4. SEPARATED AND PROPER MORPHISMS 31 (Week 9, two classes, next week is spring break.) (13) Example of fiber product: (a) Base change. 0 0 Let S be a scheme, X be an S-scheme and S an S-scheme. Then XS0 := X ×S S is a S0-scheme, this process is called base change by S0 → S. n n Let S be a scheme. Define := ×Spec S. PS PZ Z (b) Fiber over a point. Let f : X → Y be a morphism of schemes, y ∈ Y be a point with residue field k(y). Let Spec k(y) → Y be the natural morphism. Then we define the fiber of f over the point y to be the scheme Xy := X ×Y Spec k(y) Example: Consider Spec k[x, y, t]/(ty − x2). Explain the reduced structure at t = 0. 4. Separated and proper morphisms I will focus on applications instead, and will be rather sketchy on the proof. First we define separated morphism. Definition 4.1. Let f : X → Y be a morphism of schemes. (1) The diagonal morphism is the unique morphism ∆ : X → X ×Y X such p1 ◦ ∆ = p2 ◦ ∆ = idX . (2) f is separated if ∆ is a closed immersion. Example: The line over k with double origins is not separated over k. Since the diagonal is not closed: any function vanish on the diagonal must vanish on 4 origins. Glue two varieties along open subsets is not separated in general. (explain the geometry of valuative criterion). Example: A morphism of affine schemes is separated. Let X = Spec A and Y = Spec B, then X → X ×Y X corresponds to the ring morphism B ×A B → A, which is surjective, so X → X ×Y X is a closed immersion. Remark: Surjection of rings ⇒ Closed immersion of affine schemes. But injection of rings ; surjection of schemes. For example Z → Q. Proposition 4.2. A morphism f : X → Y is separated iff the set-theoretic image of the diagonal morphism ∆ is a closed subset of X × X. Proof. Obviously separatedness implies the ∆(X) is closed. So we need to prove that if ∆(X) is closed then (1) X → ∆(X) is a homeomorphism, (2) the induced morphism OX×Y X → ∆∗OX is surjective. For (1), notice that the composition X → ∆(X) → X is identity. 32 2. SCHEMES For (2), it is local in X ×Y X. So we can restrict to an open affine subset U = Spec A of X, with its image f(U) in an open affine subset V = Spec B. Then it boils down to the affine case which we have proved. Before we give the valuative criterion for separatedness and properness, let us review basic facts on valuation ring: Definition 4.3. Let K be a field and G a totally ordered abelian group. A valuation of K with values in G is a map v : K \{0} → G s.t. (1) v(xy) = v(x) + v(y), ∀x, y ∈ K, x, y 6= 0; (2) v(x + y) ≥ min(v(x), v(y)), ∀x, y ∈ K, x, y, x + y 6= 0. If v is a valuation ring, R = {x ∈ K|v(x) ≥ 0} ∪ {0} is called the valuation ring of v. If R is a valuation ring with values in Z, then R is called a discrete valuation ring. We need the following Fact: Theorem 4.4. [Atiyah,Macdonald, Thm 5.21, ex 27] A local ring R in a field K is a valuation ring if it is maximal of the set of local rings contained in K w.r.t. the domination relation, i.e. we say B dominates A if A ⊆ B and mB ∩ A = mA. Theorem 4.5 (Valuative criterion of separatedness, EGA II, 7.2.3). Let Y be a scheme (resp. locally noetherian scheme), f : X → Y be a morphism (resp. a morphism which is locally of finite type). Then f is separated iff the following holds: X → X ×Y X is quasi-compact, and for any valuation ring R (resp. discrete valuation ring R) with quotient field K, in the following commute diagram Spec K / X Spec R / Y there is at most one lifting Spec R → X making the whole diagram commute. Proof. One direction: suppose f is separated, i.e. ∆ : X → X ×Y X is a closed immersion. Let h1, h2 : Spec R → X be two lifts making the whole diagram commute. By definition of fiber product we have h = (h1, h2) : Spec R → X ×Y X. The restriction of h to Spec K is in the diagonal, hence the image of h(Spec R) is in the diagonal set-theoretically. But Spec R is reduced (reduced structure is minimal), so h factors through the diagonal X: Spec R → X → X ×Y X, which means that h1 = h2. The other direction: suppose the lifting condition is satisfied, it suffices to show that ∆(X) is a closed subset in X ×Y X.