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.
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.
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). 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
mean a notation a - b where a, b are elements
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 --> ZU and Z --> ZV be the pushouts of u and v along f, then the induced morphism Z --> ZU ZV is an isomorphism.
Remark 3. Clearly the image of any unit (resp. invertible) difference
is a unit (resp. invertible). Thus (Axiom G1) - (Axiom G3)
are equivalent to the following conditions (G1') - (G3')
Remark 4. In the short note [Idempotent]
we introduced the notion of an idempotent. (Axiom G2) 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 G3) is equivalent to the following
Any functor U: A --> Set satisfying the six axioms (A1) - (A3) and (G1) - (G3) 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,
The following three conditions are important for the classification of
Remark 7. (a) An algebraic geometry is the opposite of an analytic
Example 7.1. The following categories are algebraic geometries:
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