5.2. Localizations  An epi f: Y ® X is called essential provided that for any map g: Z ® Y, if f°g 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, h°q: Q ® P(X) is epic. Since p is epic, p°h°q = f°g°q is also epic. Since f is an essential epi, g° q: 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). n  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 Í f-1(V) implies that V = f+1(W) Í 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 p°s°q = u°r°q and the left side is epic, thus the right side u°r°q is epic. Let W = (r°q)+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). n  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). n  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) Ê f+1(M) Ê 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). n  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. n  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 Ç 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' Ç 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' Í 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) Í  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 Í S' Í 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 Ç V, thus XV Ç V is non-initial, which is a fraction of V as XV  is a fraction of X. It suffices to prove that XV Ç V is simple. Consider a proper strong subobject S of XV Ç V. Since XV Ç V is a coflat subobject of V, S is induced by a proper strong subobject S' of V, i.e. S = (XV Ç V) Ç 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 Ç V is a proper subobject of V, and Tc Ç V = (T Ç V)c. Since V is reduced and T Ç V is proper, (T Ç V)cis a non-initial analytic subobject of V. Since by definition XV Í Tc Ç V = (T Ç V)c, XV is disjoint with T Ç V. As S Í S' Í T Ç V, we see that XV is disjoint with S. Since XV is non-initial and by construction S Í XV, this means that S is initial. We have proved that any proper strong subobject S of XV Ç V is initial. Thus XV Ç V is simple. n  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). n  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 Y is a  small strong cogenerating set formed by finitely copresentable objects. If W is a Y-principal regular subobject of an object X, we say that Wc is a Y-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  Y-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 Ç 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. n  Proposition 5.2.11. Let f: Y ® X be a map. Suppose W Î 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' Î Spec(X) such that V Í V', and for any W Î Spec(Y) with f+1(W) = V, there exists W' Î Spec(Y) with f+1(W') = V' such that W Í 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' Î Spec(X) such that V Í V', and W Î Spec(B) with f+1(W) = V. Consider the localization lW: YW  ® Y, which is coflat. Then f°lW: YW ® X is coflat. By (5.2.3.b) V' is in the image of f°lW. Thus there is a prime Q of YW such that f°lW+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.). n        [Next Section][Content][References][Notations][Home] 