minimalBettiNumbers -- compute the minimal Betti numbers of a given graded ideal

i1 : E=QQ[e_1..e_4,SkewCommutative=>true]

o1 = E

o1 : PolynomialRing, 4 skew commutative variables

i2 : I=ideal(e_1*e_2,e_1*e_3,e_2*e_3)

o2 = ideal (e e , e e , e e )
             1 2   1 3   2 3

o2 : Ideal of E

i3 : J=ideal(join(flatten entries gens I,{e_1*e_2*e_3}))

o3 = ideal (e e , e e , e e , e e e )
             1 2   1 3   2 3   1 2 3

o3 : Ideal of E

i4 : I==J

o4 = true

i5 : betti I==betti J

o5 = false

i6 : minimalBettiNumbers I==minimalBettiNumbers J

o6 = true