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 0.00175722 seconds Computing generators in degree 2 : -- used 0.0172791 seconds Computing generators in degree 3 : -- used 0.0092793 seconds Computing generators in degree 4 : -- used 0.0147659 seconds Finding easy relations : -- used 0.0262427 seconds Computing relations in degree 1 : -- used 0.00428804 seconds Computing relations in degree 2 : -- used 0.00379777 seconds Computing relations in degree 3 : -- used 0.0385815 seconds Computing relations in degree 4 : -- used 0.00529247 seconds Computing relations in degree 5 : -- used 0.00271798 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 0.00328475 seconds Computing generators in degree 2 : -- used 0.0181284 seconds Computing generators in degree 3 : -- used 0.0181378 seconds Computing generators in degree 4 : -- used 0.0191771 seconds Finding easy relations : -- used 0.199018 seconds Computing relations in degree 1 : -- used 0.00748693 seconds Computing relations in degree 2 : -- used 0.00762757 seconds Computing relations in degree 3 : -- used 0.0152322 seconds Computing relations in degree 4 : -- used 0.0151226 seconds Computing relations in degree 5 : -- used 0.0121743 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 0.00218753 seconds Computing generators in degree 2 : -- used 0.0183838 seconds Computing generators in degree 3 : -- used 0.0168607 seconds Finding easy relations : -- used 0.0911426 seconds Computing relations in degree 1 : -- used 0.00526591 seconds Computing relations in degree 2 : -- used 0.01852 seconds Computing relations in degree 3 : -- used 0.0107966 seconds Computing relations in degree 4 : -- used 0.00863086 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 0.009823 seconds Computing generators in degree 2 : -- used 0.0246135 seconds Computing generators in degree 3 : -- used 0.0236458 seconds Computing generators in degree 4 : -- used 0.0141338 seconds Computing generators in degree 5 : -- used 0.001978 seconds Computing generators in degree 6 : -- used 0.00141648 seconds Computing generators in degree 7 : -- used 0.00171533 seconds Finding easy relations : -- used 0.0275322 seconds Computing relations in degree 1 : -- used 0.00339873 seconds Computing relations in degree 2 : -- used 0.0031155 seconds Computing relations in degree 3 : -- used 0.00330512 seconds Computing relations in degree 4 : -- used 0.00320386 seconds Computing relations in degree 5 : -- used 0.00255848 seconds Computing relations in degree 6 : -- used 0.0026973 seconds Computing relations in degree 7 : -- used 0.00302365 seconds Computing relations in degree 8 : -- used 0.0025093 seconds Computing relations in degree 9 : -- used 0.00290527 seconds Computing relations in degree 10 : -- used 0.002946 seconds Computing relations in degree 11 : -- used 0.00273 seconds Computing relations in degree 12 : -- used 0.00314089 seconds Computing relations in degree 13 : -- used 0.00302471 seconds Computing relations in degree 14 : -- used 0.00297877 seconds o21 = HB o21 : PolynomialRing |