3.4. Local Objects
Definition 3.4.1. (a) A non initial
object X is called local
if non initial strong subjects of X has a non initial intersection
M.
(b) An epic simple fraction
of an integral object X is called a generic
residue of X.
(c) A mono (or subject) p: P >
X is called a residue
of X if P > p^{+1}(P) is a generic residue of p^{+1}(P).
Proposition 3.4.2. Suppose X
is a local object with the strong subject M as above.
(a) M is the unique simple
prime of X.
(b) Any proper fraction U of X is disjoint with M.
(c) The inclusion M ® X
is a local map.
Proof. (a) Since M is the smallest strong subject of X,
and any non initial strong subject V of M is also a strong
subject of X, we must have V = M, i.e. M is simple.
Any other simple strong subject P of X must contain M
as the non initial simple prime, thus M = P.
(b) Consider a proper fraction U of X. There is a non
initial strong subject V that is disjoint with U by (3.3.2.d).
Since M V, M
is disjoint with U.
(c) M is contained in any non initial strong subject of X,
thus the inclusion M > X is local. ––
Proposition 3.4.3. Suppose f:
Y > X is a map and Y is a local object with the simple prime
M.
(a) f is disjoint with a strong subject V of X
if the induced map M > X does not factor through V.
(b) Suppose Y is simple. Then f is disjoint with a fraction
U of X if it does not factor through U.
(c) f factors through a fraction U of X if the
induced map M > X is not disjoint with U.
(d) Suppose {U_{i}} is an analytic cover on X.
Then f factors through some U_{i}.
Proof. (a) If the induced map M > X does not factor
through V then f^{1}(V) is a proper strong
subject not containing M. Since Y is local this means that
f^{1}(V) is initial. Thus f is disjoint with
V. The other direction is trivial.
(b) If f does not factor through V then f^{1}(V)
is a proper fraction of Y, therefore it is initial because any simple
object is quasi simple by (3.3.8). Thus
f is disjoint with V. The other direction is trivial.
(c) If M > X is not disjoint with U then it factors
through U by (b). Thus f^{1}(U) is a fraction
contains M, therefore f^{1}(U) = Y
as M > Y is quasi local by (3.4.2) and (3.3.4.a).
It follows that f factors through U. The other direction
is trivial.
(d) If {U_{i}} is an analytic cover on X, the
induced map M > X is not disjoint with at least one U_{i},
thus by (b) M > X factors through U_{i}, and hence
f factors through U_{i} by (c).
Proposition 3.4.4. (a) Any simple fraction
is a residue.
(b) Any simple prime is a residue.
(c) The unique simple prime of a local object is a residue.
(d) Any residue of an object is a maximal simple subject (i.e. it is
not contained in any other simple subject).
(e) Any integral object has at most one generic residue, which is the
intersection of all the non initial fractions; any generic residue is a
generic subject in the sense of (3.3.4.b).
(f) Any two distinct residues of an object are disjoint with each other.
Proof. (a) Suppose P is a simple fraction of an object
X. The epic map P >. p^{+1}(P) is also fractional
by (3.3.2.b), thus P is a residue.
(b) is obvious and (c) follows from (b).
(d) Suppose q: Q > X is a simple object which contains
a residue p: P > X. Since P > Q is epic, by the
uniqueness of epistrongmono factorization we have p^{+1}(P)
= q^{+1}(Q), so Q
f^{+1}(P). Now P > p^{+1}(P) is a fraction
implies that P > Q is fractional by (3.3.2.f).
But Q has exactly two fractions, so P = Q as desired.
(e) Suppose P is a generic residue and Q is a non initial
fraction of an integral object X . Then P
Q is a non initial fraction of P, so P = P
Q as P has exactly two fractions. Thus P is the intersection
of all the non initial fractions of X, therefore is unique. This
also shows that P > X is a generic map.
(f) Suppose P and Q are two residues of X and
P Q is non initial.
Since P Q > P and
P Q > Q are epic,
P Q, P and Q
have the same strong image V in X. Since P and Q
are both generic residues of V, we have P = Q by (e).
Proposition 3.4.5. Suppose p:
P > U is a residue and u: U > X is a fraction (resp.
strong mono). Then u_{°}p:
P > U is a residue of X.
Proof. t: P > p^{+1}(P) is an epic fraction
as p is a residue. By (3.3.2.b)
s: p^{+1}(P) > u^{+1}(p^{+1}(P))
is an epic fraction. Thus slt: P > (up)^{+1}(P)
is an epic fraction. This shows that the mono up: P > X
is a residue. The assertion for strong monos is obvious by the definition
of a residue.
