Let A be a coherent analytic geometry. Recall that a mono (or subobject) is called a fraction if it is coflat and normal. A fraction with a local domain is called a localization. Suppose V is a prime of an object X with the generic residue P(V). Denote by X_{V} the intersection of all the analytic subobjects of X containing P(V). Then X_{V} is a local object with P(V) as the simple prime, which is the localization of X at V. X_{V} is also the intersection of all the analytic subobjects of X which is not disjoint with V. Definition 5.3.1. (a) A map f:
Y > X is called a local
isomorphism if for any localization v: V > Y,
the composition fv: V > X is a localization.
Proposition 5.3.2. (a) The class of
local isomorphisms is closed under composition.
Proof. (a)  (c) follows directly from the fact that any composite
of fractions is a fraction..
Proposition 5.3.3. (a) Any fraction
is semisingular.
Proof. (a) Suppose u: U > X is fraction. Since
u is normal, u
is generated by u. But u
is generated by the set S of strong monos to X which is disjoint
with U by (1.5.2) as u is
coflat. This shows that u is the complement of S, thus is
semisingular.
Denote by Spec_{r}(X) the set of residues of an object X. Denote by Spec_{l}(X) the set of localizations of X. We have a bijection Spec_{r}(X) > Spec(X) sending each residue to its strong image, and a bijection Spec_{r}(X) > Spec_{l}(X) sending each residue to the localization at it. Thus Spec_{r}(X) and Spec_{l}(X) are natural topological spaces. We obtain another two functors to the category of topological spaces, which are equivalent to the analytic topology on A. Proposition 5.3.4. (a) Any complementary
mono is finitely copresentable.
Proof. Let u: U > X be the complement of a mono
v: V > X. Since A/X
is locally finitely copresentable, u is an inverse limit of a systems
of finitely copresentable objects {t_{i}: U_{i}
> X  iI} in A/X
with the maps u_{i}: U > U_{i}, where I
is a cofiltered category. Let W_{i} be the pullback of V
along t_{i}. The pullback of V along u is
0, which is the cofiltered limit of W_{i}. Since
0 is finitely copresentable, there is some i
I such that W_{i} = 0. Thus V is disjoint
with U_{i}, so t_{i} factors through u
in a map g: U_{i} > U. The relation ugu_{i}
= t_{i}u_{i} = u implies that gu_{i}
is the identity of U. Thus the object U in A/X
is a split subobject of the object t_{i}. Since the subcategory
of finitely copresentable objects in A/X
is closed under colimits, this implies that u: U > X is
a finitely copresentable object in A/X.
Recall that a map f: Y > X is a finitely copresentable if it is a finitely copresentable object in the category A/X. Proposition 5.3.5. If f: Y > X is a finitely copresentable local isomorphism, then Spec(f): Spec(Y) > Spec(X) is an open map. Proof. (a) We first prove that the image of Spec(f) of a finitely copresentable local isomorphism is an open subset of Spec(X). Suppose V > Y is a prime and let l_{V}: Y_{V} > Y be the localization of Y at V. Since f is a local isomorphism, fl_{V}: Y_{V} > X is a localization, thus the local object Y_{V} is the intersection of a collection {u_{i}: U_{i} > X  iI} of analytic subobjects of X, and we may assume that I is is cofiltered. Thus Y_{V} > X is the inverse limit of {u_{i}: U_{i} > X  iI}_{ }in A/X. Since f is is a finitely copresentable object in A/X, there exists some t I and a map h: U_{t} > Y in A/X (i.e. fh = u_{t}) such that l_{V }= hv_{t}, where v_{t}: Y_{V} > U_{t} is the inclusion. Then the open subset Spec(u_{t})(Spec(U_{t}) = Spec(f)Spec(h)(Spec(U_{t})) is in the image of Spec(f) which contains f^{+1}(V). This shows that Spec(f)(Spec(Y)) is an open subset of Spec(X). Proposition 5.3.6. A fraction u: U > X is analytic iff the image of Spec(u) is an open subset. Proof. If u is analytic then Spec(u) is an open embedding by (3.6.9.b), so the condition is necessary. Conversely, assume the image Spec(u)(Spec(U) is an open subset of Spec(X), which is the complement of the closed subset defined by a strong subobject V of X. Clearly U is disjoint with V as A is spatial. We prove that the fraction U is a complement of V. Since U is normal and the class of simple objects is unidense, it suffices to prove that any map p: P > X from a simple object to X, which is disjoint with V, factors through U. Let W = p^{+1}(P) and Consider the epi P > p^{+1}(P) = W. Since p is disjoint with V, the prime W is not contained in V, i.e. W is not in the closed subset determined by V, thus W is contained in its complement Spec(u)(Spec(U)). So W U is not initial. Since the inclusion W U > W is coflat and p: P > W is epic, p is not disjoint with W U, thus p is not disjoint with U, thus p factors through U as P is simple and U is a fraction. Proposition 5.3.7. A mono is analytic iff it is a finitely copresentable fraction. Proof. Suppose u: U > X is a finitely copresentable
fraction. By (5.3.2.c) u is a local isomorphism,
so by (5.3.5) the image Spec(u)(Spec(U)
is an open subset of Spec(X), so u is analytic by
(5.3.6). Conversely, if u is analytic then
it is a fraction, and also finitely copresentable by (5.3.4).
