Subject: categories: Axioms of Algebraic Geometry

From: Zhaohua Luo <[email-masked]>
Date: Mon, 09 Nov 1998 21:01:09 -0500

Axioms of Algebraic Geometry I

                              Zhaohua Luo

                               (11/7/98)

                                (a draft)

The axioms of algebraic geometry given below consist of three (well
known) algebraic axioms (A1) - (A3) and three geometric axioms (G1) -
(G3), based on Diers's axioms of Zariski categories. The complete html
version of this note (with links) is available at
http://www.azd.com/axioms.html. Comments and suggestions are welcome.

Consider a functor U: A --> Set from a category A to the category Set of
sets.

Algebraic Axioms:
(Axiom A1) U has a left adjoint.
(Axiom A2) Any bijective morphism in A is an isomorphism.
(Axiom A3) Any pair of parallel morphisms in A has a surjective
coequalizer.

Recall that a functor satisfying the axioms (A1) - (A3) is an algebraic
functor, and the pair (A, U) is an algebraic category (or algebraic
construct, or quasivariety). An algebraic functor U is finitary if it
preserves direct colimits. Any algebraic functor is faithful. In the
following we shall regard A as a concrete category over Set via an
algebraic functor U, and identify an object X with its underlying set
U(X).

A difference of an object is an ordered pair (a, b) of elements of A,
denoted formally by a - b.
(a) A difference a - b is a zero if a = b.
(b) A difference a - b is a unit if for any morphism f: A --> B, f(a) =
f(b) implies that B is terminal.
(c) A difference a - b is nilpotent if for any morphism f: A --> B, f(a)
- f(b) is a unit implies that B is terminal.
(d) An object is reduced if it has no non-zero nilpotent difference; (A,
U) is reduced if any object is reduced.
(e) A morphism f: A --> B is flat if the pushout functor C/A --> C/B
along it preserves monomorphisms.
(f) A difference a - b is invertible (or disjunctable) if there is a
flat epimorphism  i: A --> A(a, b) such that i(a) - i(b) is a unit, and
any morphism j: A --> B factors through i if j(a) - j(b) is a unit.

Suppose U x V is the product of two objects U and V with the projections
u: U x V --> U and v: U x V --> V. The product U x V is co-universal if
for any morphism f: U x V --> Z, let Z --> ZU and Z --> ZV  be the
pushouts of u and v along f, then the induced morphism Z --> ZU  x ZV
is an isomorphism.

Geometric Axioms:
(Axiom G1) Any object has a unit difference.
(Axiom G2) The product of any two objects isco-universal.
(Axiom G3) Any difference of an object is invertible.

We call any functor U: A --> Set satisfying the above six axioms an
algebraic-geometric functor. An algebraic-geometric category is a pair
(A, U) consisting of a category A and an algebraic-geometric functor U
on A.

Remark. (cf. [Luo, Categorical Geometry]) (a) An algebraic-geometric
category is the opposite of an analytic geometry.
(b) A finitary algebraic-geometric category is the opposite of a
coherent analytic geometry.
(c) Any finitary algebraic-geometric category satisfies the first five
of the six axioms of Zariski categories defined in Diers's book
Categories of Commutative Algebras, Oxford University Press, 1992. The
sixth axiom simply means that the category is strict, i.e. the
Grothendieck topology defined by open subsets is subcanonical.

Example. The following categories are algebraic-geometric categories:
(a) The category of frames (non-finitary, reduced, non-strict).
(b) The category of distributive lattices (finitary, reduced,
non-strict).
(c) The category of Boolean algebras (finitary, reduced, strict).
(d) The category of commutative rings with identity (finitary,
non-reduced, strict).
(e) The category of reduced commutative rings (finitary, reduced,
strict).
(f) The category of commutative regular rings (finitary, reduced,
strict).