abstract algebraic geometry

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

Date: Fri, 19 Dec 1997 12:22:05 -0800
From: Zhaohua Luo <zack@iswest.com>

The following is the third part of "The language of analytic
categories", which is a report on my paper CATEGORICAL
GEOMETRY. Again comments and suggestions are
welcome.
 

Z. Luo
_____________________________________________
THE LANGUAGE OF ANALYTIC CATEGORIES

By Zhaohua Luo (1997)
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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
Analytic Dictionary
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9. Analytic Geometries 

An "analytic geometry" is an analytic category satisfying the
following axioms:
(Axiom 4) Any intersection of strong subobjects exists.
(Axiom 5) Any non-initial object has a non-initial reduced
strong subobject.
(Axiom 6) Any strong subobject is an intersection of
disjunctable strong subobjects.
Thus an analytic geometry is a perfect, reducible, and locally
disjunctable analytic category. 

Suppose C is an analytic geometry.

(9.1) Any object has a radical; the full subcategory of reduced
subobjects is a reduced analytic geometry.

(9.2) If X is the join of two strong subobjects U and V in
R(X), then {U, V} is a unipotent cover on X.

(9.3) If U and V are two strong subobjects of an object X,
then rad(U \vee V) = rad(U) \vee rad(V).

(9.4)  Denote by D(X) the set of reduced strong subobjects of
X.  The radical mapping rad: R(X) --> D(X) is the right
adjoint of the inclusion D(X) --> R(X), which preserves finite
joins. 

(9.5) The dual D(X)^{op} of the lattice D(X) is a locale; a
reduced strong subobject is integral if and only if it is a prime
element of D(X)^{op}. 

(9.6) The spectrum Spec(X) of an object X is homeomorphic
to the space of points of  the locale D(X)^{op} (therefore is a
sober space); an analytic geometry is spatial iff D(X)^{op} is
a spatial locale for each object X.

(9.7) The functor sending each object X to D(X)^{op} and
each map f: Y --> X to rad(f)^{-1} is equivalent to the
analytic topology on C (cf. [L4]).

(9.8) If V is a strong subobject of a non-initial object X in a
spatial analytic geometry then the join of all the primes
contained in V is the radical of V.

(9.9) A non-initial reduced object X in a spatial analytic
geometry is integral iff its spectrum is irreducible.

(9.10) Suppose f: Y --> X is a mono in a spatial analytic
geometry. If f is coflat then Spec(f) is a topological
embedding; if f is analytic then Spec(f) is an open embedding; 
if f is strong then Spec(f) is a closed embedding.

(9.11) (Chinese remainder theorem) Let X be an object in a
strict analytic geometry. Suppose U_1, U_2, ..., U_n are
strong subobjects of X such that U_i, U_j are disjoint for all i
\neq j, then the induced map \sum U_i --> \vee U_i is an
isomorphism.
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10.  Zariski Geometries

Most of the results stated in this section are due to Diers (in
the dual situation). Our purpose is to present a geometric
approach using the language of analytic categories developed
above.

A category is "coherent" if  the following three axioms are
satisfied:
(Axiom 7) It is locally finitely copresentable.
(Axiom 8) Finite sums are disjoint and stable.
(Axiom 9) The sum of its terminal object with itself is finitely
copresentable.
It is easy to see that a coherent category is an analytic
category. A "Zariski geometry" (resp. "Stone geometry") is a
locally disjunctable (resp. locally decidable) coherent
category. 

Diers proved in [D1] that a locally finitely copresentable
category is a coherent category (resp. Stone geometry) iff its
full subcategory of finitely copresentable objects is lextensive
(resp. lextensive and decidable). Note that a category is a
coherent category (resp. Stone geometry) iff its opposite is a
"locally indecomposable category" (resp. "locally simple
category") in the sense of [D1]. 

Let C be a coherent category. A map f: Y --> X is called
"indirect" if it does not factor through any proper direct mono
to X. A non-initial object is "indecomposable" if it has exactly
two direct subobjects. A maximal indecomposable subobject
is called an "indecomposable component".  

(10.1)  Any non-initial object has a simple prime and an
extremal simple subobject; a coherent category is a spatial
reducible perfect analytic category.

(10.2) Cofiltered limits and products of coflat maps are
coflat; intersections of coflat monos are coflat monos;
intersections of fractions are fractions; any map can be
factored uniquely as a quasi-local map followed by a fraction.

(10.3) Any composite of locally direct mono is locally direct;
any map can be factored uniquely as an indirect map followed
by a locally direct mono.

(10.4) Any non-initial object has an indecomposable
component; an indecomposable subobject is an
indecomposable component iff it is a locally direct subobject.

(10.5) The extensive topology is naturally a strict metric
topology, which is determined by the canonical functor to the
category of Stone spaces (preserving cofiltered limits and
colimits whose right adjoint preserving sums).

(10.6) A Stone geometry is a strict reduced Zariski geometry
whose opposite is a regular category, and its analytic
topology coincides with the extensive topology. 

Let C be a Zariski geometry. A "locality" is a fraction with a
local object as domain. A "local isomorphism" is a map f: Y --
> X such that, for any locality v: V  --> Y, the composite f.v:
V  --> X is a locality. A complement of a set of strong monos
is called a "semisingular mono". Note that  (10.13) below
implies that our definitions of reduced and integral objects
coincide with those of Diers's in a Zariski geometry. 

(10.8) A Zariski geometry is a spatial analytic geometry; The
spectrum Spec(X) of any object is a coherent space for any
object X; if f: Y --> X is a unipotent map then Spec(f) is
surjective.

(10.9) If f: Y --> X is a finitely copresentable (i.e. f is a
finitely copresentable object in C/X) local isomorphism, then
Spec(f): Spec(Y) --> Spec(X) is an open map.

(10.10) A simple subobject on an object is a residue iff it is
maximal (i.e. it is not contained in any larger simple
subobject); any integral object X has a unique generic residue.
 
(10.11) (Going Up Theorem) If f: Y --> X is a coflat map and 
V is in the image of Spec(f), any prime of X containing V is
also in the image of Spec(f) (i.e. the image of Spec(f) is
closed under generalizations).

(10.12) Any colimits and cofiltered limits of reduced objects
is reduced; the full subcategory of reduced objects is a
reduced Zariski geometry.

(10.13) An object is integral (resp. reduced) iff it is a quotient
of a simple object (resp. a coproduct of simple objects).

(10.14) A Zariski geometry is strict iff any finite analytic
cover is not contained in any proper subobject. Suppose C is
strict. A mono is analytic iff it is singular (resp. a finitely
copresentable fraction); a mono is a fraction iff it is
semisingular (resp. a local isomorphism); a mono is direct iff
it is strong and analytic.

References

[D1] Diers, Y. Categories of Boolean Sheaves of Simple
Algebras, Lecture Notes in Mathematics Vol. 1187, Springer
Verlag, Berlin, 1986.

[D2] Diers, Y. Categories of Commutative Algebras, Oxford
University Press, 1992.

[L1] Luo, Z. On the geometry of metric sites, Journal of
Algebra 176, 210-229, 1995.

[L2] Luo, Z. On the geometry of framed sites, preprint, 1995.

[L3] Luo, Z.  Categorical Geometry, preprint, 1997.

[L4] Luo, Z. Abstract Algebraic Geometry, preprint, 1997.
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END OF THIRD PART