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Home , Diagonal morphism, Morphism of schemes, Proper morphism

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 , 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 , 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 .

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 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 of the diagonal morphism ∆ is a closed 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 : 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 .

We need the following Fact: Theorem 4.4. [Atiyah,Macdonald, Thm 5.21, ex 27] A 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 , 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 ), 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 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 . 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 F on X, R f∗(F )) is also coherent. In particular, if X is a 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 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 PY → Y . (2) A morphism is quasi-projective if it is a composition of an open immersion followed by a projective morphism.

Note: in EGA II 5.5 the projective morphism is defined more general as the composition of a closed immersion followed by the projection P(E) → Y where E is a quasi-coherent OY -sheaf of finite type. As pointed out by Hartshorne, two definition coincide when Y is quasi-projective over an affine scheme. As the first application of valuative criterion, we show the following Theorem 4.11. A projective morphism of (noetherian) scheme is proper.

Proof. We need noetherian condition to keep the proof simple, but the theorem is true in general, cf. EGA II, 5.5.3. n Let f : X → Y be projective. First, since X → PY is proper and the composition of proper n morphisms is proper, so suffices to prove PY → Y is proper. Since properness is stable under base change, it suffices to prove n → Spec is proper. PZ Z Given Spec K / n PZ

  Spec R / Spec Z Let z1 be the point in Spec K, ξ1 the image of z1 in X. 4. SEPARATED AND PROPER MORPHISMS 35

(1) Existence of the lift. Recall that PZ is covered by affine charts Vi := Spec Z[x0/xi, . . . , xn/xi]. 2 (Draw the 3 lines xyz = 0 in P on blackboard and draw the ‘cloud’ ξ1). Assume that ξ is at a

‘good’ position, in the sense that all the functions xi/xj are invertible in Oξ1 . (if not, then ξ1 lies n in a coordinate hyperplane of and we can use induction on n.) Pull back xi/xj to Spec K, PZ we get functions fij ∈ K satisfying fijfjk = fik. Find fij with largest valuation, say f10. (In picture, x0 = 0 is the ‘farthest’ hyperplane to ξ1). Then all fi0 ∈ R, because if v(fi0) < 0, then v(f1i) = v(f10) − v(fi0) > v(f10) contradicts with the choice of f10. Thus we have a morphism

Z[x1/x0, . . . , xn/x0] → R which induces Spec R → X compatible with Spec K → X.

(2) For the uniqueness of the lift, note that in the above argument, Spec K → Vj extends to Spec R → Vj iff fij ∈ R for all i, and such an extension is unique. If there are a lifting Spec R → Vj and a lifting Spec R → Vj0 , then by restricting to Spec R → Vj ∩ Vj0 we can show that they will give the same map Spec R → X.