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Macaulay2Doc :: eagonNorthcott(Matrix)

eagonNorthcott(Matrix) -- Eagon-Northcott complex of a matrix of linear forms

Synopsis

Description

The Eagon-Northcott complex is an explicit chain complex that gives a minimal projective resolution of the cokernel of the matrix maximal minors of a generic matrix of linear forms.
i1 : R = QQ[a..z]

o1 = R

o1 : PolynomialRing
i2 : f = genericMatrix(R,3,5)

o2 = | a d g j m |
     | b e h k n |
     | c f i l o |

             3       5
o2 : Matrix R  <--- R
i3 : M = coker gens minors_3 f

o3 = cokernel | -ceg+bfg+cdh-afh-bdi+aei -cej+bfj+cdk-afk-bdl+ael -chj+bij+cgk-aik-bgl+ahl -fhj+eij+fgk-dik-egl+dhl -cem+bfm+cdn-afn-bdo+aeo -chm+bim+cgn-ain-bgo+aho -fhm+eim+fgn-din-ego+dho -ckm+blm+cjn-aln-bjo+ako -fkm+elm+fjn-dln-ejo+dko -ikm+hlm+ijn-gln-hjo+gko |

                            1
o3 : R-module, quotient of R
i4 : C = res M

      1      10      15      6
o4 = R  <-- R   <-- R   <-- R  <-- 0
                                    
     0      1       2       3      4

o4 : ChainComplex
i5 : D = eagonNorthcott f

      1      10      15      6
o5 = R  <-- R   <-- R   <-- R  <-- 0
                                    
     0      1       2       3      4

o5 : ChainComplex
i6 : H = prune HH D

o6 = 0 : cokernel | ikm-hlm-ijn+gln+hjo-gko fkm-elm-fjn+dln+ejo-dko
         ckm-blm-cjn+aln+bjo-ako fhm-eim-fgn+din+ego-dho
         chm-bim-cgn+ain+bgo-aho cem-bfm-cdn+afn+bdo-aeo
         fhj-eij-fgk+dik+egl-dhl chj-bij-cgk+aik+bgl-ahl
         cej-bfj-cdk+afk+bdl-ael ceg-bfg-cdh+afh+bdi-aei |

     1 : 0                                                         

     2 : 0                                                         

     3 : 0                                                         

     4 : 0                                                         

o6 : GradedModule
i7 : assert( H_0 == M and H_1 == 0 and H_2 == 0 and H_3 == 0 )
This function was written by Greg Smith.