Zariski Categories

Abstract We use the axiomatic method, which deals with categories equipped with a specified structure rather than categories described concretely by their objects and their morphisms. The so-called Zariski categories have such a structure, and we claim here that it is the basic structure of categories of commutative algebras. We shall prove the validity of this claim by developing, in an arbitrary Zariski category, elementary commutative algebra and algebraic geometry. The structure of Zariski categories relies on the notion of codisjunctors. This universal construction is introduced in order to describe the calculus of fractions and is used extensively. In Zariski categories, the congruences, also called effective equivalence relations, play the important role of ideals. A kind of De Morgan law for codisjunctors of congruences is assumed. Among the basic properties of Zariski categories are the flatness properties. We use the following definition: a morphism f:  A → B is flat in A if the pushout functor A/ A→ B/A along f preserves monomorphisms. Codisjunctors are assumed to be flat morphisms.