C.3 Syzygies and resolutions
SyzygiesLet be a quotient of and let be a submodule of . Then the module of syzygies (or 1st syzygy module, module of relations) of , syz(), is defined to be the kernel of the map .
The k-th syzygy module is defined inductively to be the module of syzygies of the -stsyzygy module.
Note, that the syzygy modules of depend on a choice of generators . But one can show that they depend on uniquely up to direct summands.
Free resolutionsLet and . A free resolution of is a long exact sequence
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.
Betti numbers and regularityLet be a graded ring (e.g., ) and let be a graded submodule. Let
be a minimal free resolution of considered with homogeneous maps of degree 0. Then the graded Betti number of is the minimal number of generators in degree of the -th syzygy module of (i.e., the -st syzygy module of ). Note that, by definition, the -th syzygy module of is and the 1st syzygy module of is .
The regularity of
is the smallest integer