does the spectrum of this correspond to? Intuitively, it seems like the spectrum contains the prime ideals of $\mathbb{C}[x,y]$ contained in $(x)$ which is just $(0)$ and $(x)$. But what's the picture for that? Is it just the y-axis? If so, how is that different from the first example? I don't quite see the translations between algebra and geometry here. Any advice would be great.
Best Answer
These things are somewhat subjective since there are many ways to view schemes.
The image for $\mathbb{C}[x,y]_x$ (ie. the localization at the element $x$) can be understood as the set of functions on $D(x) = \mathbb{A}^2 \setminus V(x)$, corresponding to the fact that rational functions of the sort $f(x,y)/x^n$ only make sense where $x \neq 0$. Algebraically, this corresponds to the fact that the prime ideals of $\mathbb{C}[x,y]_x$ are in 1-1 correspondence with those of $\mathbb{C}[x,y]$ not containing $x$.
Somewhat differently, $\mathbb{C}[x,y]_{(x)}$ is the stalk of the structure sheaf of $\mathbb{A}^2$ at $(x)$ so it can be understood as the structure of the space around $(x) \in \mathbb{A}^2$. This is a little more subtle. The key geometric fact here is that if $\operatorname{Spec}A$ is affine and $\mathfrak{p} \in \operatorname{Spec}A$, then there is a topological immersion $\operatorname{Spec} A_\mathfrak{p} \hookrightarrow \operatorname{Spec} A$. Of course, this comes from the commutative algebra fact that primes of $A_{\frak{p}}$ are in correspondence with primes of $A$ contained in $\frak{p}. $
In the case of $\operatorname{Spec} \mathbb{C}[x,y]_{(x)} \hookrightarrow \mathbb{A}^2$, we can then interpret its image as the set of (generic points of) affine varieties which contain the curve $V(x)$. That is, it is simply $\{(x), (0)\}$ which correspond to the curve $V(x)$ and the affine plane $\mathbb{A}^2$. This idea generalizes quite nicely to arbitrary schemes and gives rise to nice properties such as the following:
If $Y \subset X$ is an irreducible closed subscheme and $\eta \in Y$ is the generic point of $Y$, then $\text{codim}_X(Y) = \text{dim}\; \mathcal{O}_{\eta, X}$. (Vakil FOAG 11.1.I)