liftToDGModuleMap -- Lift an image of hom-degree-0 generators to a full DGModuleMap

Usage:

f = liftToDGModuleMap(N, M, h0)
Inputs:
- N, an instance of the type DGModule, The target DGModule. Must be acyclic in positive hom-degrees up to EndDegree so that lifts exist.
- M, an instance of the type DGModule, The source DGModule. Must be semifree (M.natural a free module).
- h0, A Matrix with one column per hom-degree-0 generator of M and (rank N.natural)-many rows whose entries lie in A.natural, or a List of Vectors (in N.natural), or a single Vector when M has exactly one hom-degree-0 generator.
Optional inputs:
- EndDegree => ..., default value -1
Outputs:
- f, an instance of the type DGModuleMap,

Description

Starting from the supplied hom-degree-0 assignment, the routine inductively constructs images for each M-generator of hom-degree $d \geq 1$ by solving $d_N(x) = f(d_M(e))$ in N.natural at hom-degree $d$. Existence of a lift at every step is what requires N to be acyclic. The lift is not unique in general: different choices of preimage at each step produce chain-homotopic lifts.

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

o1 = R

o1 : QuotientRing

i2 : A = koszulComplexDGA R

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

o2 : DGAlgebra

i3 : Mmin = minimalSemifreeResolution(A, R^1 / ideal(x, y), EndDegree => 2)

o3 = {Base ring => R                                          }
      DG algebra => R[T   ..T   ]
                       1,1   1,2
                                       3
      Natural module => (R[T   ..T   ])
                            1,1   1,2
      Generator degrees => {{0, 0}, {2, 2}, {2, 2}}
      Differentials on gens => {0, | xT_(1,1) |, | yT_(1,2) |}
                                   |     0    |  |     0    |
                                   |     0    |  |     0    |

o3 : DGModule

i4 : Mnon = semifreeResolution(A, R^1 / ideal(x, y), EndDegree => 2)

o4 = {Base ring => R                                                                                    }
      DG algebra => R[T   ..T   ]
                       1,1   1,2
                                       7
      Natural module => (R[T   ..T   ])
                            1,1   1,2
      Generator degrees => {{0, 0}, {1, 1}, {1, 1}, {2, 1}, {2, 1}, {2, 2}, {2, 2}}
      Differentials on gens => {0, | x |, | y |, | -T_(1,1) |, | -T_(1,2) |, | xT_(1,1) |, | yT_(1,2) |}
                                   | 0 |  | 0 |  |     1    |  |     0    |  |     0    |  |     0    |
                                   | 0 |  | 0 |  |     0    |  |     1    |  |     0    |  |     0    |
                                   | 0 |  | 0 |  |     0    |  |     0    |  |     0    |  |     0    |
                                   | 0 |  | 0 |  |     0    |  |     0    |  |     0    |  |     0    |
                                   | 0 |  | 0 |  |     0    |  |     0    |  |     0    |  |     0    |
                                   | 0 |  | 0 |  |     0    |  |     0    |  |     0    |  |     0    |

o4 : DGModule

i5 : quism = liftToDGModuleMap(Mnon, Mmin, {(Mnon.natural)_0}, EndDegree => 2)

                               3
o5 = {Source => (R[T   ..T   ])  }
                    1,1   1,2
                               7
      Target => (R[T   ..T   ])
                    1,1   1,2
      Natural => {0, 0} | 1 0 0 |
                 {1, 1} | 0 0 0 |
                 {1, 1} | 0 0 0 |
                 {2, 1} | 0 0 0 |
                 {2, 1} | 0 0 0 |
                 {2, 2} | 0 1 0 |
                 {2, 2} | 0 0 1 |

o5 : DGModuleMap

i6 : isWellDefined quism

o6 = true

Ways to use liftToDGModuleMap:

liftToDGModuleMap(DGModule,DGModule,List)
liftToDGModuleMap(DGModule,DGModule,Matrix)
liftToDGModuleMap(DGModule,DGModule,Vector)

For the programmer

The object liftToDGModuleMap is a method function with options.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/DGAlgebras/doc.m2:481:0.

liftToDGModuleMap -- Lift an image of hom-degree-0 generators to a full DGModuleMap

Description

See also

Ways to use liftToDGModuleMap:

For the programmer