i1 : R = ZZ/101[a,b,c,d]/ideal{a^4,b^4,c^4,d^4}
o1 = R
o1 : QuotientRing
|
i2 : A = koszulComplexDGA(R)
o2 = {Ring => R }
Underlying algebra => R[T , T , T , T ]
1 2 3 4
Differential => {a, b, c, d}
isHomogeneous => true
o2 : DGAlgebra
|
i3 : apply(maxDegree A + 1, i -> numgens prune homology(i,A))
o3 = {1, 4, 6, 4, 1}
o3 : List
|
i4 : HA = homologyAlgebra(A) Computing generators in degree 1 : -- used 0.00125506 seconds Computing generators in degree 2 : -- used 0.00889616 seconds Computing generators in degree 3 : -- used 0.00828974 seconds Computing generators in degree 4 : -- used 0.00758134 seconds Finding easy relations : -- used 0.0142152 seconds Computing relations in degree 1 : -- used 0.00178136 seconds Computing relations in degree 2 : -- used 0.00180282 seconds Computing relations in degree 3 : -- used 0.0017946 seconds Computing relations in degree 4 : -- used 0.00175833 seconds Computing relations in degree 5 : -- used 0.00161858 seconds o4 = HA o4 : PolynomialRing |
i5 : R = ZZ/101[a,b,c,d]/ideal{a^4,b^4,c^4,d^4,a^3*b^3*c^3*d^3}
o5 = R
o5 : QuotientRing
|
i6 : A = koszulComplexDGA(R)
o6 = {Ring => R }
Underlying algebra => R[T , T , T , T ]
1 2 3 4
Differential => {a, b, c, d}
isHomogeneous => true
o6 : DGAlgebra
|
i7 : apply(maxDegree A + 1, i -> numgens prune homology(i,A))
o7 = {1, 5, 10, 10, 4}
o7 : List
|
i8 : HA = homologyAlgebra(A) Computing generators in degree 1 : -- used 0.00120644 seconds Computing generators in degree 2 : -- used 0.00938421 seconds Computing generators in degree 3 : -- used 0.0100051 seconds Computing generators in degree 4 : -- used 0.0107081 seconds Finding easy relations : -- used 0.097418 seconds Computing relations in degree 1 : -- used 0.00904957 seconds Computing relations in degree 2 : -- used 0.0188779 seconds Computing relations in degree 3 : -- used 0.00891923 seconds Computing relations in degree 4 : -- used 0.00909257 seconds Computing relations in degree 5 : -- used 0.00924881 seconds o8 = HA o8 : QuotientRing |
i9 : numgens HA o9 = 19 |
i10 : HA.cache.cycles
3 3 3 3 2 3 3 3 2 3 3 3 3 2 3 3
o10 = {a T , b T , c T , d T , a b c d T , a b c d T T , a b c d T T ,
1 2 3 4 1 1 2 1 2
-----------------------------------------------------------------------
2 3 3 3 2 3 3 3 2 3 3 3 3 2 3 3
a b c d T T , a b c d T T , a b c d T T T , a b c d T T T ,
1 3 1 4 1 2 3 1 2 3
-----------------------------------------------------------------------
3 3 2 3 2 3 3 3 3 2 3 3 2 3 3 3
a b c d T T T , a b c d T T T , a b c d T T T , a b c d T T T ,
1 2 3 1 2 4 1 2 4 1 3 4
-----------------------------------------------------------------------
2 3 3 3 3 2 3 3 3 3 2 3 3 3 3 2
a b c d T T T T , a b c d T T T T , a b c d T T T T , a b c d T T T T }
1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4
o10 : List
|
i11 : Q = ZZ/101[x,y,z] o11 = Q o11 : PolynomialRing |
i12 : I = ideal{y^3,z*x^2,y*(z^2+y*x),z^3+2*x*y*z,x*(z^2+y*x),z*y^2,x^3,z*(z^2+2*x*y)}
3 2 2 2 3 2 2 2 3
o12 = ideal (y , x z, x*y + y*z , 2x*y*z + z , x y + x*z , y z, x , 2x*y*z +
-----------------------------------------------------------------------
3
z )
o12 : Ideal of Q
|
i13 : R = Q/I o13 = R o13 : QuotientRing |
i14 : A = koszulComplexDGA(R)
o14 = {Ring => R }
Underlying algebra => R[T , T , T ]
1 2 3
Differential => {x, y, z}
isHomogeneous => true
o14 : DGAlgebra
|
i15 : apply(maxDegree A + 1, i -> numgens prune homology(i,A))
o15 = {1, 7, 7, 1}
o15 : List
|
i16 : HA = homologyAlgebra(A) Computing generators in degree 1 : -- used 0.00119896 seconds Computing generators in degree 2 : -- used 0.00924717 seconds Computing generators in degree 3 : -- used 0.00976708 seconds Finding easy relations : -- used 0.0544223 seconds Computing relations in degree 1 : -- used 0.00641342 seconds Computing relations in degree 2 : -- used 0.0176505 seconds Computing relations in degree 3 : -- used 0.00624856 seconds Computing relations in degree 4 : -- used 0.0166423 seconds o16 = HA o16 : QuotientRing |
i17 : R = ZZ/101[a,b,c,d] o17 = R o17 : PolynomialRing |
i18 : S = R/ideal{a^4,b^4,c^4,d^4}
o18 = S
o18 : QuotientRing
|
i19 : A = acyclicClosure(R,EndDegree=>3)
o19 = {Ring => R }
Underlying algebra => R[T , T , T , T ]
1 2 3 4
Differential => {a, b, c, d}
isHomogeneous => true
o19 : DGAlgebra
|
i20 : B = A ** S
o20 = {Ring => S }
Underlying algebra => S[T , T , T , T ]
1 2 3 4
Differential => {a, b, c, d}
isHomogeneous => true
o20 : DGAlgebra
|
i21 : HB = homologyAlgebra(B,GenDegreeLimit=>7,RelDegreeLimit=>14) Computing generators in degree 1 : -- used 0.0161953 seconds Computing generators in degree 2 : -- used 0.0129876 seconds Computing generators in degree 3 : -- used 0.0122561 seconds Computing generators in degree 4 : -- used 0.00748261 seconds Computing generators in degree 5 : -- used 0.000936633 seconds Computing generators in degree 6 : -- used 0.00089976 seconds Computing generators in degree 7 : -- used 0.000917754 seconds Finding easy relations : -- used 0.0144449 seconds Computing relations in degree 1 : -- used 0.00187171 seconds Computing relations in degree 2 : -- used 0.0018369 seconds Computing relations in degree 3 : -- used 0.00187761 seconds Computing relations in degree 4 : -- used 0.00184377 seconds Computing relations in degree 5 : -- used 0.00169423 seconds Computing relations in degree 6 : -- used 0.00169853 seconds Computing relations in degree 7 : -- used 0.00168861 seconds Computing relations in degree 8 : -- used 0.00169251 seconds Computing relations in degree 9 : -- used 0.0016797 seconds Computing relations in degree 10 : -- used 0.00166362 seconds Computing relations in degree 11 : -- used 0.00167631 seconds Computing relations in degree 12 : -- used 0.0016679 seconds Computing relations in degree 13 : -- used 0.00160964 seconds Computing relations in degree 14 : -- used 0.00160904 seconds o21 = HB o21 : PolynomialRing |