6.4 Valuation Objects  In this section we assume C is a simple spectral site.  A valuation object V of C is an irreducible locality such that any birational morphism of irreducible localities f: X ® V is an isomorphism.  We shall denote by T and K the closed point and the generic point of a valuation object V respectively. Let i: K ® V be the effective inclusion.  Definition 6.4.1. We say C has enough valuations if for any morphism f: X ® Y, a point x Î |X|, and a specialization y of f(x), there exist a valuation object V and morphisms g: V ® Y, h: K ® X with g(T) = y, h(K) = x, such that the following diagram commutes:  and one can choose (V, g, h) such that h induces an isomorphism from K to the spot x.  In the following we assume C has enough valuation objects.  Consider the following conditions on a morphism f: X ® Y:  (V1) For any valuation object V and g, h as above (6.4.1), there exists at most one morphism from V to X making the diagram commutative.   (V2) For any valuation object V and g, h as above (6.4.1), there exists a morphism V ® X making the diagram commutative.  Remark 6.4.2. (a) The composition of two morphisms satisfies (V1) or (V2) if each of them satisfies (V1) or (V2).  (b) Any monomorphism f satisfies (V1).  (c) Any closed effective morphism satisfies (V2).  (d) (V1) and (V2) are stable under base extensions.  Lemma 6.4.3. A morphism f: X ® Y satisfies (V1) if and only if the diagonal morphism Df: X ® X ×Y X satisfies (V2).  Proof. Suppose Df: X ® X ×Y X satisfies (V2). Let g, h be as in diagram (6.4.1). Suppose j, j': V ® X are two morphisms making the diagram commutes. Then the restrictions j, j' on the generic point K of V are equal. Consider the canonical morphism (j, j'): V ® X ×Y X. Since the composition K ® V ® X ×Y X can be factored through Df: X ® X ×Y X, applying (V2) we see that (j, j') can also be factored through Df. This implies that j = j'.  Conversely we assume (V1) holds for f. Suppose g: V ® X ×Y X is a morphism such that K ® V ® X ×Y X factors through Df: X ® X ×Y X. Composing with the projections p, q we obtain two morphisms V ® X making the diagram commute. By (V1) These two morphisms must be equal. This shows that g factors through Df. Thus (V2) holds for Df.   Lemma 6.4.4. Suppose f: X ® V is a stable morphism and V is a valuation object. Suppose h: K ® X is a V-morphism forming a commutative diagram:   Then there exists a section V ® X of f making the whole diagram commutative.  Proof. We may assume |X| is the closure of h(K). Since K ® h(K) ® K is an isomorphism, K ® h(K) is a section, hence an isomorphism because C is simple. Since f is stable, we have |f(X)| = V as any point of V is a specialization of K. Let x Î |X| be a point above T. Then the restriction of f on the locality xo is an isomorphism xo ® V since V is a valuation object. The composition V ® xo ® X is a section of f making the diagram commutative.   Theorem 6.4.5. A morphism f: X ® Y is universally stable if and only if the condition (V2) holds for f.  Proof. (a) First we assume the condition (V2) holds for f. To prove that f is universally stable it suffices to prove that f is stable since the property that a morphism satisfies (V2) is universal. We verify the condition (6.2.4.b) for f, which will then imply that f is stable. Let x Î |X| and y = f(x). Let y' be a specialization of y. By (6.4.1) there is a valuation object V, a morphism g: V ® Y and a morphism h: K ® Y such that g(T) = y', h(K) = x, and h: K ® x is an isomorphism of spots such that the diagram of (6.4.1) commutes. Applying (V2) we find a morphism j: V ® X making that diagram commute. Let x' = j(T) be the image of T Î |V| in |X|. Then f(x') = y', and x' is a specialization of x.   (b) Now suppose f is universally stable. Consider the diagram of (6.4.1). Let X' = V ×Y X. Then the projection X' ® V is a stable morphism. We can apply (6.4.4) to obtain a section V ® X' extending the morphism (i, h)Y: K ® X'. Composing with the projection X' ® X gives the desired morphism from V to X. Thus (V2) holds for f.  Theorem 6.4.6. (Valuation Criterion for Separated Morphisms). Suppose f: X ® Y is a morphism. The following conditions are equivalent:  (a) f is separated.  (b) Df is a closed special morphism.  (c) Df is quasi-compact and universally stable.  (d) Df is quasi-compact and satisfies (V2).   (e) f is quasi-separated and satisfies (V1).  Proof. (a) and (b) are equivalent by (6.3.3b) and (5.3.3).  (b) and (c) are equivalent by (5.4.7) and (6.2.8);   (c) and (d) are equivalent by (6.4.5);  (d) and (e) are equivalent by (5.4.4), (6.1.14.a) and (6.4.3).  Theorem 6.4.7, (Valuation Criterion for Universally Closed Morphisms). Suppose f: X ® Y is a quasi-compact morphism. Then the following conditions are equivalent:  (a) f is universally closed.  (b) f is universally stable.  (c) Condition (V2) holds for f.  Proof. (a) and (b) are equivalent by (6.1.14.a) and (6.2.8);  (b) and (c) are equivalent by (6.4.5).   Proposition 6.4.8. Suppose C is a spectral site with a simple model C' which is a spectral site having enough valuation objects. Then   (a) For any morphism f: X ® Y in C, a point x Î |X|, and a specialization y of f(x), there is a valuation object V e C' and morphisms f': V ® Y, h: K ® X such that g(T) = y, h(K) = x, and the following diagram commutes:  where i is the canonical effective morphism.  (b) Consider the following conditions on a morphism f: X ® Y in C:  (U1) For any valuation object V and g, h as above, there exists at most one morphism from V to X making the diagram commutative.   (U2) For any valuation object V and g, h as above, there exists a morphism V ® X making the diagram commutative.  Then f is separated if and only if f is quasi-separated and satisfies (U1). If f: X ® Y is a quasi-compact morphism, then f is universally closed if and only if (U2) holds for f.  Proof. Denote by J: C ® C' the isometric coreflector. Applying (6.4.1) to J(f): J(X) ® J(Y) we see that (a) holds. (b) follows easily from (6.4.6) and (6.4.7).   Example 6.4.9. The site GSch of reduced schemes is a simple spectral site with enough valuation objects: the spectrum Spec(A) of any valuation ring A is a valuation object of GSch. Thus (6.4.6) and (6.4.7) hold for GSch.  Example 6.4.10. The site LNGSch of locally noetherian reduced schemes with morphisms of finite type is a simple spectral site having enough valuation objects: the spectrum Spec(A) of any discrete valuation ring A is a valuation object of LNGSch. Thus (6.4.6) and (6.4.7) hold for LNGSch.  Example 6.4.11. The site Sch of schemes is a spectral site with GSch as a simple model. The site LNSch of locally noetherian schemes with morphisms of finite type is a spectral site with LNGSch as a simple model. Hence we can apply (6.4.8) to these two sites.      [Content][References][Notations][Home]