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An abstract model of the [[Category|category]] of Jacobson-commutative algebras (cf. [[Commutative algebra|Commutative algebra]]). It is a special instance of a [[Zariski category|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110020/j1100201.png" /> satisfying the following five axioms:
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1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110020/j1100202.png" /> is co-complete, i.e., has all small colimits;
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2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110020/j1100203.png" /> has a strong generating set of objects whose objects are finitely presentable and directly co-disjunctable;
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An abstract model of the [[Category|category]] of Jacobson-commutative algebras (cf. [[Commutative algebra|Commutative algebra]]). It is a special instance of a [[Zariski category|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 $
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satisfying the following five axioms:
  
3) regular epimorphisms in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110020/j1100204.png" /> are universal, i.e., stable under pull-backs along any morphism (cf. also [[Morphism|Morphism]]);
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1) $  \mathbf A $
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is co-complete, i.e., has all small colimits;
  
4) the [[terminal object]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110020/j1100205.png" /> is finitely presentable and has no proper subobject;
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2) $  \mathbf A $
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has a strong generating set of objects whose objects are finitely presentable and directly co-disjunctable;
  
5) the product of two objects in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110020/j1100206.png" /> is co-universal, i.e., stable under pushouts along any morphism.
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3) regular epimorphisms in $  \mathbf A $
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are universal, i.e., stable under pull-backs along any morphism (cf. also [[Morphism|Morphism]]);
  
The notion of a directly co-disjunctable object used in 2) is defined as follows. A [[Morphism|morphism]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110020/j1100207.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110020/j1100208.png" /> is a direct factor morphism if there is a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110020/j1100209.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110020/j11002010.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110020/j11002011.png" /> is a product of objects in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110020/j11002012.png" />. An object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110020/j11002013.png" /> is directly co-disjunctable if the pair of inductions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110020/j11002014.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110020/j11002015.png" /> into its [[Coproduct|coproduct]] by itself has a co-disjunctor (cf. [[Zariski category|Zariski category]]) that is a direct factor morphism.
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4) the [[terminal object]] of  $  \mathbf A $
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is finitely presentable and has no proper subobject;
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5) the product of two objects in  $  \mathbf A $
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is co-universal, i.e., stable under pushouts along any morphism.
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The notion of a directly co-disjunctable object used in 2) is defined as follows. A [[Morphism|morphism]] $  d : A \rightarrow D $
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in $  \mathbf A $
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is a direct factor morphism if there is a morphism $  {d  ^  \prime  } : A \rightarrow {D  ^  \prime  } $
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in $  \mathbf A $
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such that $  ( d : A \rightarrow D , {d  ^  \prime  } : A \rightarrow {D  ^  \prime  } ) $
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is a product of objects in $  \mathbf A $.  
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An object $  A \in \mathbf A $
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is directly co-disjunctable if the pair of inductions $  A \aRR A \amalg A $
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of $  A $
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into its [[Coproduct|coproduct]] by itself has a co-disjunctor (cf. [[Zariski category|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)|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.
 
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)|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110020/j11002016.png" /> of algebraic commutative algebras over a commutative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110020/j11002017.png" /> (cf. also [[Algebraic algebra|Algebraic algebra]]), i.e., algebras whose elements are algebraic over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110020/j11002018.png" />. More generally, the category of algebraic objects of any Zariski category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110020/j11002019.png" /> is a Jacobson category, and is indeed the universal Jacobson category associated to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110020/j11002020.png" /> [[#References|[a2]]].
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A typical example of a Jacobson category is the category $  { \mathop{\rm Alg} } \mathbf C { \mathop{\rm Alg} } ( k ) $
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of algebraic commutative algebras over a commutative ring $  k $(
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cf. also [[Algebraic algebra|Algebraic algebra]]), i.e., algebras whose elements are algebraic over $  k $.  
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More generally, the category of algebraic objects of any Zariski category $  \mathbf A $
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is a Jacobson category, and is indeed the universal Jacobson category associated to $  \mathbf A $[[#References|[a2]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  Y. Diers,  "Jacobson categories"  ''J. Algebra'' , '''168''' :  3  (1994)  pp. 949–967</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  Y. Diers,  "Clear objects in categories of commutative algebras"  ''J. Pure Appl. Algebra'' , '''10'''  (1995)  pp. 155–171</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  Y. Diers,  "Jacobson categories"  ''J. Algebra'' , '''168''' :  3  (1994)  pp. 949–967</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  Y. Diers,  "Clear objects in categories of commutative algebras"  ''J. Pure Appl. Algebra'' , '''10'''  (1995)  pp. 155–171</TD></TR></table>

Latest revision as of 22:14, 5 June 2020


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
How to Cite This Entry:
Jacobson category. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Jacobson_category&oldid=47461
This article was adapted from an original article by Y. Diers (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article