A very important special case are schemes that are of finite type over a field. Thus before we progress with the general abstract theory of schemes we focus in this and the next chapter on the case of schemes of finite type over a field (although some of the definitions and results are formulated and proved in greater generality). In fact this is also an important building block for the study of arbitrary morphism of schemes f : X → S because we have seen how we may attach to each s ∈ S its fiber Xs = f−1 (s) (4.8). Thus f yields a family of schemes over various fields and we may study f by first studying its fibers and then how these fibers vary.
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