5.5. Local Properties of Maps

Let A be a strict coherent analyitc geometires. 

Proposition 5.5.1. (a) Suppose X is a proper strong subobject. There is a prime V of X such that the localization XV is not contained in W
(b} Let L(X) be the coproducts of all localization lP: XP --> X and let e: L(X) --> X be the map induced by lP. Then e is epic. 

Proof. Since any proper strong subobject is contained in a proper regular subobject, and any proper regular subobject is an intersection of proper finitely cogenerated regular subobjects by (4.1.3), it suffices to prove the assertion for a proper finitely cogenerated regular subobject W. First we note that there is a prime V of X such that any analytic neighborhood U of V is not contained in W. Otherwise W contains an analytic cover {Ui} of X, which is not the case as the category is strict, therefore X is the colimit of {Ui  Uj}, and this only happens if X = W. Thus let V be such a prime. The localization XV of X at V is the cofiltered limit of all the analytic neighborhoods of V . Assume XV is contained in W. Suppose W is the equalizer of a pair (r1, r2): X --> T of maps with T finitely copresentable. There is an open neighborhood u: U --> X of V such that r1v = r2v (see the proof of (4.1.4)). Then U which contradicts to the choice of V. This shows that XV is not contained in W
(b) e does not factors through any proper strong subobject by (a), thus is epic. 

Proposition 5.5.2. An object is reduced iff each of its localization is reduced. 

Proof. Suppose X is reduced. By (3.1.3.e) analytic subobjects of X are reduced. Any localization of X is a cofiltered limit of analytic subobjects of X, thus is reduced by (5.1.5.b). 
Conversely, assume each localization is reduced. Consider a proper strong subobject W of X. By (5.5.1) there is a localization L of X which is not contained in W, i.e. W is a proper strong subobject of L. As L is reduced, W is not unipotent. This implies that W is not unipotent as the pullback of any unipotent map is unipotent. Thus X is reduced by (3.1.2.a). 

Proposition 5.5.3. A map f: Y --> X is epic iff its pullback along any localization of X is so. 

Proof. The pullback of any epi along a localization is epic as any localization is coflat. 
Conversely assume f is not epic. Then f factors through a proper regular subobject v: V --> X in a map g: Y --> V. According to (5.5.1.a) there is a localization l: L --> X which does not factor through V. Let (u: L  V --> L, w: V --> V) be the pullback of (v, l). Then u: V --> L is a proper strong subobject of L. Let m: M --> L  V be the pullback of g along w. Then um is the pullback of f = vg. Since um factors through the proper strong subobject V of L, it is not epi. 

Proposition 5.5.4. Suppose f: Y --> X is a mono. 
(a) If the residue maps of f are isomorphisms, then Spec(f) is injective. 
(b) If f is a local isomorphism then Spec(f) is injective. 

Proof. (a) Suppose Spec(Y) and Spec(Y) are two primes over the same Spec(X). Then their  residues k(P) and k(Q) are isomorphic to the residue k(O) of O by assumption. Thus there are isomorphisms u: k(P) --> k(O) and v: k(Q) --> k(O). Let s (resp. t) be the compositions of u-1: k(O) --> k(P) (resp. v-1: k(O) --> k(Q)) with the inclusions k(P) --> Y (resp. k(Q) --> Y ). Then fs = ft is the inclusion k(O) --> X. Since f is a mono, we have s = t. So P = Q. This shows that Spec(f) is injective. 
(b) follows from (a) as the residue maps of a local isomorphism are isomrophisms. 

Proposition 5.5.5. Suppose f: Y --> X is a  map. 
(a) If f is unipotent then Spec(f) is surjective. 
(b) If f is coflat and Spec(f) is surjective then it is unipotent. 
(c) Assume f is coflat and X is local. Then f is unipotent iff the simple prime of X is contained in the image of Spec(f). 

Proof. (a) If Spec(f) is not surjective we can find a residue P --> X which does not factors through f, then P --> X is disjoint with f, thus f is not unipotent. 
(b) We may assume that f is non-initial. Consider a non-initial map t: T --> X. Let p: P --> X be the generic residue of a prime in the image of Spec(t). Let u: F --> P be the pullback of f along p, and let v: G --> P be the pullback of t along p. Then F and G are not initial as P is in the image of Spec(f) and Spec(t). Since u as the pullback of a coflat map is coflat, and v is epic as a non-initial map to a simple object, the fibre product W of u and v is epic over F, thus is non-initial. The following commutative diagram implies that t and f are not disjoint, thus f is unipotent. 

(c) The condition is clearly necessary by (a). Conversely assume the simple prime of X is contained in the image of Spec(f). By (5.2.3.b) the image of Spec(f) is closed under generalization. Thus Spec(f) is surjective, which implies that f is unipotent by (b). 

Proposition 5.5.6. Any unipotent local isomorphic mono f: Y --> X is an isomorphism. 

Proof. First note that Spec(f) is bijective by (5.5.4) and (5.5.5). Let Z be the coproducts of all localization lP: YP --> Y and let z: Z --> Y be the map induced by lP, then z is an epi by (5.5.1.b). Since f is a local isomorphism and Spec(f) is bijective, Z is also naturally the coproducts of localizations of X with fz: Z --> X as the canonical map. Thus fz is epic by (5.5.1.b), which implies that f is epic. Since finitely copresentable objects form a strong generating set of A, to see that f is an isomorphism, it suffices to prove that any map t: Y --> C from Y to a finitely copresentable object C factors through f. Suppose p: YP --> Y is a localization of Y at a prime P. Since f is a local isomorphism, fp: YP --> X is the localization of X at f+1(P). Thus YP is the cofiltered limit of a collection of analytic neighborhoods Vi of f+1(P) with the maps (fp)s: YP --> Vi. Since C is finitely copresentable, there exists an analytic neighborhood Vs of f+1(P) and a map gs: Vs --> C such that ts ps = gs fsps, where ts: f-1(Vs) --> C, ps: YP --> f-1(Vs) and fs: f-1(Vs) --> Vs are the induced maps. Since C is finitely copresentable and YP is the filtered limits of open analytic neighborhood of P, and f-1(Vs) is one of such analytic neighborhood, there is a small analytic neighborhood UP contained in f-1(Vs) such that the restrictions of ts and gs fs on UP are the same, denoted this restriction by tP . We have proved that for any prime P we can find a small open neighborhood UP of P such that the restriction of t on UP can be factored through the restriction of f on UP. Since the analytic category is strict, these factorization can be glue together to obtain a global factorization for t by f

Proposition 5.5.7. A mono is a local isomorphism iff it is a fraction. 

Proof. Suppose u: U --> X is a local isomorphic mono. Any unipotent pullback of u is a unipotent local isomorphic, therefore is an isomorphism by (5.5.6). Thus u is normal. So we only need to prove that u is coflat. Since the class of local isomorphisms is closed under pullback, it suffices to prove that u is precoflat. Suppose t: T --> X is an epi and r: Z --> U, s: Z --> T is the pullback of f and t. For any localization g: G --> U the map r-1(U) --> U is epic because it is the pullback of the epic map t along the localization ug: G --> X. This shows that if r factors through a strong mono w: U, then W contains all the localization of U. This is only possible if W = U by (5.5.1.a). This shows that r is epic, which means that  u is precoflat as desired. The other direction has been proved in (5.3.3).