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Difference between revisions of "Syzygy"

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An astronomical term denoting the disposition of three celestial bodies on a line.
 
An astronomical term denoting the disposition of three celestial bodies on a line.
  
In algebra it is used in the sense of a relationship. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091990/s0919901.png" /> be a left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091990/s0919902.png" />-module, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091990/s0919903.png" /> be a family of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091990/s0919904.png" />; a relationship, or syzygy, between the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091990/s0919905.png" /> is a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091990/s0919906.png" /> of elements of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091990/s0919907.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091990/s0919908.png" />. Thus there arises the module of syzygies, the chain complex of syzygies, etc. See [[Hilbert theorem|Hilbert theorem]] on syzygies.
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In algebra it is used in the sense of a relationship. Let $M$ be a left $A$-module, and let $(m_i)_{i\in I}$ be a family of elements of $M$; a relationship, or syzygy, between the $m_i$ is a set $(a_i)_{i\in I}$ of elements of the ring $A$ such that $\sum_{i\in I} a_i m_i = 0$. Thus there arises the module of syzygies, the chain complex of syzygies, etc. See [[Hilbert theorem]] on syzygies.
  
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====Comments====
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Syzygies appear in the definition of syzygetic ideals and the theory of regular algebras and regular sequences, cf. [[Koszul complex]]; [[Depth of a module]].
  
 
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====Comments====
 
Syzygies appear in the definition of syzygetic ideals and the theory of regular algebras and regular sequences, cf. [[Koszul complex|Koszul complex]]; [[Depth of a module|Depth of a module]].
 

Revision as of 11:47, 20 August 2016

An astronomical term denoting the disposition of three celestial bodies on a line.

In algebra it is used in the sense of a relationship. Let $M$ be a left $A$-module, and let $(m_i)_{i\in I}$ be a family of elements of $M$; a relationship, or syzygy, between the $m_i$ is a set $(a_i)_{i\in I}$ of elements of the ring $A$ such that $\sum_{i\in I} a_i m_i = 0$. Thus there arises the module of syzygies, the chain complex of syzygies, etc. See Hilbert theorem on syzygies.

Comments

Syzygies appear in the definition of syzygetic ideals and the theory of regular algebras and regular sequences, cf. Koszul complex; Depth of a module.

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How to Cite This Entry:
Syzygy. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Syzygy&oldid=18722
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article