
7.4.3 Syzygies and resolutions (plural)SyzygiesLet be a GRalgebra. A left (resp. right) syzygy between elements is a tuple satisfyingThe set of all left (resp. right) syzygies between is a left (resp. right) submodule of .
Note, that
If 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 sequencewith and where the columns of the matrix generate . Note, that resolutions over factoralgebras need not to be of finite length. Generalized Hilbert Syzygy TheoremFor a algebra , generated by variables, there exists a free resolution of length smaller or equal than .
