3.2. Integral Objects
Definition 3.2.1. (a) A noninitial
object is primary
if any noninitial analytic mono to it is epic.
(b) A noninitial object is quasiprimary
if any two noninitial 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 noninitial 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 quasiprimary.
Proof. (a) Let f: Y > X be an epi with Y
primary. Consider any noninitial analytic mono t: T > X.
Let (g: W > Y, h: W > T) be the pullback of (t,
f). Then h is noninitial epic as it is the pullback of an epi
along the noninitial coflat map t, and g as the pullback
of an analytic mono is a noninitial 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 noninitial 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 noninitial.
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 noninitial
analytic subobject of a primary object is primary.
(b}Any noninitial analytic subobject of an integral object is integral.
Proof. (a) Suppose u: U > X is a noninitial
analytic subobject of a primary object X. Consider a noninitial
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 U^{c}
and V^{c} 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) U^{c} V^{c}
= (U V)^{c}
= X^{c} is initial by (1.6.4).
Proposition 3.2.6. Suppose A
is locally
disjunctable. The following are equivalent for a noninitial reduced
object X:
(a) Any noninitial coflat map to X is epic.
(b) X is primary (i.e. X is integral).
(c) X is quasiprimary.
(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, U^{c} and V^{c}
are noninitial, and they are disjoint by (3.2.5).
Thus X is not quasiprimary. Hence (c) implies (d).
Next we show that (d) implies (b). Assume X is irreducible.
Consider a noninitial 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 = V^{c} 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 noninitial 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, V^{c}
is a noninitial analytic mono, therefore epic by (b). Since f is
noninitial and coflat, f is not disjoint with V^{c}.
Since V^{c} 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 quasiprimary.
(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 noninitial object X is quasiprimary iff its radical is integral.
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