Proposition 3.4.6. Suppose f:
P > Z is a local map with P simple. Then Z is local
and f^{+1}(P) is the simple prime of Z.
Proof. Since f is local, any non initial strong subject
V of Z is not disjoint with f. Thus f factors
through V by (3.4.3.b). Hence V contains
f^{+1}(P). It follows that f^{+1}(P)
is the intersection of non initial strong subject of X. Thus X
is local with f^{+1}(P) as the simple prime.
Proposition 3.4.7. (a) Suppose f:
X > Z is a local map and X is local with the simple prime
P. Then Z is local with f^{+1}(P) as
the simple prime.
(b) Suppose f: Y > X is a map of local objects. Then
f is local iff f^{+1} sends the simple prime of Y
to that of X.
Proof. (a) Let M be the simple prime of X. The
composite of M > X with X > Z is a local map (as M
> X is local by (3.4.2.c)), the assertion follows
from (3.4.6).
(b) The condition is necessary by (a). The other direction is obvious.
Proposition 3.4.8. Suppose f:
P > X is a map and P is simple.
(a) f is a local epi if X is simple.
(b) f is a local strong mono if X is local with the simple
prime P.
(c) f is an epic fraction if X is integral with the generic
residue P.
Proof. (a) If f is a local epi then X is local
and f^{+1}(P) = X as the simple prime of X
by (3.4.6), so X is simple. The other direction
is obvious.
(b) If f is a local strong mono then X is local with
f^{+1}(P) = P as the simple prime by (3.4.6).
The other direction follows from (3.4.2).
(c) is obvious.
Proposition 3.4.9. Suppose A
is locally
disjunctable reducible.
(a) Suppose f: P > Z is a prelocal map with P
simple. Then f is a local map; Z is a local object with f^{+1}(P)
as the simple prime of Z.
(b) Suppose f: X > Z is a prelocal map and X
is local. Then f is a local map and Z is a local object.
Proof. (a) By (3.4.6) it suffices to prove
that f is local. Assume v: V > Z is a strong subject
and f is disjoint with V. Since A
is locally disjunctable, v is the intersection of a set {v_{i}:
V_{i} > Z} of disjunctable strong monos. Then by
(3.1.10) we have {(v_{i})^{c}}
= v. Since f v,
f is not disjoint with some v_{i}^{c}, so
f factor through the proper analytic fraction v_{i}^{c}
by (3.4.3.b). Since f is prelocal, we have
(V_{i})^{c} = X. Since (V_{i})^{c}
is disjoint with V, V is initial. This shows that f
is local.
(b) Let M be the simple prime of X. The composite t:
M > Z of M > X with X > Z is a local map (as M
> X is local by (3.4.2.c)), so t is local
and Z is a local object with t^{+1}(M) as
the simple prime by (a). Any non initial strong subject of Z contains
t^{+1}(M). Thus f is not disjoint with any
non initial strong subject of Z. This shows that f is local.
Definition 3.4.10. (a) A non initial
object X is called cprimary
(resp. fprimary) if any non initial
coflat map (resp. fraction) to X is epic.
(b) A non initial object X is called cquasiprimary
(resp. fquasiprimary) if the intersection
of any two non initial coflat maps (resp. fractions) to X is not
initial.
(c) A reduced and cprimary (resp. fprimary) object is called a cintegral
object (resp. fintegral object).
Proposition 3.4.11. (a) Any quotient
of a cprimary (resp. fprimary) object is cprimary (resp. fprimary).
(b) Any cprimary object is cquasiprimary; any fprimary object is
fquasiprimary.
(c) Any cprimary object is fprimary; any fprimary object is primary.
(d) Any cquasiprimary object is fquasiprimary; any fquasiprimary
object is quasiprimary.
(e) Any cintegral object is fintegral; any fintegral object is integral.
(f) Any simple object is cintegral; any quotient of a simple object
is cintegral.
(g) Any quasi simple object is fprimary; any quotient of a quasi simple
object is fprimary.
(h) Assume A is locally disjunctable reducible.
Any presimple object is quasi simple. Any quotient of a presimple object
is fprimary.
(i) Assume A is locally disjunctable. A
non initial reduced object X is fprimary if it is cquasiprimary.
Proof. The proof for (a) and (b) is similar to that of (3.2.2.)
(c)  (e) are obvious.
(f) The first assertion is obvious; the second follows from (a).
(g) The first assertion is trivial; the second follows from (a).
(h) follows from by (3.3.10) and (g).
(i) follows from (3.2.5).
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