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 $ \mathbf A $ satisfying the following six axioms:

1) $ \mathbf A $ is co-complete, i.e., has all small colimits;

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

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

4) the terminal object of $ \mathbf A $ is finitely presentable and has no proper subobject;

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

6) for any finite sequence of co-disjunctable congruences $ r _ {1} \dots r _ {n} $ on any object $ A \in \mathbf A $, with respective co-disjunctors $ d _ {1} \dots d _ {n} $, one has

$$ r _ {1} \lor \dots \lor r _ {n} = 1 _ {A \times A } \Rightarrow d _ {1} \lor \dots \lor d _ {n} = 1 _ {A \times A } , $$

where $ \lor $ denotes union of congruences on $ A $ on the left-hand side and co-union of quotient objects of $ A $ 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 $ ( g,h ) : C \aRR A $ be an arbitrary pair of parallel morphisms in $ \mathbf A $. It is said to be co-disjointed if its co-equalizer is a strict terminal object. A morphism $ f : A \rightarrow B $ co-disjoints $ ( g,h ) $ if $ ( fg,fh ) $ is co-disjointed. A co-disjunctor of $ ( g,h ) $ is a morphism $ d : A \rightarrow D $ that co-disjoints $ ( g,h ) $ and through which any morphism $ f $ that co-disjoints $ ( g,h ) $ factors uniquely. The pair $ ( g,h ) $ is co-disjunctable if it has a co-disjunctor. The object $ A $ is co-disjunctable if the pair of inductions $ A \aRR A \amalg A $ of $ A $ into its coproduct by itself is co-disjunctable. It is flatly co-disjunctable if, moreover, the co-disjunctor $ d $ of $ A \aRR A \amalg A $ is a flat morphism, i.e., the pushout functor along $ d $ 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