5.1. Coherent Analytic Geometries
Any coherent analytic
category is spatial
(4.2.2) and perfect
(4.1.2.b). Thus a locally
disjunctable coherent analytic category is a spatial analytic
geometry, which is simply called a coherent
analytic geometry.
In the following we assume A is a coherent
analytic geometry. Recall that (3.6.9)
if X is an object. an open subset U of Spec(X)
is affine open if it is determined by an analytic subobject of X.
The set of affine open subsets is a base for the topology on Spec(X).
Proposition 5.1.1. (a) Spec(X)
is a coherent space for any object
X.
(b} For any map f: Y > X the continuous mapping Spec(f):
Spec(Y) > Spec(X) is a mapping
of coherent spaces.
Proof. (a) Suppose {V_{i}} is a set of disjunctable
strong subobjects such that {(V_{i})^{c }
iI}
is an analytic cover for X. Let V be the intersection of
these {V_{i}}. Then by (3.1.10)
we have X = {(V_{i})^{c}}
= V, so V = 0. Since
0
is finitely copresentable by (4.3.2.c),
there is a finite subset J
I such that {V_{i}}^{ } iI}
has intersection 0. Applying (3.1.10)
again we see that {(V_{i})^{c}} is a finite
analytic cover for X. This shows that Spec(X) is quasicompact.
Thus any affine open subset is also quasicompact. Since affine open subsets
form a base for Spec(X), any finite intersection of quasicompact
open subsets is quasicompact. This means that Spec(X) is
coherent.
(b) The pullback of analytic subobject of X along f is
an analytic subobject of Y. Since affine open subsets form a base
for Spec(X), this implies that inverse image of quasicompact
open subset is quasicompact open. Thus Spec(f) is a mapping
of coherent spaces. –
Proposition 5.1.2. (a) If X is
quasiprimary then the fraction hull
P(X) of X is the intersection of all noninitial analytic
monos.
(b) An object is primary iff
its rational hull R(X)
is quasisimple; if X
is primary then R(X) = Q(X) = P(X);
(c) An object is integral
iff its rational hull is simple;
any integral object X has a generic
residue P(X) > X which is an epic coflat simple
fraction.
Proof. (a) Consider the intersection P of noninitial
analytic subobjects of X, which is a noninitial fraction of X
containing
P(X). We have to prove that P is quasisimple, or
equivalently, that P is pseudosimple
(3.3.10). By (3.3.8.a)
it suffices to prove that any noninitial strong subobject of P
is unipotent. Since P > X
is coflat, any strong subobject of
P
is induced from X by (1.5.4). Since
any unipotent strong subobject of X induces a unipotent strong subobject
of P, we only need to prove that P is disjoint with any nonunipotent
strong subobject V of X. Since A
is locally
disjunctable, V is an intersection of proper disjunctable
strong subobjects {V_{i}}. Since V is nonunipotent,
at least one complement (V_{i})^{c} is noninitial,
and P (V_{i})^{c}.
Thus P V = 0. This
shows that P is pseudosimple, thus P = P(X).
(b) If X is primary then any noninitial analytic mono is epic.
Thus R(X) = P(X) by (a), and P(X)
is primary by (3.3.8). Any object X
is
a quotient of its rational hull R(X). If R(X)
is quasisimple, it is primary by (3.3.8),
thus X as a quotient of R(X) is primary by (3.2.2.a)
(c) If X is integral then any proper strong subobject of the
reduced
object X is nonunipotent by (3.1.2).
According to the proof of (a),
P(X) is disjoint
with any nonunipotent strong subobject V of X. Since P(X)
> X is coflat, any strong subobject of P(X) is induced
from X.
This means that the only proper strong subobject of P(X)
is 0 , thus P(X) is simple, which is a generic residue
of X; and P(X) > X is an epic coflat fraction
as P(X) = R(X) by (b)and
R(X) > X is epic coflat. Conversely if P(X) is simple
the X as a quotient of integral object R(X)
= P(X) is integral by (3.2.3.a).
Proposition 5.1.3. Suppose X
is a quasiprimary object with the fractional hull p_{X}:
P(X) > X. Let s: S(X) > X
be the strong image of p_{X}.
(a) P(X) is generic quasisimple and any generic map
from a quasisimple object to X factors through p_{X}
uniquely.
(b) S(X) is generic primary and any generic map from
a primary object to X factors through s uniquely.
(c) Any quasiprimary object has a unique generic primary strong subobject.
Proof. (a) P(X) is generic quasisimple by
(4.2.5.c). It is easy to see that any generic
map from a quasisimple object to X factors through any noninitial
analytic subobject of X, therefore it also factors through the intersection
p_{X}
of these analytic monos, which is P(X) by (5.1.2.a).
