Relative Affine Schemes
نویسنده
چکیده
1 Affine Morphisms Definition 1. Let f : X −→ Y be a morphism of schemes. Then we say f is an affine morphism or that X is affine over Y , if there is a nonempty open cover {Vα}α∈Λ of Y by open affine subsets Vα such that for every α, fVα is also affine. If X is empty (in particular if Y is empty) then f is affine. Any morphism of affine schemes is affine. Any isomorphism is affine, and the affine property is stable under composition with isomorphisms on either end. Example 1. Any closed immersion X −→ Y is an affine morphism by our solution to (Ex 4.3). Remark 1. A scheme X affine over S is not necessarily affine (for example X = S) and if an affine scheme X is an S-scheme, it is not necessarily affine over S. However, if S is separated then an S-scheme X which is affine is affine over S. Lemma 1. An affine morphism is quasi-compact and separated. Any finite morphism is affine. Proof. Let f : X −→ Y be affine. Then f is separated since any morphism of affine schemes is separated, and the separatedness condition is local. Since any affine scheme is quasi-compact it is clear that f is quasi-compact. It follows from (Ex. 3.4) that a finite morphism is an affine morphism of a very special type.
منابع مشابه
Math 632 Notes Algebraic Geometry
1 Affine schemes 4 1.1 Motivation and review of varieties . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 First attempt at defining an affine scheme . . . . . . . . . . . . . . . . . . . 4 1.3 Affine schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 The Zariski topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.5 The ringed space s...
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