Revisions to A reducible connected scheme with pairwise disjoint irreducible components

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edited Apr 13, 2017 at 12:58

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In the answers to Non-integral scheme having integral local rings Non-integral scheme having integral local rings , and in particular in t3suji's beautiful geometric construction, it is shown that there is a (non-empty) affine scheme $X=\mathrm{Spec}(A)$ whose underlying topological space is connected, such that all local rings of $X$ are integral domains and such that $X$ is however not integral. This scheme fulfills the conditions you are requiring: It is reduced and not integral, hence reducible. Assume two irreducible components $Y_1$ and $Y_2$ of $X$ met in a point $x\in X$. The point $x$ corresponds to a prime ideal $\mathfrak{q} \subset A$, and the irreducible components $Y_1$ and $Y_2$ are of the form $Y_1=V(\mathfrak{p}_1)$ and $Y_2=V(\mathfrak{p}_2)$ for minimal prime ideals $\mathfrak{p}_1$ and $\mathfrak{p}_2$ of $A$. As $x\in Y_1,Y_2$, we would have $\mathfrak{p}_1, \mathfrak{p}_2 \subset \mathfrak{q}$.

Then $\mathfrak{p}_1$ and $\mathfrak{p}_2$ would correspond to distinct minimal prime ideals of $A_\mathfrak{q}=\mathcal{O}_{X,x}$, which is impossible, as $\mathcal{O}_{X,x}$ is by construction a domain, so that its only minimal prime ideal is $\{0\}$.

In the answers to Non-integral scheme having integral local rings , and in particular in t3suji's beautiful geometric construction, it is shown that there is a (non-empty) affine scheme $X=\mathrm{Spec}(A)$ whose underlying topological space is connected, such that all local rings of $X$ are integral domains and such that $X$ is however not integral. This scheme fulfills the conditions you are requiring: It is reduced and not integral, hence reducible. Assume two irreducible components $Y_1$ and $Y_2$ of $X$ met in a point $x\in X$. The point $x$ corresponds to a prime ideal $\mathfrak{q} \subset A$, and the irreducible components $Y_1$ and $Y_2$ are of the form $Y_1=V(\mathfrak{p}_1)$ and $Y_2=V(\mathfrak{p}_2)$ for minimal prime ideals $\mathfrak{p}_1$ and $\mathfrak{p}_2$ of $A$. As $x\in Y_1,Y_2$, we would have $\mathfrak{p}_1, \mathfrak{p}_2 \subset \mathfrak{q}$.

Then $\mathfrak{p}_1$ and $\mathfrak{p}_2$ would correspond to distinct minimal prime ideals of $A_\mathfrak{q}=\mathcal{O}_{X,x}$, which is impossible, as $\mathcal{O}_{X,x}$ is by construction a domain, so that its only minimal prime ideal is $\{0\}$.

In the answers to Non-integral scheme having integral local rings , and in particular in t3suji's beautiful geometric construction, it is shown that there is a (non-empty) affine scheme $X=\mathrm{Spec}(A)$ whose underlying topological space is connected, such that all local rings of $X$ are integral domains and such that $X$ is however not integral. This scheme fulfills the conditions you are requiring: It is reduced and not integral, hence reducible. Assume two irreducible components $Y_1$ and $Y_2$ of $X$ met in a point $x\in X$. The point $x$ corresponds to a prime ideal $\mathfrak{q} \subset A$, and the irreducible components $Y_1$ and $Y_2$ are of the form $Y_1=V(\mathfrak{p}_1)$ and $Y_2=V(\mathfrak{p}_2)$ for minimal prime ideals $\mathfrak{p}_1$ and $\mathfrak{p}_2$ of $A$. As $x\in Y_1,Y_2$, we would have $\mathfrak{p}_1, \mathfrak{p}_2 \subset \mathfrak{q}$.

Then $\mathfrak{p}_1$ and $\mathfrak{p}_2$ would correspond to distinct minimal prime ideals of $A_\mathfrak{q}=\mathcal{O}_{X,x}$, which is impossible, as $\mathcal{O}_{X,x}$ is by construction a domain, so that its only minimal prime ideal is $\{0\}$.