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 ]
1 4
Differential => {a, b, c, d}
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) Finding easy relations : -- used 0.0301912 seconds o4 = HA o4 : PolynomialRing, 4 skew commutative variables |
Note that HA is a graded commutative polynomial ring (i.e. an exterior algebra) since R is a complete intersection.
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 ]
1 4
Differential => {a, b, c, d}
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) Finding easy relations : -- used 0.155154 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 2 3 3 3 2 3 3 3 2 3 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 , a b c d T T , a b c d T T , a b c d T T T ,
1 2 3 4 1 1 2 1 2 1 3 1 4 1 2 3
---------------------------------------------------------------------------------------------------------------------------
3 2 3 3 3 3 2 3 2 3 3 3 3 2 3 3 2 3 3 3 2 3 3 3 3 2 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 , a b c d T T T , a b c d T T T T , a b c d T T T T ,
1 2 3 1 2 3 1 2 4 1 2 4 1 3 4 1 2 3 4 1 2 3 4
---------------------------------------------------------------------------------------------------------------------------
3 3 2 3 3 3 3 2
a b c d T T T T , a b c d T T T T }
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 3
o12 = ideal (y , x z, x*y + y*z , 2x*y*z + z , x y + x*z , y z, x , 2x*y*z + 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 ]
1 3
Differential => {x, y, z}
o14 : DGAlgebra
|
i15 : apply(maxDegree A + 1, i -> numgens prune homology(i,A))
o15 = {1, 7, 7, 1}
o15 : List
|
i16 : HA = homologyAlgebra(A) Finding easy relations : -- used 0.120438 seconds o16 = HA o16 : QuotientRing |
One can check that HA has Poincare duality since R is Gorenstein.
If your DGAlgebra has generators in even degrees, then one must specify the options GenDegreeLimit and RelDegreeLimit.
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 ]
1 4
Differential => {a, b, c, d}
o19 : DGAlgebra
|
i20 : B = A ** S
o20 = {Ring => S }
Underlying algebra => S[T ..T ]
1 4
Differential => {a, b, c, d}
o20 : DGAlgebra
|
i21 : HB = homologyAlgebra(B,GenDegreeLimit=>7,RelDegreeLimit=>14) Finding easy relations : -- used 0.0304143 seconds o21 = HB o21 : PolynomialRing, 4 skew commutative variables |
The object homologyAlgebra is a method function with options.