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 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 {Ui} is an analytic cover on X. Then f factors through some Ui

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 {Ui} is an analytic cover on X, the induced map M --> X is not disjoint with at least one Ui, thus by (b) M --> X factors through Ui, and hence f factors through Ui 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 epi-strong-mono factorization we have p+1(P) = q+1(Q), so 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 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 Q is non initial. Since Q --> P and Q --> Q are epic, 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 {vi: Vi --> Z} of disjunctable strong monos. Then by (3.1.10) we have {(vi)c}v. Since  v, f is not disjoint with some vic, so f factor through the proper analytic fraction vic by (3.4.3.b). Since f is prelocal, we have (Vi)c = X. Since (Vi)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 c-primary (resp. f-primary) if any non initial coflat map (resp. fraction) to X is epic. 
(b) A non initial object X is called c-quasi-primary (resp. f-quasi-primary) if the intersection of any two non initial coflat maps (resp. fractions) to X is not initial. 
(c) A reduced and c-primary (resp. f-primary) object is called a c-integral object (resp. f-integral object). 

Proposition 3.4.11. (a) Any quotient of a c-primary (resp. f-primary) object is c-primary (resp. f-primary). 
(b) Any c-primary object is c-quasi-primary; any f-primary object is f-quasi-primary. 
(c) Any c-primary object is f-primary; any f-primary object is primary. 
(d) Any c-quasi-primary object is f-quasi-primary; any f-quasi-primary object is quasi-primary. 
(e) Any c-integral object is f-integral; any f-integral object is integral. 
(f) Any simple object is c-integral; any quotient of a simple object is c-integral. 
(g) Any quasi simple object is f-primary; any quotient of a quasi simple object is f-primary. 
(h) Assume A is locally disjunctable reducible. Any presimple object is quasi simple. Any quotient of a presimple object is f-primary. 
(i) Assume A is locally disjunctable. A non initial reduced object X is f-primary if it is c-quasi-primary. 

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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