3.3. Simple Objects
Definition 3.3.1. (a) A mono (or subobject)
is called a fraction
if it is coflat and normal.
(b) A map to an object X is called generic
if it is not disjoint with any noninitial analytic subobject of X.
Proposition 3.3.2. (a) The class of
fractions is closed under composition and stable under pullback.
(b) If f: Y > X is a fraction and U is a strong
subobject of Y, then U > f^{+1}(U) is a
fraction.
(c) If f: Y > X is a fraction which factors through
a mono u: U > X in a map v: Y > U, then
v is a fraction.
(d) Any proper fraction of an object is disjoint with a noninitial
strong subobject.
Proof. (a) The classes of coflat maps and normal monos are closed
under composition and stable under pullback.
(b) Since f is coflat, we have f^{1}f^{+1}(U)
= U by (1.5.4). Thus U > f^{+1}(U)
is the pullback of f along u: f^{+1}(U)
> X, therefore is a fraction by (a).
(c) The mono v is the pullback of f along u, thus
is fractional by (a).
(d) Suppose U is a proper fraction of an object X. Since
U is normal, U is disjoint with a noninitial map t:
T > X, and we may assume t is a strong mono by (1.5.2).
––
Definition 3.3.3. (a) A map to an object
X is called local
if it is not disjoint with any noninitial strong subobject of X;
a noninitial object is called pseudosimple
if any noninitial map to it is local.
(c) A map to an object X is called quasilocal
if it does not factor through any proper fraction to X; a noninitial
object is called quasisimple
if any noninitial map to it is quasilocal.
(d) A map to an object X is called prelocal
if it does not factor through any proper analytic mono to X; a noninitial
object is called presimple
if any noninitial map to it is prelocal.
Proposition 3.3.4. (a) Any local map
is quasilocal; any quasilocal map is prelocal.
(b) The class of local (resp. generic, resp. quasilocal, resp. prelocal)
maps is closed under composition.
(c) A quasilocal fraction (resp. prelocal analytic mono) is an isomorphism.
(d) If f: Y > X and g: Z > Y are two
maps and gf is local (resp. generic, resp. quasilocal, resp. prelocal)
then f is local (resp. generic, resp. quasilocal, resp. prelocal).
(e) Any unipotent map is both local and generic.
(f) Any epi is generic.
(g) If f: Y > X is a generic map and Y is quasiprimary
then X is quasiprimary.
Proof. (a) Any proper fraction u: U > X is disjoint
with a noninitial strong subobject V of X by (3.3.2.d).
Any local map f: Y > X is not disjoint with V, therefore
f does not factor through any proper fraction U. The second
assertion is trivial.
(b) Consider two local maps f: Y > X and g:
Z > Y. Suppose fg is disjoint with a strong mono v:
V > X. Then f^{1}(V) >
Y is disjoint with g: Since g: Z >
Y is local this implies that f^{1}(V) is initial.
Thus f is disjoint with V. Since f is local, V
is initial. This shows that fg is local. The proof for generic maps
is similar.
Consider two quasilocal maps f: Y > X and g:
Z > Y. Suppose f_{°}g
factors through a fraction v: V > X. Then g factors
through f^{1}(V). Since g is quasilocal
and f^{1}(V) > Y is a fraction, we have
f^{1}(V) = Y. Thus f factors through
V, so V = X as f is quasilocal. This shows that gf
is quasilocal. The proof for prelocal maps is similar.
(c)  (e) are obvious.
(f) If f: Y > X is an epi then its pullback along any
noninitial analytic mono V > X
is a noninitial epi W > V,
thus f is not disjoint with V > X.
Hence f is generic.
(g) Consider two noninitial analytic subobjects U and V
of X. Since f is generic, the analytic subobjects f^{1}(U)
and f^{1}(V) of Y are noninitial. Since
Y is quasiprimary, f^{1}(V)
f^{1}(U) is noninitial. Since the induced map f^{1}(V)
f^{1}(U) > X of
f factors through U
V, it follows that U
V is noninitial. Thus X is quasiprimary.
Recall that an epi e is called extremal
provided that it does not factors through a proper mono. Every regular
epi is an extremal epi.
Definition 3.3.5. A noninitial object
is called simple
(resp. extremal
simple, resp. unisimple)
if any noninitial map to it is epic (resp. extremal epic, resp. unipotent).
Proposition 3.3.6. An object X
is simple (resp. extremal simple, resp. unisimple, resp. quasisimple,
resp. presimple) iff it has exactly two strong subobjects (resp. subobjects,
resp. normal sieves, resp. fractions, resp. analytic subobjects).
Proof. A noninitial strong subobject V of a simple object
is determined by an epic strong mono, which must be an isomorphism by (1.1.2.b),
thus V = X. Conversely, assume X has exactly two strong subobjects;
then X is noninitial. If t: T > X is any noninitial
map then t^{+1}(T) is a noninitial strong subobject
of X, thus t^{+1}(T) = X, so t
is epic.
