Find irreducible and connected components of $\operatorname{Spec}(\mathbb{C}[x] \times \mathbb{C}[y])$

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Find irreducible and connected components of $\operatorname{Spec}(\mathbb{C}[x] \times \mathbb{C}[y])$

That's the problem 11.1 from commutative algebra course

As answered here

we can see the bijection between $\operatorname{Spec}(A_1\times A_2)$ and $\operatorname{Spec}(A_1)\sqcup\operatorname{Spec}(A_2)$ as follows: the prime ideals of $A_1\times A_2$ are of the form $\mathfrak{p}\times A_2$ where $\mathfrak p$ is a prime ideal of $A_1$, or $A_1\times\mathfrak q$ where $\mathfrak q$ is a prime ideal of $A_2$.

That means that we should find irreducible and connected components of $\operatorname{Spec}(\mathbb{C}[x])$

As discussed here:
Let $A$ be a commutative ring with unit, $X = \operatorname{Spec}(A) $ with the Zariski topology. The irreducible components are $\lbrace V(p) : p\subset A \ \text{minimal prime ideal} \rbrace$ where $V(P) =\lbrace q \ \text{prime ideal } \mid p\subset q\rbrace$.

An ideal is called prime if the complement is a multiplicative set.

I know that prime ideals of $\mathbb{C}[x]$ are the principal ideals generated by $\lbrace (x-a) | a \in \mathbb{C}\rbrace$ and $0$. $0$ is contained in all the other ideals.

The prime ideals $\mathfrak{p}$ of $\mathbb{C}[x]$ except $0$ are all minimal, so the irreducible components of $\operatorname{Spec}\mathbb{C}[x]$ are all $\lbrace V(p) \rbrace$.

Is the final answer that the connected component are the same as irreducible components and are in the bijection with $I_{x-a} \times \mathbb{C}[y]$ and
$\mathbb{C}[x] \times I_{y-a}$ ?

Could you explain how this topology can be seen "in a picture"?

Best Answer

Your proof is correct!

The topology of $\displaystyle\mathrm{Spec}(\mathbb{C}[x])$ is the following:

  1. on $\mathrm{Spec}(\mathbb{C}[x])\setminus\{(0)\}$ one has the cofinite topology, I mean that all and only closed sets are the whole of the space, the empty set, and the finite subsets;
  2. $\{(0)\}$ is a fat point, I mean that $\overline{\{(0)\}}=\mathrm{Spec}(\mathbb{C}[x])$.

As a picture: $\mathrm{Spec}(\mathbb{C}[x])\setminus\{(0)\}$ is the affine (complex) line $\mathbb{A}^1_{\mathbb{C}}$ plus a dense point, that is a point arbitrally closed to any other point.

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