Back to Categorical Geometry Homepage
 Axioms of Algebraic Geometry Zhaohua Luo (11/7/98) (revised 11/20/98) 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.  Algebraic Axioms: (Axiom A1) U has free objects (or equivalently, 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). 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.  (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 a terminal object.  (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 a terminal object. (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 called flat if the pushout functor A/A --> A/B along it preserves monomorphisms. (f) An epimorphism i: A --> A(a, b) is called a localization of A at a difference a - b of A if i(a) - i(b) is a unit, and any morphism j: A --> B factors through i if j(a) - j(b) is a unit. (g) A difference a - b is invertible (or disjunctable) if it has a flat localization. Suppose U V is the product of two objects U and V with the projections u: U V --> U and v: U V --> V. The product U V is co-universal (or costable) if for any morphism f: U V --> 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. Geometric Axioms:  (Axiom G1) Any object has a unit difference. (Axiom G2) The product of any two objects is co-universal. (Axiom G3) Any difference of an object is invertible. 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') respectively: (a) (G1') The initial object Z has a unit difference (usually denoted by (0, 1)). (b) (G2') The "generic" product Z Z is co-universal. (c) (G3') The "generic" difference x - y of the free object Z[x, y] over the set of two elements x, y is invertible. Remark 4. In the short note [Idempotent] we introduced the notion of an idempotent. (Axiom G2) is equivalent to the following two conditions: (I1) The image of an idempotent under any morphism is an idempotent. (I2) If A = U V is the product of two objects U and V then the element [0U, 1V] of U V is an idempotent of A. 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 two conditions: (H1) Any difference of an object has a localization. (H2) If A is a subobject of an object B and a - b is a difference of A, then the induced morphism from the localization of A at a - b to the a localization of B at a - b is injective.  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, y]/(x-y). The following three conditions are important for the classification of algebraic geometries: (D1) q is a direct product factor. (D2) i is a direct product factor. (D3) i is a coequalizer. Note that (A, U) is reduced if it satisfies (D1). If (A, U) is finitary then it satisfies D2 iff its subcategory of reduced objects (which is also an algebraic geometry) satisfies D1.  Remark 7. (a) An algebraic geometry is the opposite of an analytic geometry. (b) A finitary algebraic geometry is the opposite of a coherent analytic geometry.  (c) Any finitary algebraic geometry satisfies the first five of the six axioms of Zariski categories. The sixth axiom simply means that the opposite of the finitary algebraic geometry is a strict coherent analytic geometry.  Example 7.1. The following categories are algebraic geometries: (a) The category of frames (non-finitary, reduced, with D3). (b) The category of distributive lattices (finitary, reduced, with D3). (c) The category of Boolean algebras (finitary, reduced, with D1). (d) The category of commutative rings with identity (finitary, non-reduced). (e) The category of reduced commutative rings (finitary, reduced, with D1). 