abstract algebraic geometry

[categories <cat-dist@mta.ca>]

Date: Thu, 20 Nov 1997 12:40:02 -0800
From: Zhaohua Luo <zack@iswest.com>

The following is the second part of "The language of analytic
categories", which is a report on my paper CATEGORICAL
GEOMETRY. Please note that Section 6 on integral objects
(which was included in the first part) has been modified in
order to conform with the notion of a primary object by
Diers. The fact is that there are several ways to introduce a
primary object in a general analytic category, and the one give
by Diers (for a Zariski category) happens to be the weakest
one. The new definition of an integral object given below
(being a reduced primary object) is therefore weaker than the
old one given in the first part of this note, but the basic
properties remain the same (see (6.1) - (6.3)). On the other
hand, Diers's definition of an integral object in a Zariski
category (being a quotient of a simple object) is the strongest
one. In practice these definitions agree in most cases (for
instance, see (6.4) and (6.5) below). 

Z. Luo
----------------------------------------------------------------
The opposite RING^op of the category RING of
commutative rings (with unit) is an analytic category, which
is equivalent to the category of affine schemes. Following
Diers we have the following list: 

RING^op 		RING

simple			field	
integral		integral domain
reduced                 without non-null nilpotent 	
			elements
radical			the residue class ring with 	
			respect to its radical
pseudo-simple		exactly one prime ideal
quasi-primary		ab = 0 => (a or b is nilpotent)
primary 		any zero divisor is nilpotent
analytically closed	total ring of quotients
irreducible		the ideal {0} is irreducible with
			respect to intersection
regular			von Neumann regular ring
local			local ring
generic residue		quotient field
----------------------------------------------------------------
THE LANGUAGE OF ANALYTIC CATEGORIES

By Zhaohua Luo (1997)

Content

1. Analytic Categories
2. Distributive Properties
3. Coflat Maps
4. Analytic Monos
5. Reduced Objects
6. Integral Objects
7. Simple Objects
8. Local Objects
9. Analytic Geometries 
10. Zariski Geometries
References
Appendix: Analytic Dictionary
----------------------------------------------------------------------
SECOND PART
------------------------------------------------------------------------
6. Integral Objects

Let C be an analytic category (i.e. a lextensive category with
epi-strong-mono factorizations).

A non-initial object is "primary" if any non-initial analytic
subobject is epic. A non-initial object is "quasi-primary" if 
any two non-initial analytic subobjects has a non-initial
intersection. An "integral" object is a reduced primary object.
A "prime" of an object is an integral strong subobject.  A
non-initial object is "irreducible" if it is not the join of two
proper strong subobjects.

For any object X denote by Spec(X) the set of primes of X. If
U is any analytic subobject of X we denote by X(U) the set of
primes of X which is not disjoint with U, called an "affine
subset" of X. Using (4.3) one can show that the class of affine
subsets is closed under intersection. Thus affine subsets form
a base for a topology on Spec(X). The resulting topological
space Spec(X) is called the "spectrum" of X. Since the
pullback of an analytic mono is analytic, it follows from (6.2)
below that Spec is naturally a functor from C to the
(meta)category of topological spaces. For instance, if C is the
category of affine schemes or affine varieties then Spec
coincides with the classical Zariski topology.

(6.1) Any quotient of a primary object is primary; any primary
object is quasi-primary.

(6.2) Any quotient of an integral object is integral; if f: Y  --> 
X is a map and U a prime of Y, then f^{+1}(U) is a prime of
X.    

(6.3) Any non-initial analytic subobject of a primary object is
primary; any non-initial analytic subobject of an integral
object is integral.

(6.4) Suppose C is locally disjunctable. The following are
equivalent for a non-initial reduced object X:
(a) Any non-initial coflat map to X is epic.
(b) X is primary.
(c) X is quasi-primary.
(d) X is irreducible.
 
(6.5) Suppose C is locally disjunctable. Then
(a) An object is integral iff it is reduced and quasi-primary.
(b) An object is integral iff it is reduced and irreducible.