(note here that we don’t need to prove it is a subscheme, which save us lots of strength). For this, we use the following fact: ∆(X) is a closed subset in X ×Y X if it is stable under specialization, i.e. let z1 be a point in ∆(X), z0 be a point in z1, then z0 should be in ∆(X). The idea is to construct a valuation ring R and f : Spec R → X ×Y X s.t. the generic point sends to z1 and the closed point sends to z0. Then p1 ◦ f, p2 ◦ f : Spec R → X give two lifting, and 4. SEPARATED AND PROPER MORPHISMS 33 by the lifting condition we have p1 = p2. Hence f factors through ∆ as ∆ f : Spec R → X → X ×Y X then z0 is in the diagonal ∆(X). So to finish the proof, we need to find a valuation ring that works. Let K = k(z1) the residue field of z1. Let O be the local ring of z0 on z1 and let R be a valuation ring of K dominating O. The existence of such an R is guaranteed by Theorem 4.4. Proposition 4.6. Assuming all schemes below are noetherian. (1) Open and closed immersions are separated. (2) A composition of two separated morphism is separated. (3) Separatedness is stable under base change. (4) If f : X → Y , g : X0 → Y 0 are separated morphisms over a base scheme S, then f × f 0 : 0 0 X ×S X → Y ×S Y are separated. (5) If f : X → Y and g : Y → Z are morphisms, if g ◦ f is separated then f is separated. −1 (6) f : X → Y is separated iff Y has a covering {Vi} such that f (Vi) → Vi are separated. Proof. Show (5) in class using valuative criterion. Similar to the separatedness, the properness for a morphism of schemes cannot be defined by requiring the inverse image of a quasi-compact set is quasi-compact, since there are too many quasi-compact sets. (recall that all affine schemes are quasi-compact). Amazingly, we can use closed morphism to define proper morphism. Definition 4.7. (1) A morphism is closed if the image of any closed subset is closed. A morphism is universally closed if the morphism is closed for every base change. (2) A morphism is proper if it is separated, or finite type, and universally closed. 1 Example: A → pt. Although it is closed since a point is a closed subset of itself, but it is not ’closed in heart’. And the way to find it out is by base-change. Theorem 4.8 (valuative criterion for properness, EGA II, 7.3.8). Let Y be a scheme (resp. locally noetherian scheme), f : X → Y be quasi-compact separate (resp. of finite type). Then f is universally closed (resp. proper) iff for any valuation ring R (resp. discrete valuation ring R) with quotient field K, in the following commute diagram Spec K / X Spec R / Y there is at least (resp. exactly) one lifting Spec R → X making the whole diagram commute. The proof is skipped. Again, we have the following properties: 34 2. SCHEMES Corollary 4.9. Assuming all schemes below are noetherian. (1) A closed immersion is proper. (2) Composition of proper morphisms is proper. (3) Proper morphisms are stable under base change. (4) Products of proper morphisms is proper. (5) Given f : X → Y , g : Y → Z, if g ◦ f is proper and g is separated, then f is proper. (6) Properness is local on the base. Why do we care about properness? Here are some good properties. (a) We use it to define a complete k-variety if it is proper over k. q (b) For a proper f : X → Y and a coherent sheaf F on X, R f∗(F )) is also coherent. In particular, if X is a complete variety over k, then Hq(X, F ) is finite dimensional. Very often, we require a morphism f : X → S to be proper and flat to form a good family. So, what are the examples of proper morphisms? We will give a large class of proper morphisms: projective morphisms. You may think of them as families of projective schemes. Here is the n n definition. Recall that the projective space is defined by ×Spec Y . PY PZ Z Definition 4.10. (1) A morphism f : X → Y is projective if it factors into a closed immersion n n i : X → PY for some n, followed by the projection PY → Y . (2) A morphism is quasi-projective if it is a composition of an open immersion followed by a projective morphism.