Commutative algebra: Difference between revisions

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Commutative algebra is a theory in mathematics having the aim of translating classical geometric ideas into an algebraic framework, pioneered by [[David Hilbert]] and [[Emily Noether]] at the beginning of the 20th century.   
Commutative algebra developed as a theory in mathematics having the aim of translating classical geometric ideas into an algebraic framework, pioneered by [[David Hilbert]] and [[Emmy Noether]] at the beginning of the 20th century.   


==Definitions and major results==
==Definitions and major results==
The notion of ''commutative [[ring]]'' assumes commutativity of the multiplication operation and usually the existence of a multiplicative identity in addition.
The notion of ''commutative [[ring (mathematics)|ring]]'' assumes commutativity of the multiplication operation and usually the existence of a multiplicative identity in addition.


The category of commutative rings has  
The [[Category theory|category]] of commutative rings has  
#commutative rings as its objects
#commutative rings as its objects
#ring homomorphisms as its morphisms; i.e., functions <math>\phi:R\to R'</math> such that <math>\phi</math> is a morphism of abelian groups (with respect to the additive structure of the rings <math>R</math> and<math>R'</math>), <math>\phi(r_1r_2)=\phi(r_1)\phi(r_2)</math> for all <math>r_1,r_2\in R</math>, and <math>\phi(1_R)=1_{R'}</math>.
#ring homomorphisms as its morphisms; i.e., functions <math>\phi:R\to R'</math> such that <math>\phi</math> is a morphism of abelian groups (with respect to the additive structure of the rings <math>R</math> and<math>R'</math>), <math>\phi(r_1r_2)=\phi(r_1)\phi(r_2)</math> for all <math>r_1,r_2\in R</math>, and <math>\phi(1_R)=1_{R'}</math>.


==Affine Schemes==
==Affine Schemes==
The theory of [[Affine scheme|affine schemes]] was initiated with the definition of the ''prime spectrum'' of a ring, the set of all prime ideals of a given ring.  For curves defined by polynomial equations over a ring <math>A</math>, the object to consider would be the prime spectrum of a polynomial ring in sufficiently many variables modulo the principal ideal generated by the polynomials in question.  The Zariski topology (together with a structural sheaf of rings) on this set endows a geometric structure for which many illuminating algebro-geometric correspondences manifest themselves. For example, for a noetherian ring <math>A</math>, primary decomposition of an  
The theory of [[Affine scheme|affine schemes]] was initiated with the definition of the ''prime spectrum'' of a ring, the set of all [[prime ideal]]s of a given ring.  For curves defined by [[polynomial equation]]s over a ring <math>A</math>, the object to consider would be the prime spectrum of a [[polynomial ring]] in sufficiently many variables modulo the ideal generated by the polynomials in question.  The [[Zariski topology]] (together with a structural sheaf of rings) on this set endows a geometric structure for which many illuminating algebro-geometric correspondences manifest themselves. For example, for a [[noetherian ring]] <math>A</math>, primary decomposition of an [[ideal]] <math>I</math> translates exactly into a decomposition of the closed subset <math>V(I)</math> into irreducible components.  
ideal <math>I</math> translates exactly into a decomposition of the closed subset <math>V(I)</math> into irreducible components.  


Formally speaking, the assignment of a ring <math>A</math> to its prime spectrum <math>Spec(A)</math> is functorial, and is in fact an equivalence of categories between the category of commutative rings and affine schemes.  It is this mechanism, in addition to a number of correspondence theorems, which allows us to change between the language of algebra and geometry.  
Formally speaking, the assignment of a ring <math>A</math> to its prime spectrum <math>Spec(A)</math> is functorial, and is in fact an equivalence of categories between the category of commutative rings and affine schemes.  It is this mechanism, in addition to a number of correspondence theorems, which allows us to change between the language of algebra and geometry.


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==Computational commutative algebra==
==Computational commutative algebra==


==Combinatorial commutative algebra==
==Combinatorial commutative algebra==
 
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Commutative algebra developed as a theory in mathematics having the aim of translating classical geometric ideas into an algebraic framework, pioneered by David Hilbert and Emmy Noether at the beginning of the 20th century.

Definitions and major results

The notion of commutative ring assumes commutativity of the multiplication operation and usually the existence of a multiplicative identity in addition.

The category of commutative rings has

  1. commutative rings as its objects
  2. ring homomorphisms as its morphisms; i.e., functions such that is a morphism of abelian groups (with respect to the additive structure of the rings and), for all , and .

Affine Schemes

The theory of affine schemes was initiated with the definition of the prime spectrum of a ring, the set of all prime ideals of a given ring. For curves defined by polynomial equations over a ring , the object to consider would be the prime spectrum of a polynomial ring in sufficiently many variables modulo the ideal generated by the polynomials in question. The Zariski topology (together with a structural sheaf of rings) on this set endows a geometric structure for which many illuminating algebro-geometric correspondences manifest themselves. For example, for a noetherian ring , primary decomposition of an ideal translates exactly into a decomposition of the closed subset into irreducible components.

Formally speaking, the assignment of a ring to its prime spectrum is functorial, and is in fact an equivalence of categories between the category of commutative rings and affine schemes. It is this mechanism, in addition to a number of correspondence theorems, which allows us to change between the language of algebra and geometry.