View source for 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]].

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