toComplexMap(DGModuleMap,ZZ) -- The per-hom-degree component of a DG module map as a matrix

Function: toComplexMap
Usage:

m = toComplexMap(f, n)
Inputs:
- f, an instance of the type DGModuleMap, A DG module map M --> N.
- n, an integer, A homological degree.
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:
- m, a matrix, The hom-degree-n component F_n(M) --> F_n(N) as an A.ring-linear matrix.

Description

Builds the A.ring-linear matrix that f induces on the hom-degree-n pieces of M.natural and N.natural. This is the per-degree primitive used by toComplexMap(DGModuleMap) when assembling a full ComplexMap.

i1 : R = QQ[x, y] / ideal(x^2, y^2)

o1 = R

o1 : QuotientRing

i2 : Mdg = minimalSemifreeResolution(koszulComplexDGA R, R^1 / ideal(x, y), EndDegree => 3)

o2 = {Base ring => R                                                                                    }
      DG algebra => R[T   ..T   ]
                       1,1   1,2
                                       6
      Natural module => (R[T   ..T   ])
                            1,1   1,2
      Generator degrees => {{0, 0}, {2, 2}, {2, 2}, {4, 4}, {4, 4}, {4, 4}}
      Differentials on gens => {0, | xT_(1,1) |, | yT_(1,2) |, |     0    |, |     0    |, |     0    |}
                                   |     0    |  |     0    |  | xT_(1,1) |  | yT_(1,2) |  |     0    |
                                   |     0    |  |     0    |  |     0    |  | xT_(1,1) |  | yT_(1,2) |
                                   |     0    |  |     0    |  |     0    |  |     0    |  |     0    |
                                   |     0    |  |     0    |  |     0    |  |     0    |  |     0    |
                                   |     0    |  |     0    |  |     0    |  |     0    |  |     0    |

o2 : DGModule

i3 : idM = identityDGModuleMap Mdg

                               6
o3 = {Source => (R[T   ..T   ])        }
                    1,1   1,2
                               6
      Target => (R[T   ..T   ])
                    1,1   1,2
      Natural => {0, 0} | 1 0 0 0 0 0 |
                 {2, 2} | 0 1 0 0 0 0 |
                 {2, 2} | 0 0 1 0 0 0 |
                 {4, 4} | 0 0 0 1 0 0 |
                 {4, 4} | 0 0 0 0 1 0 |
                 {4, 4} | 0 0 0 0 0 1 |

o3 : DGModuleMap

i4 : all(0..3, n -> (
         cm := toComplexMap(idM, n);
         source cm == target cm and cm == id_(source cm)
         ))

o4 = true

Ways to use this method:

toComplexMap(DGModuleMap,ZZ) -- The per-hom-degree component of a DG module map as a matrix

The source of this document is in DGAlgebras/doc.m2:7963:0.

toComplexMap(DGModuleMap,ZZ) -- The per-hom-degree component of a DG module map as a matrix

Description

See also

Ways to use this method: