# Syzygy

From Encyclopedia of Mathematics

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://www.encyclopediaofmath.org/index.php?title=Syzygy&oldid=39057

This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article