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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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