3.2. Integral Objects Definition 3.2.1. (a) A non-initial object is primary if any non-initial analytic mono to it is epic.  (b) A non-initial object is quasi-primary if any two non-initial analytic monos are not disjoint with each other.  (c) An integral object is a reduced primary object.  (d) A prime of an object is an integral strong subobject.  (e) A non-initial object is irreducible if it is not the join of two proper strong subobjects.  Proposition 3.2.2. (a) Any quotient of a primary object is primary.  (b) If f: Y --> X is a map and U a primary subobject of Y, then the strong image f+1(U) of f in X is a primary subobject of X.  (c) Any primary object is quasi-primary.  Proof. (a) Let f: Y --> X be an epi with Y primary. Consider any non-initial analytic mono t: T --> X. Let (g: W --> Y, h: W --> T) be the pullback of (t, f). Then h is non-initial epic as it is the pullback of an epi along the non-initial  coflat map t, and g as the pullback of an analytic mono is a non-initial analytic mono, so g is epic as Y is primary. It follows that fg = th is epic, and so t is epic.  (b) follows from (a)  (c) Suppose U and V are two non-initial analytic subobject of a primary object X. Since U and V are coflat and epic, U  V is an epic analytic mono of U and V (cf. (1.6.6.b)), thus is non-initial.  Corollary 3.2.3. (a) Any quotient of an integral object is integral.  (b) If f: Y --> X is a map and U a prime of Y, then f+1(U) is a prime of X.  Proof. (a) follows from (3.2.2.a) and (3.1.3.a).  (b) follows from (3.2.2.b) and (3.1.3.b).   Proposition 3.2.4. (a) Any non-initial analytic subobject of a primary object is primary.  (b}Any non-initial analytic subobject of an integral object is integral.  Proof. (a) Suppose u: U --> X is a non-initial analytic subobject of a primary object X. Consider a non-initial analytic subobject t: T --> U. Since ut is analytic and X is primary, ut is epic, so t is epic (as the pullback of the epi ut along the coflat mono u), thus U is primary as desired.  (b) follows from (a) and (3.1.3.e).   Proposition 3.2.5. Suppose X = U  V is the join of two strong subobjects U and V of X.  (a) If t: T --> X is a coflat map which is disjoint with V then it factors through U.  (b) If U and V are disjunctable then Uc and Vc are disjoint.  Proof. (a) By (1.5.3) we have  T = t-1(X) = t-1(U  V) = t-1(U)  t-1(V) = t-1(U). So t factors through U.  (b) Uc  Vc = (U  V)c = Xc is initial by (1.6.4).   Proposition 3.2.6. Suppose A is locally disjunctable. The following are equivalent for a non-initial reduced object X:  (a) Any non-initial coflat map to X is epic.  (b) X is primary (i.e. X is integral).  (c) X is quasi-primary.  (d) X is irreducible.  Proof. Clearly (a) implies (b), and (b) implies (c) by (3.2.2.c). We prove that (c) implies (d). Assume X is not irreducible. Then X = U  V is the join of two proper strong subobjects U and V. Since A is locally disjunctable, we may assume U and V are disjunctable. Since X is reduced, Uc and Vc are non-initial, and they are disjoint by (3.2.5). Thus X is not quasi-primary. Hence (c) implies (d).  Next we show that (d) implies (b). Assume X is irreducible. Consider a non-initial analytic subobject U, which is the complement of a proper strong object V of X . We prove that U is epic. Consider any strong subobject W of X containing U. Since W  V contains U = Vc and V, it is unipotent, thus X = W  V as X is reduced. Since V is proper, by (d) we have W = X. This shows that U is an epic subobject by (1.1.3.e). This shows that (d) implies (b).  Finally we prove that (b) implies (a). Consider a non-initial coflat map f: Y --> X. As A is locally disjunctable, any proper strong subobject is contained in a proper disjunctable strong subobject. To see that f is epic by (1.1.3.e)  it suffices to show that it does not factor through any proper disjunctable strong subobject V of X. Since X is reduced, Vc is a non-initial analytic mono, therefore epic by (b). Since f is non-initial and coflat, f is not disjoint with Vc. Since Vc is disjoint with V, this implies that f: Y --> X does not factor through any proper disjunctable strong subobject V, so it is epic. This shows that (b) implies (a).   Corollary 3.2.7. Suppose A is locally disjunctable.  (a) An object is integral iff it is reduced and quasi-primary.  (b) An object is integral iff it is reduced and irreducible.  Proof. By (3.2.6).   Later we shall prove in Section 5 that if A is an analytic geometry then a non-initial object X is quasi-primary iff its radical is integral.      [Next Section][Content][References][Notations][Home]