(b) The quasisimple object P(X) is primary (3.3.8),
thus its quotient S(X) is also primary. S(X)
is a generic strong subobject of X because any noninitial analytic
subobject of X contains P(X), therefore not disjoint
with S(X). We prove that S(X) has the required
universal property. Consider a generic map t: T > X with
T
primary. Let p_{T}: P(T) > T be the
fractional hull of T. Since p_{T} is generic, so
is the composite tp_{T}. By (a) the generic map t_{°}p_{T}:
P(X) > X factors through p_{X}. Thus
tp_{T}.
factors through the strong mono s. Suppose
me = t
is the epistrongmono factorization of t. Since
p_{T}
is epic, mep_{T} = tp_{T} is the epistrongmono
factorization of tp_{T}. Thus m is the smallest strong
mono such that tp_{T} can be factored through. Since tp_{T}
factors through s, m factors throughout s, thus t
= me factors through
s as desired.
(c) It follows from (b) that S(X) is the unique generic
primary strong subobject of X.
Proposition 5.1.4. (a) Any map f:
S > X with a simple domain factors through a unique residue.
(b) Any simple subobject is contained in a unique residue.
(c) A simple subobject is a residue iff it is maximal.
Proof. (a) The epi S > f^{+1}(S)
is generic by (3.3.4.f), so it factors
through the generic residue of f^{+1}(S). The uniqueness
follows from (3.4.4.f).
(b) follows from (a).
(c) It follows from (b) that any maximal simple subobject is a residue.
The other direction has been noticed in (3.4.4.d).
Proposition 5.1.5. (a) Any colimits
of reduced objects is reduced.
(b) Any cofiltered limits of reduced object is reduced.
Proof. (a) Since the full subcategory of reduced objects is a
coreflective subobject of A , it is closed
under colimits.
(b) Let {r_{i}: X > X_{i } iI}
be a cofiltered limits of reduced objects in A.
We have to prove that any proper strong subobject U of X
is nonunipotent. Since any proper strong subobject is contained in a proper
regular subobject, and any proper regular subobject is an intersection
of proper finitely cogenerated regular subobject by (4.1.3.a),
we may assume that U is a finitely cogenerated regular subobject.
So let us assume that U is the equalizer of a pair of distinct maps
(m, n): X > T where T is finitely copresentable.
Since X is a cofiltered limits and T is finitely copresentable,
we can find some t in I and a pair (m_{t}, n_{t}):
X_{t} > T of maps such that m_{t}r_{t}
= m and n_{t}r_{t} = n. We may assume that t
is an initial object in I. Let U_{t} be the equalizer
of (m_{t}, n_{t}). Then the pullback of U_{t}
along r_{t}: X > X_{t} is U. Since the
proper regular subobject U_{t} is an intersection of proper
disjunctable strong subobject, and r_{t} does not factors
through U_{t}, we can find a proper disjunctable subobject
V
of X_{t} containing U_{t} such that
r_{t}
does not factor through r_{t}. Let
V be the pullback
of V_{t} along r_{t}, and V_{i}
be the pullback of V_{t} along
X_{i} > X_{t}.
Then V_{i} and V are proper disjunctable strong subobjects
and U V, and V^{c}
is the cofiltered limit of (V_{i})^{c}. Since
each X_{i} is reduced and V_{i} is proper,
each (V_{i})^{c} is noninitial. Since the
initial object is finitely copresentable, this implies that V^{c}
is noninitial. Thus V is not unipotent, and hence U is not
unipotent as desired.
Proposition 5.1.6. (a) An object is
integral iff it is a quotient of a simple object.
(b) A noninitial object is primary iff it is a quotient of a quasisimple
object.
(c) An object is reduced iff it is a quotient of coproducts of simple
objects.
Proof. (a) The condition is sufficient because any quotient of a simple
object is integral. Conversely any integral object is a quotient of its
rational hull, which is simple (5.1.2.c).
(b) The condition is sufficient because any quotient of a quasisimple
object is primary (3.3.8). Conversely any
primary object is a quotient of its rational hull, which is quasisimple
(5.1.2.b).
(c) Any coproducts of simple objects is reduced (5.1.5.a).
Conversely, assume X is a reduced object. Let T be the coproduct
of all the residues p_{i}: P_{i} > X of
X.
Denote by t: T > X the map induced by p_{i}.
Then T is reduced (5.1.5.a). We prove that
t
is epic. It suffices to show that t is unipotent as by assumption
X is reduced. Any map s: S > X with a simple domain
factors through a unique residue of X by (5.1.4.a).
So s factors through t. Since the class of simple objects
is unipotent dense, it follows that t is unipotent by (2.2.10).
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