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7.4.3 Syzygies and resolutions (plural)

Syzygies


Remark: With respect to the definitions of ideal and module (see PLURAL), PLURAL works with left syzygies only (by syz we understand a left syzygy). If S is a matrix of a left syzygy module of left submodule given by matrix M, then transpose(S)*transpose(M) = 0 (but, in general,

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 resolutions


with

and where the columns of the matrix generate . Note, that resolutions over factor-algebras need not to be of finite length.

Generalized Hilbert Syzygy Theorem

variables, there exists a free resolution of length smaller or equal than .

Example:
 


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