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Macaulay2Doc :: syz(GroebnerBasis)

syz(GroebnerBasis) -- retrieve the syzygy matrix

Synopsis

Description

Warning: the result may be zero if syzygies were not to be retained during the calculation, or if the computation was not continued to a high enough degree.

The matrix of syzygies is returned without removing non-minimal syzygies.

i1 : R = QQ[a..g];
i2 : I = ideal"ab2-c3,abc-def,ade-bfg"

               2    3
o2 = ideal (a*b  - c , a*b*c - d*e*f, a*d*e - b*f*g)

o2 : Ideal of R
i3 : G = gb(I, Syzygies=>true);
i4 : syz G

o4 = {3} | -abc+def 0        -ade+bfg -d2e2f+b2cfg |
     {3} | ab2-c3   -ade+bfg 0        c3de-b3fg    |
     {3} | 0        abc-def  ab2-c3   -bc4+b2def   |

             3       4
o4 : Matrix R  <--- R
There appear to be 4 syzygies, but the last one is a combination of the first three:
i5 : syz gens I

o5 = {3} | -abc+def 0        -ade+bfg |
     {3} | ab2-c3   -ade+bfg 0        |
     {3} | 0        abc-def  ab2-c3   |

             3       3
o5 : Matrix R  <--- R
i6 : mingens image syz G

o6 = {3} | -abc+def 0        -ade+bfg |
     {3} | ab2-c3   -ade+bfg 0        |
     {3} | 0        abc-def  ab2-c3   |

             3       3
o6 : Matrix R  <--- R

See also