5.2. Localizations  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. (b) This follows from (a) as any coflat bimorphism is an essential epi. by (1.4.4.e).   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). (b) If V Í W are two primes of X and V is in the image of Spec(f), then by (a) f-1(V) is non-initial. This implies that f-1(W) is non-initial, so W is in the image of Spec(f) by (a).  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 XV 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) XV is the intersection of all the analytic subobjects of X containing P(V).  (b) XV is a local object whose simple prime contains P(V).  (c) XV is the intersection of all the fractions of X containing P(V).  (d) If V is a prime then P(V) = XV 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) XV contains P(V) by (a), thus is non-initial. We prove that any non-initial strong subobject S of XV contains P(V), which would implies that the intersection P'(V) of all the non-initial strong subobjects of XV is non-initial, and therefore XV 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 XV is a coflat subobject of X, S is induced by a strong subobject S' such that S = S' XV. 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) Tc, thus Tc is not disjoint with V, so XV as the intersection of such analytic subobjects is contained in Tc, which is absurd as S is in XV but not in Tc as S S' T. This shows that any non-initial strong subobject S of XV contains P(V), so XV  is local.  (c) As P(V) ® XV is local (thus quasi-local by (3.3.4.a) and XV --> X is a faction, this pair of maps is the quasi-local-fraction factorization of the inclusion P(V) --> X. By (4.2.9) XV is the intersection of fractions of X containing P(V).  (d) Clearly P(V) is contained in XV V, thus XV V is non-initial, which is a fraction of V as XV  is a fraction of X. It suffices to prove that XV V is simple. Consider a proper strong subobject S of XV V. Since XV V is a coflat subobject of V, S is induced by a proper strong subobject S' of V, i.e. S = (XV 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 Tc 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 XV Tc V = (T V)c, XV is disjoint with T V. As S S' T V, we see that XV is disjoint with S. Since XV is non-initial containing S, this means that S is initial. We have proved that any proper strong subobject S of XV V is non-initial. Thus XV V is simple.  Proposition 5.2.7. If P is a simple subobject of X then XP is a local object, and XP  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 lV:  XV --> 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 Wc 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 XV  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 --> XV is also local and the inclusion XV  --> 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 = XV 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 fW: YW --> XV of local objects such that the following diagram commutes: Proof. Since XV is the intersection of analytic subobjects of X which is not disjoint with V, f-1(XV) contains YW, which is the intersection of analytic subobjects of Y that is not disjoint with W. Thus the composition W --> X of the inclusion W --> Y and f factors through the inclusion XV --> X in a unique map fW: YW --> XV. Clearly f sends YW --> W = P(W) into XV --> V = P(V), where P(W) and P(V) are the generic residues of W and V respectively. Thus fW is local by (3.4.7.b).  With the notation of (5.2.11) we say that XV is the local image of the localization YW. 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 lW: YW  --> Y, which is coflat. Then flW: YW --> X is coflat. By (5.2.3.b) V' is in the image of flW. Thus there is a prime Q of YW such that flW+1(Q) = V. Let W' = lW+1(Q). Then W' contains W and f(W') = W as desired (alternatively the theorem follows from the fact that the induced local map YW --> XV is local, thus the mapping Spec(YW) --> Spec(XV) is surjective, and the fact that the mappings Spec(YW) --> Spec(Y) and Spec(XV) --> X are embeddings.).        [Next Section][Content][References][Notations][Home] 