An epi f: Y --> X is called essential provided that for any map g: Z --> Y, if fg is epic then g is epic. 

Proposition 5.2.1. Suppose f: Y --> X is a coflat map and X is integral. 
(a} If f is an essential epi then Y is integral. 
(b} If f is a bimorphism then Y is integral. 

Proof. Let p: P(X) -->  X be the generic residue of the integral object X, which is an epi coflat fraction by (5.1.2.c). Let (g: Z --> Y, h: Z --> P(X)) be the pullback of (f: Y --> X, p: P(X) --> X). 
(a) Assume f is an essential epi. Since f and p are coflat and epic, g and h are epic. Thus Z is non-initial, which has a simple strong subobject q: Q --> Z (4.2.2.d). Since P is simple, hq: Q --> P(X) is epic. Since p is epic, phq = fgq is also epic. Since f is an essential epi, gq: Q --> Y is epic. Then Y, as a quotient of a simple object, is integral. 

Proposition 5.2.2. Suppose f: Y --> X is any map. The image of Spec(f) is {V Î Spec(X): f⁺¹(f^-1(V)) = V}. 

Proof. Suppose V = Spec(f)(W) = f⁺¹(W) for a prime W of Y. Then f⁺¹f^-1(V) is a subobject of X contained in V. But W  f^-1(V) implies that V = f⁺¹(W)  f⁺¹f^-1(V). Thus V = f⁺¹f^-1(V). 
Conversely, assume V = f⁺¹f^-1(V) holds. Consider the pullback (r: Z --> f^-1(V), s: Z --> P(V) of the maps (u: f^-1(V) --> V, p: P(V) --> V). Since u is epic and p is coflat epic, s: Z  --> p(V) is epic, thus Z is non-initial, which has a simple subobject q: Q --> Z. But psq = urq and the left side is epic, thus the right side urq is epic. Let W = (rq)⁺¹(Q). Then u⁺¹(W) = V. But u is the restriction of f on f^-1(V). This shows that f⁺¹(W) = V, i.e. V is lying in the image of Spec(f). 

 

Proposition 5.2.3. Suppose f: Y --> X is a coflat map. 
(a} The image of f is {V Î Spec(X): f^-1(V) is non-initial}. 
(b} If V is in the image of Spec(f) and W is a integral strong subobject containing V, then W is in the image (i.e. the image of Spec(f) is closed under generalizations). 

Proof. (a) If V is in the image then by (5.2.2) f⁺¹f^-1(V) = V, so f^-1(V) is non-initial. 
Conversely, assume f^-1(V) is non-initial. The induced map t: f^-1(V) --> V is the pullback of f along V --> X, thus is also coflat. Since V is integral and f^-1(V) is non-initial, t is epic by (3.2.6.a). Thus f⁺¹f^-1(V) = V, which implies that V is in the image of Spec(f) by (5.2.2). 

Proposition 5.2.4. Suppose Y is a local object with the simple prime M and f: Y --> X is a coflat mono. Then Spec(f) induces a homeomorphism between Spec(Y) and the subspaces of Spec(X) consisting of primes which contains M. 

Proof. Since f is a coflat mono, Spec(f) is a topological embedding by (3.6.9.a). Since any prime V of Y contains M, we have f⁺¹(V)  f⁺¹(M)  M. Conversely, if W is a prime of X which contains M, then it contains f⁺¹(M). Thus by (5.2.3.b) it is in the image of Spec(f). 

Proposition 5.2.5. Suppose f: Y --> X is a coflat local map of local objects. Then Spec(f): Spec(Y) --> Spec(X) is surjective. 

Proof. By (3.4.7) the simple prime P of X is in the image of f. Since any prime of X contains P, by (5.2.3.b) it is in the image of f.  

Suppose V is an integral subobject of an object X. Denote by X_V the intersection of all the analytic subobjects of X which is not disjoint with V.  

Proposition 5.2.6. Assume V is an integral subobject of X with the generic residue P(V). 
(a) X_V is the intersection of all the analytic subobjects of X containing P(V). 
(b) X_V is a local object whose simple prime contains P(V). 
(c) X_V is the intersection of all the fractions of X containing P(V). 
(d) If V is a prime then P(V) = X_V  V. 

