# 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 . 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 .

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

$f_{1}=xyz+x^{2}z,\ f_{2}=xy^{2}+x^{2}y$ have a syzygy with coefficient polynomials

$y,-z;$ this means that

$yf_{1}+(-z)f_{2}=0.$ In his article, Cayley makes use, in a special case, of what was later  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 , an introductory textbook on computations with polynomials, and , a more advanced text by the same authors which includes a computational approach to modules and resolutions.

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