Difference between revisions of "Syzygy"
From Encyclopedia of Mathematics
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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 | + | 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 also [[Hilbert syzygy theorem]]. |
| + | ====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]]. | ||
| + | ====References==== | ||
| + | * David Eisenbud, ''The Geometry of Syzygies. A second course in commutative algebra and algebraic geometry'', Graduate Texts in Mathematics '''229''', Springer-Verlag (2005) {{ISBN|0-387-22232-4}} {{ZBL|1066.14001}} | ||
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Latest revision as of 16:48, 23 November 2023
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 also Hilbert syzygy theorem.
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.
References
- David Eisenbud, The Geometry of Syzygies. A second course in commutative algebra and algebraic geometry, Graduate Texts in Mathematics 229, Springer-Verlag (2005) ISBN 0-387-22232-4 Zbl 1066.14001
How to Cite This Entry:
Syzygy. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Syzygy&oldid=18722
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