# Syzygy (mathematics)

In algebra, the **syzygy** refer to relations between generators of modules, then relations between those relations and so forth.

Hilbert's syzygy theorem states the existence of free syzygy for ideals of a polynomial ring.

## History

The word syzygy came into mathematics in the work of Arthur Cayley [1]. In that paper, Cayley applied the theory to the study of resultants and discriminants; a modern account, including a reprint of his paper, is found in [2].

In astronomy, the word syzygy is used to denote a linear relation between planets; Cayley used it to denote a linear relation between polynomials in several variables with polynomial coefficients. For example, the polynomials

have a *syzygy* with coefficient polynomials

this means that

In his article, Cayley makes use, in a special case, of what was later [3] called the Koszul complex after a similar construction in differential geometry by the mathematician Jean-Louis Koszul.

## Definitions

The notation of this section appears in [4], an introductory textbook on computations with polynomials, and [5], a more advanced text by the same authors which includes a computational approach to modules and resolutions.

If we think of as a row vector whose elements are polynomials, then every **syzygy** on corresponds to a column vector (whose elements are themselves polynomials) that satisfies . The set of all syzygies, i.e. all such that , is an -submodule of , called the (first) **syzygy module** of , and denoted For a Noetherian ring , any submodule of a finitely generated -module is finitely generated; thus, this syzygy module is finitely generated.

## See also

## Notes

- 1847[Cayley 1847] A. Cayley, “On the theory of involution in geometry”, Cambridge Math. J. 11 (1847), 52–61. See also Collected Papers, Vol. 1 (1889), 80–94, Cambridge Univ. Press, Cambridge.
- [Gel’fand et al. 1994] I. M. Gel’fand, M. M. Kapranov, and A. V. Zelevinsky, Discriminants, resultants, and multidimensional determinants, Mathematics: Theory & Applications, Birkhäuser, Boston, 1994.
- Serre, Jean-Pierre Algèbre locale. Multiplicités. (French) Cours au Collège de France, 1957–1958, rédigé par Pierre Gabriel. Seconde édition, 1965. Lecture Notes in Mathematics, 11 Springer-Verlag, Berlin-New York 1965 vii+188 pp.; this is the published form of mimeographed notes from Serre's lectures at the College de France in 1958.
- Cox, David; Little, John; O’Shea, Donal (2007). "Ideals, Varieties, and Algorithms".
*Undergraduate Texts in Mathematics*. New York, NY: Springer New York. doi:10.1007/978-0-387-35651-8. ISBN 978-0-387-35650-1. ISSN 0172-6056. - "Using Algebraic Geometry".
*Graduate Texts in Mathematics*. New York: Springer-Verlag. 2005. doi:10.1007/b138611. ISBN 0-387-20706-6.

## References

- Eisenbud, David (1995).
*Commutative Algebra with a View Toward Algebraic Geometry*. Graduate Texts in Mathematics.**150**. Springer-Verlag. doi:10.1007/978-1-4612-5350-1. ISBN 0-387-94268-8. - David Eisenbud, The Geometry of Syzygies, Graduate Texts in Mathematics, vol. 229, Springer, 2005.