, we denote by $\beta_{i,j}(M)=\dim_K\mathrm{Tor}_{i}^{E}(M,K)_j$ the graded Betti numbers of
.
i1 : E=QQ[e_1..e_4,SkewCommutative=>true]
o1 = E
o1 : PolynomialRing, 4 skew commutative variable(s)
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i2 : F=E^{0,0}
2
o2 = E
o2 : E-module, free
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i3 : I_1=ideal(e_1*e_2,e_1*e_3,e_2*e_3)
o3 = ideal (e e , e e , e e )
1 2 1 3 2 3
o3 : Ideal of E
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i4 : I_2=ideal(e_1*e_2,e_1*e_3)
o4 = ideal (e e , e e )
1 2 1 3
o4 : Ideal of E
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i5 : M_1=createModule({I_1,I_2},F)
o5 = image | e_2e_3 e_1e_3 e_1e_2 0 0 |
| 0 0 0 e_1e_3 e_1e_2 |
2
o5 : E-module, submodule of E
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i6 : J=ideal(join(flatten entries gens I_1,{e_1*e_2*e_3}))
o6 = ideal (e e , e e , e e , e e e )
1 2 1 3 2 3 1 2 3
o6 : Ideal of E
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i7 : M_2=createModule({J,I_2},F)
o7 = image | e_2e_3 e_1e_3 e_1e_2 0 0 |
| 0 0 0 e_1e_3 e_1e_2 |
2
o7 : E-module, submodule of E
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i8 : M_1==M_2
o8 = true
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i9 : betti M_1==betti M_2
o9 = true
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i10 : minimalBettiNumbers M_1==minimalBettiNumbers M_2
o10 = true
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