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In mathematics, a syzygy (from Greek συζυγία 'pair') is a relation between the generators of a module ''M''. The set of all such relations is called the "first syzygy module of ''M''". A relation between generators of the first syzygy module is called a "second syzygy" of ''M'', and the set of all such relations is called the "second syzygy module of ''M''". Continuing in this way, we derive the ''n''th syzygy module of ''M'' by taking the set of all relations between generators of the (''n'' − 1)th syzygy module of ''M''. If ''M'' is finitely generated over a polynomial ring over a field, this process terminates after a finite number of steps; i.e., eventually there will be no more syzygies (see Hilbert's syzygy theorem). The syzygy modules of ''M'' are not unique, for they depend on the choice of generators at each step. The sequence of the successive syzygy modules of a module ''M'' is the sequence of the successive images (or kernels) in a free resolution of this module. Buchberger's algorithm for computing Gröbner bases allows the computation of the first syzygy module: The reduction to zero of the S-polynomial of a pair of polynomials in a Gröbner basis provides a syzygy, and these syzygies generate the first module of syzygies. ==Further reading== * * 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Syzygy (mathematics)」の詳細全文を読む スポンサード リンク
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