7. Simple Objects

A mono (or subobject) is called a "fraction" if it is coflat
normal. A map to an object X is called "local" (resp.
"generic") if it is not disjoint with any non-initial strong
subobject (resp. analytic subobject). A map to an object X is
called "quasi-local" if it does not factor through any proper
fraction to X. A map to an object X is called "prelocal" if it
does not factor through any proper analytic mono to X. A
non-initial object is called  "simple" (resp. "extremal simple", 
resp. "unisimple", resp. "pseudo-simple", resp. "quasi-
simple", resp. "presimple") if any non-initial map to it is epic
(resp. extremal epic, resp. unipotent, resp. local, resp. quasi-
local, resp. prelocal).

(7.1) The class of fractions is closed under composition and
stable under pullback.

(7.2) Any local map is quasi-local; any quasi-local map is
prelocal; the class of local (resp. generic, resp. quasi-local,
resp. prelocal) maps is closed under composition; a quasi-
local fraction (resp. prelocal analytic mono) is an
isomorphism.

(7.3) Any unipotent map is both local and generic; any epi is
generic.

(7.4) An object X is simple (resp. extremal simple, resp.
unisimple, resp. quasi-simple, resp. presimple) iff it has
exactly two strong subobjects (resp. subobjects, resp. normal
sieves, resp. fractions, resp. analytic subobjects).

(7.5) Any simple object is integral; any extremal simple object
and any reduced unisimple object is simple.

(7.6) A non-initial object is pseudo-simple iff any non-initial
strong subobject is unipotent; any simple object, extremal
simple object, and unisimple object is pseudo-simple; any
pseudo-simple object is quasi-simple; any quasi-simple object
is presimple; any presimple object is primary.

(7.7) Any reduced pseudo-simple object is simple; the radical
of any pseudo-simple object is simple.

(7.8)  Suppose C is locally disjunctable reducible. The
following are equivalent for an object X:
(a) X is pseudo-simple.
(b) X is quasi-simple. 
(c) X is presimple.
(d) The radical of X is simple.

(7.9) Suppose any coflat unipotent map 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.

8. Local Objects

A non-initial object X is called "local" if non-initial strong
subobjects of X has a non-initial intersection M. An epic
simple fraction of an integral object X  is called a "generic
residue" of X. A mono (or subobject) p: P --> X is called a
"residue" of X if P is a generic residue of  a prime of X. An
object is called "regular" if any disjunctable strong mono to it
is direct. An object is "analytically closed" if any epic analytic
mono to it is an isomorphism. 

(8.1) Suppose X is a local object with the strong subobject M
as above. Then M is the unique simple prime of X; any proper
fraction U of X is disjoint with M; M --> X is a local map.

(8.2) 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 subobject. 

(8.3) Any simple fraction and any simple prime is a residue;
any residue of an object is a maximal simple subobject (i.e. it
is not contained in any other simple subobject); any two
distinct residues of an object are disjoint with each other.

(8.4) Suppose p: P --> U is a residue and u: U --> X is a
fraction (resp. strong mono). Then u.p: P --> X is a residue
of X.

(8.5) 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.

(8.6) Suppose f: X --> Z is a local map and X is local. Then Z
is local.

(8.7)  Suppose f: P --> X is a map and P is simple. Then
(a) f is a local epi iff X is simple. 
(b) f is a local strong mono iff X is local with the simple
prime P.
(c) f is an epic fraction iff X is integral with the generic
residue P.

(8.8) Suppose C 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.

(8.9) Any sum of regular objects is regular; any extremal
quotient of a regular object is regular; any regular and
presimple object is analytically closed.

(8.10) Suppose C is a complete and cocomplete, well-
powered and co-well-powered analytic category. Then 
(a) The union of any family of subobjects consisting of 
regular objects is regular.
(b) The full subcategory of regular objects is a coreflective
subcategory.

(8.11) Suppose C is a locally disjunctable analytic category.
Then
(a) Any regular object is reduced.
(b) A regular object is integral iff it is simple.
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END OF SECOND PART