If A is a DGAlgebra, and z is a cycle of A, then left multiplication of A by z gives a chain map from A to A. This command converts A to a complex using toComplex, and constructs a ChainComplexMap that represents left multiplication by z. This command is used to determine the module structure that is computed in homologyModule.
i1 : R = QQ[x,y,z]/ideal{x^3,y^3,z^3}
o1 = R
o1 : QuotientRing
|
i2 : KR = koszulComplexDGA R
o2 = {Ring => R }
Underlying algebra => R[T ..T ]
1 3
Differential => {x, y, z}
o2 : DGAlgebra
|
i3 : z1 = x^2*T_1
2
o3 = x T
1
o3 : R[T ..T ]
1 3
|
i4 : phi = dgAlgebraMultMap(KR,z1)
3 1
o4 = 1 : R <-------------- R : 0
{1} | x2 |
{1} | 0 |
{1} | 0 |
3 3
2 : R <------------------- R : 1
{2} | 0 x2 0 |
{2} | 0 0 x2 |
{2} | 0 0 0 |
1 3
3 : R <------------------ R : 2
{3} | 0 0 x2 |
o4 : ChainComplexMap
|
As you can see, the degree of phi is the homological degree of z:
i5 : degree phi == first degree z o5 = true |
Care is also taken to ensure the resulting map is homogeneous if R and z are:
i6 : isHomogeneous phi o6 = true |
One may then view the action of multiplication by the homology class of z upon taking the induced map in homology:
i7 : Hphi = prune HH(phi); (Hphi#0,Hphi#1,Hphi#2)
o8 = ({3} | 1 |, {6} | 0 1 0 |, {9} | 0 0 1 |)
{3} | 0 | {6} | 0 0 1 |
{3} | 0 | {6} | 0 0 0 |
o8 : Sequence
|
The object dgAlgebraMultMap is a method function.