Proof. (a) Since V is integral and the simple object  P(V) is the intersection of non-initial fractions of V, an analytic subobject of X is not disjoint with V iff it contains P(V). 
(b) X_V contains P(V) by (a), thus is non-initial. We prove that any non-initial strong subobject S of X_V contains P(V), which would implies that the intersection P'(V) of all the non-initial strong subobjects of X_V is non-initial, and therefore X_V is local whose simple prime P'(V) contains P(V). If this is not the case, then the simple object P(V) is disjoint with S. Since X_V is a coflat subobject of X, S is induced by a strong subobject S' such that S = S'  X_V. Then P(V) is disjoint with S'. Since A is locally disjunctable, we can find a disjunctable strong subobject T of X such that S'  T and P(V) is not contained in T. Then the simple object P(V) is disjoint with the strong subobject T, so P(V)   T^c, thus T^c is not disjoint with V, so X_V as the intersection of such analytic subobjects is contained in T^c, which is absurd as S is in X_V but not in T^c as S  S'  T. This shows that any non-initial strong subobject S of X_V contains P(V), so X_V  is local. 
(c) As P(V) ® X_V is local (thus quasi-local by (3.3.4.a) and X_V --> X is a faction, this pair of maps is the quasi-local-fraction factorization of the inclusion P(V) --> X. By (4.2.9) X_V is the intersection of fractions of X containing P(V). 
(d) Clearly P(V) is contained in X_V  V, thus X_V  V is non-initial, which is a fraction of V as X_V  is a fraction of X. It suffices to prove that X_V  V is simple. Consider a proper strong subobject S of X_V  V. Since X_V  V is a coflat subobject of V, S is induced by a proper strong subobject S' of V, i.e. S = (X_V  V)  S'. Since V is a proper strong subobject of X, S' is also a proper strong subobject of X. Since A is locally disjunctable, we can find a disjunctable strong subobject T of X containing S' but not V. Then T  V is a proper subobject of V, and T^c  V = (T  V)^c. Since V is reduced and T  V is proper, (T  V)^cis a non-initial analytic subobject of V. Since by definition X_V  T^c  V = (T  V)^c, X_V is disjoint with T  V. As S  S'  T  V, we see that X_V is disjoint with S. Since X_V is non-initial containing S, this means that S is initial. We have proved that any proper strong subobject S of X_V  V is non-initial. Thus X_V  V is simple. 

Proposition 5.2.7. If P is a simple subobject of X then X_P is a local object, and X_P  is the intersection of all the analytic subobjects (or fractions) of X which contains P. 

Proof. The assertion follows from (5.2.6). 

Definition 5.2.8. (a) A fraction Y --> X with a local domain Y is called a localization. 
(b) Suppose V is an integral subobject of an object X. With the notations of (5.2.6), the localization l_V:  X_V --> X is called the localization of X at V. 

Remark 5.2.9. Suppose  is a  small strong cogenerating set formed by finitely copresentable objects. If W is a -principal regular subobject of an object X, we say that W^c is a -principal analytic subobject of X. Similar to (5.2.6) and (5.2.7) one can show that for any integral subobject V of X, the localization X_V  of  X at V is the intersection of -principal analytic subobjects of X which is not disjoint with V (the key step is (5.2.6.c) to prove that the intersection is a local object).  

Proposition 5.2.10. Suppose f: Y --> X is a localization and P is the simple prime of the local object Y and V = f⁺¹(P). Then Y is the localization of X at the prime V. 

Proof. Since f is coflat, P = f^-1f⁺¹(P) = Y  f⁺¹(P) implies that P is a epic simple fraction, thus P is the generic residue of the prime V = f⁺¹(P). Since the inclusion P --> Y is local and f: Y --> X is a fraction, this pair of maps  is the quasi-local-fraction factorization of P --> X. But the inclusion P --> X_V is also local and the inclusion X_V  --> X is a fraction, we obtain another pair of maps which is also a quasi-local-fraction factorization of the inclusion P --> X. By the uniqueness of such a factorization we see that Y = X_V as subobjects of X. 

Proposition 5.2.11. Let f: Y --> X be a map. Suppose W  Spec(Y) and V = f⁺¹(W). Then there is a unique local map f_W: Y_W --> X_V of local objects such that the following diagram commutes: 

With the notation of (5.2.11) we say that X_V is the local image of the localization Y_W. of Y (under f). 

Definition 5.2.12. Let f: Y --> X be a map. We say that the going-up theorem holds for f  if the following condition is satisfied: 
For any V, V'  Spec(X) such that V  V', and for any W  Spec(Y) with f⁺¹(W) = V, there exists W'  Spec(Y) with f⁺¹(W') = V' such that W  W'. 

Now we can prove a generalization of the dual version of the going-down theorem for a flat homomorphism of commutative rings (cf. [Matsumura 1980, p.33, Theorem 4]): 

Proposition 5.2.13. (Going Up Theorem) The going-up theorem holds for any coflat map f: Y --> X. 

Proof. Suppose V, V'  Spec(X) such that V  V', and W  Spec(B) with f⁺¹(W) = V. Consider the localization l_W: Y_W  --> Y, which is coflat. Then fl_W: Y_W --> X is coflat. By (5.2.3.b) V' is in the image of fl_W. Thus there is a prime Q of Y_W such that fl_W⁺¹(Q) = V. Let W' = l_W⁺¹(Q). Then W' contains W and f(W') = W as desired (alternatively the theorem follows from the fact that the induced local map Y_W --> X_V is local, thus the mapping Spec(Y_W) --> Spec(X_V) is surjective, and the fact that the mappings Spec(Y_W) --> Spec(Y) and Spec(X_V) --> X are embeddings.).