Syzygy

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.

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