toComplexMap(DGModuleMap) -- Export a DG module map to a ComplexMap

Function: toComplexMap
Usage:

cm = toComplexMap f
Inputs:
- f, an instance of the type DGModuleMap, A DG module map M --> N.
Optional inputs:
- AssertWellDefined => ..., default value true, Construct the ComplexMap associated to a DGAlgebraMap
- EndDegree => ..., default value -1, Construct the ComplexMap associated to a DGAlgebraMap
Outputs:
- cm, a map of complexes, The induced ComplexMap toComplex M --> toComplex N.

Description

toComplexMap f assembles the per-degree pieces of f into a ComplexMap. When either side has infinite hom-degree, an EndDegree bound must be supplied; the option AssertWellDefined (default true) causes isWellDefined f to be checked up front. When the target is a DG quotient module only the canonical projection is supported; the result is the induced quotient chain map. Use toComplexMap(DGModuleMap,ZZ) to extract a single per-degree component instead.

i1 : R = QQ[a, b] / ideal(a^2, b^2)

o1 = R

o1 : QuotientRing

i2 : A = koszulComplexDGA R

o2 = {Ring => R                          }
      Underlying algebra => R[T   ..T   ]
                               1,1   1,2
      Differential => {a, b}

o2 : DGAlgebra

i3 : phi = dgAlgebraMap(A, A, matrix {gens A.natural})

o3 = map (R[T   ..T   ], R[T   ..T   ], {T   , T   , a, b})
             1,1   1,2      1,1   1,2     1,1   1,2

o3 : DGAlgebraMap

i4 : cm = toComplexMap phi

          1             1
o4 = 0 : R  <--------- R  : 0
               | 1 |

          2                   2
     1 : R  <--------------- R  : 1
               {1} | 1 0 |
               {1} | 0 1 |

          1                 1
     2 : R  <------------- R  : 2
               {2} | 1 |

o4 : ComplexMap

i5 : assert(instance(cm, ComplexMap))

Ways to use this method:

toComplexMap(DGModuleMap) -- Export a DG module map to a ComplexMap

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/DGAlgebras/doc.m2:7922:0.

toComplexMap(DGModuleMap) -- Export a DG module map to a ComplexMap

Description

See also

Ways to use this method: