Zhaohua Luo (11/7/98) (revised 11/20/98) The axioms of algebraic geometry given below consist of three (well known) algebraic axioms (A_{1}) - (A_{3}) and three geometric axioms (G_{1}) - (G_{3}), based on Diers's axioms of Zariski categories. Consider a faithful functor U: A --> Set from a category A to the category Set of sets. In the following we shall regard A as a concrete category over Set via the faithful functor U, and identify an object X with its underlying set U(X). A subset S of an object A is called a free generating set of A if for any function t: S --> B from S to an object B there is a unique morphism A --> B whose restriction on S is t, in which case we say that X is a free object on the set S. We say U has free objects if the free objects on any set exists. Algebraic Axioms:
Recall that a functor satisfying the axioms (A_{1}) - (A_{3}) is an algebraic functor, and the pair (A, U) is an algebraic category (or algebraic construct, or quasivariety). It is well known that any algebraic category is complete, cocomplete and regular (see [Luo, Algebraic Categories). An algebraic functor U is finitary if it preserves direct colimits. Remark 1. Note that although we assume U is faithful at the beginning, in fact any algebraic functor is necessary faithful (the best reference for the theory of algebraic functors is [Herrlich and Strecker]). Definition 2. By a difference of an object
A
we
mean a notation a - b where a, b are elements
of A.
Suppose UV is the product of two objects U and V with the projections u: U V --> U and v: UV --> V. The product UV is co-universal (or costable) if for any morphism f: UV --> Z, let Z --> Z_{U} and Z --> Z_{V} be the pushouts of u and v along f, then the induced morphism Z --> Z_{U} Z_{V} is an isomorphism. Geometric Axioms:
Remark 3. Clearly the image of any unit (resp. invertible) difference
is a unit (resp. invertible). Thus (Axiom G_{1}) - (Axiom G_{3})
are equivalent to the following conditions (G_{1}') - (G_{3}')
respectively:
Remark 4. In the short note [Idempotent]
we introduced the notion of an idempotent. (Axiom G_{2}) is equivalent
to the following two conditions:
Suppose f: A --> B is a morphism and a - b is a difference of A. Suppose i: A --> A_{(a, b) } and j: B -->B_{(f(a), f(b))} are the localizations. Since jf(a) - jf(b) is a unit of B_{(f(a), f(b)),} by the universal property of localization there is a unique morphism k: A_{(a, b) }-->B_{(f(a), f(b))} such that jf = ki, called the induced morphism. Remark 5. (Axiom G_{3}) is equivalent to the following
two conditions:
Any functor U: A --> Set satisfying the six axioms (A_{1}) - (A_{3}) and (G_{1}) - (G_{3}) is an algebraic-geometric functor. An algebraic geometry is a pair (A, U) consisting of a category A and an algebraic-geometric functor U on A. Remark 6. Suppose (A, U) is an algebraic geometry.
Consider the generic localization i: Z[x, y]
--> Z[x, y]_{x - y} and the generic
coequalizer q: Z[x, y] --> Z[x,
y]/(x-y).
The following three conditions are important for the classification of
algebraic geometries:
Remark 7. (a) An algebraic geometry is the opposite of an analytic
geometry.
Example 7.1. The following categories are algebraic geometries:
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