Zariski category

An abstract model of the category of commutative algebras (cf. Commutative algebra) in which basic commutative algebra and algebraic geometry can be performed. Zariski categories are axiomatically defined as categories satisfying the following six axioms:

1) is co-complete, i.e., has all small colimits;

2) has a strong generating set of objects whose objects are finitely presentable and flatly co-disjunctable;

3) regular epimorphisms in are universal, i.e., stable under pull-backs along any morphism (cf. Morphism);

4) the terminal object of is finitely presentable and has no proper subobject;

5) the product of two objects in is co-universal, i.e., stable under pushouts along any morphism;

6) for any finite sequence of co-disjunctable congruences on any object , with respective co-disjunctors , one has

where denotes union of congruences on on the left-hand side and co-union of quotient objects of on the right-hand side.

These axioms rely on the notions of limits, colimits, finitely presentable objects, and co-disjunctors [a1]. This latter notion describes in a universal way the calculus of fractions, and is defined as follows. Let be an arbitrary pair of parallel morphisms in . It is said to be co-disjointed if its co-equalizer is a strict terminal object. A morphism co-disjoints if is co-disjointed. A co-disjunctor of is a morphism that co-disjoints and through which any morphism that co-disjoints factors uniquely. The pair is co-disjunctable if it has a co-disjunctor. The object is co-disjunctable if the pair of inductions of into its coproduct by itself is co-disjunctable. It is flatly co-disjunctable if, moreover, the co-disjunctor of is a flat morphism, i.e., the pushout functor along preserves monomorphisms.

The whole of basic commutative algebra and algebraic geometry can be formally developed in any Zariski category as if its objects were commutative rings. Any result proven in an arbitrary Zariski category has various interpretations in various concrete categories of commutative algebras of different kinds, possible equipped with some extra structure such as an order, lattice, gradation, filtration, differentiation, etc.

References

[a1]	Y. Diers, "Categories of commutative algebras" , Oxford Univ. Press (1992) MR1181196 Zbl 0772.18001
[a2]	Y. Diers, "The Zariski category of graded commutative rings" Canad. Math. Soc. Conf. Proc. , 13 (1992) pp. 171–181 MR1192146 MR1186946 Zbl 0798.18008 Zbl 0777.18003