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.