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: 

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 D_f: X ® X ×_Y X satisfies (V₂). 

Proof. Suppose D_f: 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 D_f: X ® X ×_Y X, applying (V₂) we see that (j, j') can also be factored through D_f. 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 D_f: 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 D_f. Thus (V₂) holds for D_f.  

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:  

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 x^o is an isomorphism x^o ® V since V is a valuation object. The composition V ® x^o ® 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) D_f is a closed special morphism. 
(c) D_f is quasi-compact and universally stable. 
(d) D_f 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: 

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.