View source for Zariski category

An abstract model of the [[Category|category]] of commutative algebras (cf. [[Commutative algebra|Commutative algebra]]) in which basic commutative algebra and [[Algebraic geometry|algebraic geometry]] can be performed. Zariski categories are axiomatically defined as categories <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z110/z110010/z1100101.png" /> satisfying the following six axioms:

1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z110/z110010/z1100102.png" /> is co-complete, i.e., has all small colimits;

2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z110/z110010/z1100103.png" /> has a strong generating set of objects whose objects are finitely presentable and flatly co-disjunctable;

3) regular epimorphisms in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z110/z110010/z1100104.png" /> are universal, i.e., stable under pull-backs along any morphism (cf. [[Morphism|Morphism]]);

4) the terminal object of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z110/z110010/z1100105.png" /> is finitely presentable and has no proper subobject;

5) the product of two objects in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z110/z110010/z1100106.png" /> is co-universal, i.e., stable under pushouts along any morphism;

6) for any finite sequence of co-disjunctable congruences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z110/z110010/z1100107.png" /> on any object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z110/z110010/z1100108.png" />, with respective co-disjunctors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z110/z110010/z1100109.png" />, one has

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z110/z110010/z11001010.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z110/z110010/z11001011.png" /> denotes union of congruences on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z110/z110010/z11001012.png" /> on the left-hand side and co-union of quotient objects of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z110/z110010/z11001013.png" /> on the right-hand side.

These axioms rely on the notions of limits, colimits, finitely presentable objects, and co-disjunctors [[#References|[a1]]]. This latter notion describes in a universal way the calculus of fractions, and is defined as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z110/z110010/z11001014.png" /> be an arbitrary pair of parallel morphisms in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z110/z110010/z11001015.png" />. It is said to be co-disjointed if its co-equalizer is a strict terminal object. A morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z110/z110010/z11001016.png" /> co-disjoints <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z110/z110010/z11001017.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z110/z110010/z11001018.png" /> is co-disjointed. A co-disjunctor of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z110/z110010/z11001019.png" /> is a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z110/z110010/z11001020.png" /> that co-disjoints <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z110/z110010/z11001021.png" /> and through which any morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z110/z110010/z11001022.png" /> that co-disjoints <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z110/z110010/z11001023.png" /> factors uniquely. The pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z110/z110010/z11001024.png" /> is co-disjunctable if it has a co-disjunctor. The object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z110/z110010/z11001025.png" /> is co-disjunctable if the pair of inductions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z110/z110010/z11001026.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z110/z110010/z11001027.png" /> into its [[Coproduct|coproduct]] by itself is co-disjunctable. It is flatly co-disjunctable if, moreover, the co-disjunctor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z110/z110010/z11001028.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z110/z110010/z11001029.png" /> is a [[Flat morphism|flat morphism]], i.e., the pushout functor along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z110/z110010/z11001030.png" /> 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====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> Y. Diers, "Categories of commutative algebras" , Oxford Univ. Press (1992) {{MR|1181196}} {{ZBL|0772.18001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> Y. Diers, "The Zariski category of graded commutative rings" ''Canad. Math. Soc. Conf. Proc.'' , '''13''' (1992) pp. 171–181 {{MR|1192146}} {{MR|1186946}} {{ZBL|0798.18008}} {{ZBL|0777.18003}} </TD></TR></table>