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C.3 Syzygies and resolutions

Syzygies

The k-th syzygy module is defined inductively to be the module of syzygies of the syzygy module.

Example:
 

Free resolutions


where the columns of the matrix generate . Note that resolutions need not to be finite (i.e., of finite length). The Hilbert Syzygy Theorem states that for there exists a ("minimal") resolution of length not exceeding the number of variables.

Example:
 

Betti numbers and regularity

The regularity of is the smallest integer such that

Example:
 

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            User manual for Singular version 3-1-6, Dec 2012, generated by texi2html.