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

${\displaystyle f_{1}=xyz+x^{2}z,\ f_{2}=xy^{2}+x^{2}y}$

have a syzygy with coefficient polynomials

${\displaystyle y,-z;}$

this means that

${\displaystyle yf_{1}+(-z)f_{2}=0.}$

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 ${\displaystyle (f_{1},\ldots ,f_{t})}$ as a row vector whose elements are polynomials, then every syzygy on ${\displaystyle (f_{1},\ldots ,f_{t})}$ corresponds to a column vector ${\displaystyle (a_{1},\ldots ,a_{t})^{T}}$ (whose elements ${\displaystyle a_{i}}$ are themselves polynomials) that satisfies ${\displaystyle (f_{1},\ldots ,f_{t})\cdot (a_{1},\ldots ,a_{t})^{T}=0}$. The set of all syzygies, i.e. all ${\displaystyle (a_{1},\ldots ,a_{t})^{T}\in R^{t}}$ such that ${\displaystyle a_{1}f_{1}+\cdots +a_{t}f_{t}=0}$, is an ${\displaystyle R}$-submodule of ${\displaystyle R^{t}}$, called the (first) syzygy module of ${\displaystyle (f_{1},\ldots ,f_{t})}$, and denoted ${\displaystyle Syz(f_{1},\ldots ,f_{t}).}$ For a Noetherian ring ${\displaystyle R}$, any submodule of a finitely generated ${\displaystyle R}$-module ${\displaystyle M}$ is finitely generated; thus, this syzygy module is finitely generated.

Notes

1. 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.
2. [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.
3. 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.
4. 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.
5. "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.