A noninitial subobject V of an extremal simple object X
is an extremal epic mono, so V = X. Conversely if X has exactly
two subobjects, any noninitial map t: T > X must be extremal
epic.
Any noninitial map to a unisimple object is unipotent, thus generates
the maximal normal sieve. This implies that X has exactly two normal
sieves. Conversely if X has exactly two normal sieves, any noninitial
map to X generates the maximal normal sieve, thus it is unipotent.
A noninitial fraction to a quasisimple object is a quasilocal fraction,
thus is an isomorphism by (3.3.4.c). Conversely, if
X has exactly two fractions then any noninitial map is quasilocal,
so it is quasisimple. The proof for the case of presimple objects is similar.
Proposition 3.3.7. (a) Any simple object
is integral.
(b) Any extremal simple object and any reduced unisimple object is
simple.
Proof. (a) Any simple object is reduced and primary (cf (3.3.6),
(3.1.2.a) and (3.2.1.a)).
(b) Clearly any extremal simple object is simple by (3.3.6).
Any noninitial map f to an object X is unipotent if X
is unisimple, and is epic if furthermore X is reduced. Thus any
reduced unisimple object is simple.
Proposition 3.3.8. (a) A noninitial
object is pseudosimple iff any noninitial strong subobject is unipotent.
(b) Any simple object, extremal simple object, and unisimple object
is pseudosimple.
(c) Any pseudosimple object is quasisimple.
(d) Any quasisimple object is presimple.
(e) Any presimple object is primary.
Proof. (a) By definition a noninitial object X is quasisimple
iff any noninitial map to it is not disjoint from any noninitial strong
mono. This implies that any noninitial strong mono is unipotent.
(b) Any noninitial strong mono to a simple or extremal simple object
is an isomorphism. Any noninitial strong mono to a unisimple object is
unipotent. By (a) these objects are pseudosimple.
(c) and (d) follow from (3.3.4.a)
(e) Any noninitial analytic mono to a presimple object is an isomorphism,
thus any presimple object is primary.
Proposition 3.3.9. (a) Any reduced pseudosimple
object is simple.
(b) Any noninitial strong subobject of a pseudosimple object is pseudosimple.
(c) The radical of any
pseudosimple object is simple.
Proof. (a) Suppose X is a reduced pseudosimple object.
By (3.3.8.a) any noninitial strong mono to X
is unipotent, therefore is epic, thus is an isomorphism. It follows that
X is simple by (3.3.6).
(b) Suppose V is a noninitial strong subobject of a pseudosimple
object X . Then any noninitial strong subobject W of V
is also a strong subobject of X , thus W is a unipotent subobject
of X. It follows that W is also a unipotent subobject of
V. Thus V is pseudosimple by (3.3.8.a).
(c) follows from (a) and (b).
Proposition 3.3.10. Suppose A
is locally
disjunctable in which each object has a radical. The following are
equivalent for an object X :
(a) X is pseudosimple.
(b) X is quasisimple.
(c) X is presimple.
(d) The radical of X is simple.
Proof. (a) implies (b) and (b) implies (c) by (3.3.8).
Assume X is presimple. Any proper strong subobject V of its
radical rad(X) is not unipotent in X, thus it is disjoint
with a noninitial analytic mono u: U > X to X by
(3.1.10). Since X is presimple, U = X
by (3.3.6). Thus V is initial. This shows that
rad(X) is simple. Thus (c) implies (d). Finally assume X
is an object whose radical rad(X) is simple. We prove that
it is pseudosimple. Suppose V is a noninitial strong subobject
of X. Its radical rad(V) is a noninitial strong subobject
of rad(X), therefore we have rad(V) = rad(X)
as by assumption rad(X) is simple. Since rad(X)
is unipotent, V is unipotent. Thus X is pseudosimple by
(3.3.8.a).
Proposition 3.3.11. Suppose any coflat
unipotent map in A is regular epic and any
map to a simple object is coflat. Then
(a) Any coflat mono is normal.
(b) Any simple object is extremal simple and unisimple.
Proof. (a) Consider a coflat mono u: U > X. To
see that u is normal it suffices to prove that any pullback of u
is not proper unipotent. Consider the pullback v: V > Y
of u along a map f: Y > X. If v is unipotent
then the coflat unipotent mono v is a regular epi by assumption,
so is an isomorphism because any regular epic mono is isomorphic.
(b) Consider a simple object P. We first show that P is unisimple.
Consider two noninitial maps t: T > P and s:
S > P. By assumption s and t are coflat epic. Thus
the pullback of (s, t) is noninitial. This show that P is
unisimple. Next we prove that P is extremal simple. Any noninitial
subobject V of P is coflat unipotent, thus is a regular epi.
Again since any regular epic mono is isomorphic, we have V = P as
desired.
Example 3.3.11.1. The category of
affine schemes satisfies the conditions of (3.1.11).
Thus any coflat mono of affine schemes is normal and any simple object
(i.e. the spectrum of a field) is extremal simple and unisimple.
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