5.5. Local Properties of Maps Let A be a strict coherent analyitc geometires.  Proposition 5.5.1. (a) Suppose W 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 V 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. L W is a proper strong subobject of L. As L is reduced, L 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: L V --> V) be the pullback of (v, l). Then u: L 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 L 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 P Spec(Y) and Q Spec(Y) are two primes over the same O 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: 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).     [Content][References][Notations][Home] 