Jacobson category

An abstract model of the category of Jacobson-commutative algebras (cf. Commutative algebra). It is a special instance of a Zariski category. To be precise, it a Zariski category in which all objects are Jacobson objects, i.e., objects whose prime congruences are meets of maximal congruences. Jacobson categories are axiomatically defined as categories $ \mathbf A $ satisfying the following five 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 directly co-disjunctable;

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

The notion of a directly co-disjunctable object used in 2) is defined as follows. A morphism $ d : A \rightarrow D $ in $ \mathbf A $ is a direct factor morphism if there is a morphism $ {d ^ \prime } : A \rightarrow {D ^ \prime } $ in $ \mathbf A $ such that $ ( d : A \rightarrow D , {d ^ \prime } : A \rightarrow {D ^ \prime } ) $ is a product of objects in $ \mathbf A $. An object $ A \in \mathbf A $ is directly co-disjunctable if the pair of inductions $ A \aRR A \amalg A $ of $ A $ into its coproduct by itself has a co-disjunctor (cf. Zariski category) that is a direct factor morphism.

What makes Jacobson categories special is the fact that co-disjunctors are co-equalizers or, more concretely, that algebras of fractions are quotient algebras. It follows that integral or local reduced algebras are fields, that reduced algebras are von Neumann regular (cf. also Regular ring (in the sense of von Neumann)), that prime ideals are maximal, that prime spectra are Hausdorff and Boolean spaces, that the set of finitely generated ideals of an algebra is equipped with, next to the usual addition and multiplication of ideals, an extra multiplication of ideals, giving it the structure of a Stone semi-algebra, etc.

A typical example of a Jacobson category is the category $ { \mathop{\rm Alg} } \mathbf C { \mathop{\rm Alg} } ( k ) $ of algebraic commutative algebras over a commutative ring $ k $( cf. also Algebraic algebra), i.e., algebras whose elements are algebraic over $ k $. More generally, the category of algebraic objects of any Zariski category $ \mathbf A $ is a Jacobson category, and is indeed the universal Jacobson category associated to $ \mathbf A $[a2].

References

[a1]	Y. Diers, "Jacobson categories" J. Algebra , 168 : 3 (1994) pp. 949–967
[a2]	Y. Diers, "Clear objects in categories of commutative algebras" J. Pure Appl. Algebra , 10 (1995) pp. 155–171