Why $(0)$ and $spec(k)$ is $R$-scheme?

Question

Let $R$ be a DVR, whose residue field is $k$ and $p$ is it's uniformizer. Then, generic fiber and special fiber of a given scheme is defined as fiber product of $X$ and $(0)$ and $Spec(k)$.But fiber product is defined between $R$-schemes. Could you give me an explanation why $(0)$ and $Spec(k)$ is $R$-scheme ?

Answer 1

Let $K=\operatorname{Frac}R$ and $k=R/p$. There are natural homomorphisms $R\to K$ and $R\to k$. These induce morphisms on spectra in the usual way. In other words, $\operatorname{Spec} K$ and $\operatorname{Spec} k$ are $R$-schemes.

Now, if $X$ is an $R$-scheme, the generic fiber is defined to be $$X\times_{\operatorname{Spec}R} \operatorname{Spec} K,$$ and the special fiber is $$X\times_{\operatorname{Spec}R} \operatorname{Spec} k.$$

These are homeomorphic to the inverse image (i.e. fiber) above $(0)$ and $(p)$ respectively, under the structural morphism of $X$.