# Reduction map from the generic to the special fibre

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:

1. 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?

2. 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?

 Liu, Q., Algebraic Geometry and Arithmetic Curves, Oxford Graduate Texts in Mathematics, 6. Oxford University Press, 2002

1. 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 $\mathcal{X}$.
2. 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?
• The typography is definitely not the problem. I think the problem is that I'm having trouble seeing how to consider $X$ and $\mathcal{X}_s$ as being "in" $\mathcal{X}$. They are both $\mathcal{X}$-schemes having been formed by fibre products, so should the expression $\overline{\{x\}} \cap \mathcal{X}_s$ be interpreted as $\pi_X(\overline{\{x\}}) \cap \pi_{\mathcal{X}_s}(\mathcal{X}_s)$ where $\pi_X\colon X \rightarrow \mathcal{X}$ and $\pi_{\mathcal{X}_s}\colon \mathcal{X}_s \rightarrow \mathcal{X}$ are the canonical projections? – Hamish Apr 23 '11 at 14:46
• Call $f:\mathcal X \to S$ your morphism of schemes.Then topologically $X$ is just the inverse image $X=f^{-1}(\eta )$ of the generic point $\eta$ of $S$ and $\mathcal X_s=f^{-1}(s)$ the inverse image of the closed point $s$ of $S$. So $\mathcal X$ is just the disjoint union of its open subset $X$ and of its closed subset $\mathcal X_s$. This business of fibre products (=base change) is useful for giving a scheme structure to fibres of morphisms but obscures a little the fact that you just deal with inverse images. – Georges Elencwajg Apr 23 '11 at 17:56