I have a few basic questions about Liu's [1, Section 10.1.3] description of the reduction map from the closed points of the generic fibre of a proper scheme over a complete DVR to its special fibre. Here's (my slight paraphrasing of) the section that's bothering me:
Let $R$ be a complete DVR and let $S = \operatorname{Spec}R$. Let $s \in S$ be the closed point. Let $\mathcal{X}\rightarrow S$ be a proper scheme over $S$ with generic fibre $X$ and special fibre $\mathcal{X}_s$. For any closed point $x \in X$, the Zariski closure $\overline{\{x\}}$ is an irreducible finite scheme over $S$ and is therefore a local scheme with closed point $\overline{\{x\}} \cap \mathcal{X}_s$.
One then defines the reduction map $r\colon X^0 \rightarrow \mathcal{X}_s$, where $X^0$ is the set of closed points of $X$, by $r(x) = \overline{\{x\}} \cap \mathcal{X}_s$. Here are my questions:
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What is the point of considering the Zariski closure of a closed point? By definition $x$ is closed precisely when $\{x\} = \overline{\{x\}}$, so this part of the definition seems redundant. Am I missing something obvious?
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More importantly, I don't understand how to interpret the intersection $\overline{\{x\}} \cap \mathcal{X}_s$ since the two sets in question don't share a common "parent" space. What is the space in which the intersection is taking place?
Thanks in advance.
[1] Liu, Q., Algebraic Geometry and Arithmetic Curves, Oxford Graduate Texts in Mathematics, 6. Oxford University Press, 2002
Best Answer
The point $x$ is closed as a subset of the generic fiber $X$, but not as a subspace of the big scheme $\mathcal{X}$ over $S$ ; so it makes sense to take the closure of $x$ in the big scheme $\mathcal{X}$.
Then you can take the intersection of this closure with the special fibre $\mathcal X_s$ : this intersection takes place in the big scheme $\mathcal{X}$, which is your "parent" scheme.
Maybe the source of the confusion is the simultaneous use of $X$ and $\mathcal{X}$ , which are typographically not sufficiently distinguishable?