This function computes the homology algebra of the DGAlgebra A and determines if the multiplication on H(A) is trivial.
i1 : R = ZZ/101[a,b,c,d]/ideal{a^4,b^4,c^4,d^4} o1 = R o1 : QuotientRing |
i2 : S = R/ideal{a^3*b^3*c^3*d^3} o2 = S o2 : QuotientRing |
i3 : A = acyclicClosure(R,EndDegree=>3) o3 = {Ring => R } Underlying algebra => R[T , T , T , T , T , T , T , T ] 1 2 3 4 5 6 7 8 3 3 3 3 Differential => {a, b, c, d, a T , b T , c T , d T } 1 2 3 4 isHomogeneous => true o3 : DGAlgebra |
i4 : B = A ** S o4 = {Ring => S } Underlying algebra => S[T , T , T , T , T , T , T , T ] 1 2 3 4 5 6 7 8 3 3 3 3 Differential => {a, b, c, d, a T , b T , c T , d T } 1 2 3 4 isHomogeneous => true o4 : DGAlgebra |
i5 : isHomologyAlgebraTrivial(B,GenDegreeLimit=>6) Computing generators in degree 1 : -- used 0.0169033 seconds Computing generators in degree 2 : -- used 0.0227467 seconds Computing generators in degree 3 : -- used 0.0451891 seconds Computing generators in degree 4 : -- used 0.0889812 seconds Computing generators in degree 5 : -- used 0.354422 seconds Computing generators in degree 6 : -- used 0.561539 seconds o5 = true |
The command returns true since R --> S is Golod. Notice we also used the option GenDegreeLimit here.
i6 : R = ZZ/101[a,b,c,d]/ideal{a^4,b^4,c^4,d^4} o6 = R o6 : QuotientRing |
i7 : A = koszulComplexDGA(R) o7 = {Ring => R } Underlying algebra => R[T , T , T , T ] 1 2 3 4 Differential => {a, b, c, d} isHomogeneous => true o7 : DGAlgebra |
i8 : isHomologyAlgebraTrivial(A) Computing generators in degree 1 : -- used 0.0122479 seconds Computing generators in degree 2 : -- used 0.0262863 seconds Computing generators in degree 3 : -- used 0.0245357 seconds Computing generators in degree 4 : -- used 0.0226352 seconds o8 = false |
The command returns false, since R is Gorenstein, and so HA has Poincare Duality, hence the multiplication is far from trivial.