5.1. Coherent Analytic Geometries   Any coherent analytic category is a spatial (4.2.2) and perfect (4.1.2.b). Thus a locally disjunctable coherent analytic category is a spatial analytic geometry, which is simply called a coherent analytic geometry.  In the following we assume A is a coherent analytic geometry. Recall that (3.6.9) if X is an object. an open subset U of Spec(X) is affine open if it is determined by an analytic subobject of X. The set of affine open subsets is  a base for the topology on Spec(X).  Proposition 5.1.1. (a) Spec(X) is a coherent space for any object X.  (b} For any map f: Y ® X the continuous mapping Spec(f): Spec(Y) ® Spec(X) is a mapping of coherent spaces.  Proof. (a) Suppose {Vi} is a set of disjunctable strong subobjects such that {(Vi)c}iÎI  is an analytic cover for X. Let V be the intersection of these {Vi}. Then by (3.1.10) we have X = ØØ{(Vi)c} = ØV, so V = 0. Since 0 is finitely copresentable by (4.3.2.c), there is a finite subset J Í I such that {Vi}iÎJ has intersection 0. Applying (3.1.10) again we see that {(Vi)c} is a finite analytic cover for X. This shows that Spec(X) is quasi-compact. Thus any affine open subset is also quasi-compact. Since affine open subsets form a base for Spec(X), any finite intersection of quasi-compact open subsets is quasi-compact. This means that Spec(X) is coherent.  (b) The pullback of analytic subobject of X along f is an analytic subobject of Y. Since affine open subsets form a base for Spec(X), this implies that inverse image of quasi-compact open subset is quasi-compact open. Thus Spec(f) is a mapping of coherent spaces. n  Proposition 5.1.2. (a) If X is quasi-primary then the fraction hull P(X) of X is the intersection of all non-initial analytic monos.  (b) An object is primary iff its rational hull R(X) is quasi-simple; if X is primary then R(X) = Q(X) = P(X);  (c) An object is integral iff its rational hull is simple; any integral object X has a generic residue P(X) ® X which is an epic coflat simple fraction.  Proof. (a) Consider the intersection P of non-initial analytic subobjects of X, which is a non-initial fraction of X containing P(X). We have to prove that P is quasi-simple, or equivalently, that P is pseudo-simple (3.3.10). By (3.3.8.a) it suffices to prove that any non-initial strong subobject of P is unipotent. Since P ® X is coflat, any strong subobject of P is induced from X by (1.5.4). Since any unipotent strong subobject of X induces a unipotent strong subobject of P, we only need to prove that P is disjoint with any non-unipotent strong subobject V of X. Since A is locally disjunctable, V is an intersection of proper disjunctable strong subobjects {Vi}. Since V is non-unipotent, at least one complement (Vi)c is non-initial, and P Í (Vi)c. Thus P Ç V = 0. This shows that P is pseudo-simple, thus P = P(X).  (b) If X is primary then any non-initial analytic mono is epic. Thus R(X) = P(X) by (a), and P(X) is primary by (3.3.8). Any object X is a quotient of its rational hull R(X). If R(X) is quasi-simple, it is primary by (3.3.8), thus X as a quotient of R(X) is primary by (3.2.2.a)  (c) If X is integral then any proper strong subobject of the reduced object X is non-unipotent by (3.1.2). According to the proof of (a), P(X)  is disjoint with any non-unipotent strong subobject V of X. Since P(X) ® X is coflat, any strong subobject of P(X) is induced from X. This means that the only proper strong subobject of P(X) is 0 , thus P(X) is simple, which is a generic residue of X; and P(X) ® X is an epic coflat fraction as P(X) = R(X)  by (b) and R(X) ® X is epic coflat. Conversely if P(X) is simple the X as a quotient of integral object  R(X) = P(X) is integral by (3.2.3.a).  n  Proposition 5.1.3. Suppose X is a quasi-primary object with the fractional hull pX: P(X) ® X. Let s: S(X) ® X be the strong image of  pX.  (a) P(X) is generic quasi-simple and any generic map from a quasi-simple object to X factors through pX uniquely.  (b) S(X) is generic primary and any generic map from a primary object to X factors through s uniquely.  (c) Any quasi-primary object has a unique generic primary strong subobject.  Proof. (a)  P(X) is generic quasi-simple by  (4.2.5.c). It is easy to see that any generic map from a quasi-simple object to X factors through any non-initial analytic subobject of X, therefore it also factors through the intersection pX of these analytic monos, which is P(X) by (5.1.2.a).  (b) The quasi-simple object P(X) is primary (3.3.8), thus its quotient S(X) is also primary. S(X) is a generic strong subobject of X because any non-initial analytic subobject of X contains P(X), therefore not disjoint with S(X). We prove that S(X) has the required universal property. Consider a generic map t: T ® X with T primary. Let pT: P(T) ® T be the fractional hull of T. Since pT is generic, so is the composite t°pT. By (a) the generic map t°pT: P(X) ® X factors through pX. Thus t°pT. factors through the strong mono s. Suppose m°e = t is the epi-strong-mono factorization of t. Since pT  is epic, m° e°pT = t°pT is the epi-strong-mono factorization of t°pT. Thus m is the smallest strong mono such that t°pT can be factored through. Since t°pT factors through s, m factors throughout s, thus t =  m°e factors through s as desired.  (c) It follows from (b) that S(X) is the unique generic primary strong subobject of X. n  Proposition 5.1.4. (a) Any map f: S ® X with a simple domain factors through a unique residue.  (b) Any simple subobject is contained in a unique residue.  (c) A simple subobject is a residue iff it is maximal.  Proof. (a) The epi S ® f+1(S) is generic by (3.3.4.f), so it factors through the generic residue of f+1(S). The uniqueness follows from (3.4.4.f).  (b) follows from (a).  (c) It follows from (b) that any maximal simple subobject is a residue. The other direction has been noticed in (3.4.4.d). n  Proposition 5.1.5. (a) Any colimits of reduced objects is reduced.  (b) Any cofiltered limits of reduced object is reduced.  Proof. (a) Since the full subcategory of reduced objects is a coreflective subobject of A , it is closed under colimits.  (b) Let {ri: X ® Xi}iÎI be a cofiltered limits of reduced objects in A. We have to prove that any proper strong subobject U of X is non-unipotent. 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 subobject by (4.1.3.a), we may assume that U is a finitely cogenerated regular subobject. So let us assume that U is the equalizer of a pair of distinct maps (m, n): X ® T where T is finitely copresentable. Since X is a cofiltered limits and T is finitely copresentable, we can find some t in I and a pair (mt, nt): Xt ® T of maps such that mt°rt = m and nt°rt = n. We may assume that t is an initial object in I. Let Ut be the equalizer of (mt, nt). Then the pullback of Ut along rt: X ® Xt is U. Since the proper regular subobject Ut is an intersection of proper disjunctable strong subobject, and rt does not factors through Ut, we can find a proper disjunctable subobject V of Xt containing Ut such that rt does not factor through rt. Let V be the pullback of Vt along rt, and Vi be the pullback of Vt along Xi ® Xt. Then Vi and V are proper disjunctable strong subobjects and U Í V, and Vc is the cofiltered limit of (Vi)c. Since each Xi is reduced and Vi is proper, each (Vi)c is non-initial. Since the initial object is finitely copresentable, this implies that Vc is non-initial. Thus V is not unipotent, and hence U is not unipotent as desired. n  Proposition 5.1.6. (a) An object is integral iff it is a quotient of a simple object.  (b) A non-initial object is primary iff it is a quotient of a quasi-simple object.  (c) An object is reduced iff it is a quotient of coproducts of simple objects.  Proof. (a) The condition is sufficient because any quotient of a simple object is integral. Conversely any integral object is a quotient of its rational hull, which is simple (5.1.2.c).  (b) The condition is sufficient because any quotient of a quasi-simple object is primary (3.3.8). Conversely any primary object is a quotient of its rational hull, which is quasi-simple (5.1.2.b).  (c) Any coproducts of simple objects is reduced (5.1.5.a). Conversely, assume X is a reduced object. Let T be the coproduct of all the residues pi: Pi ® X of X. Denote by t: T ® X the map induced by pi. Then T is reduced (5.1.5.a). We prove that t is epic. It suffices to show that t is unipotent as by assumption X is reduced. Any map s: S ® X with a simple domain factors through a unique residue of X by (5.1.4.a). So s factors through t. Since the class of simple objects is unipotent dense, it follows that t is unipotent by (2.2.10). n    [Next Section][Content][References][Notations][Home] 