dgAlgebraMultMap -- Returns the chain map corresponding to multiplication by a cycle.

Synopsis

Usage:

phi = dgAlgebraMultMap(A,z)
Inputs:
- A, an instance of the type DGAlgebra,
- z, a ring element,
Outputs:
- phi, a chain complex map,

Description

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

Ways to use dgAlgebraMultMap :

dgAlgebraMultMap(DGAlgebra,RingElement)

For the programmer

The object dgAlgebraMultMap is a method function.