zeroDGModuleMap -- The zero DGModuleMap between two DG modules over a common algebra

i1 : R = ZZ/101[x] / ideal(x^2)

o1 = R

o1 : QuotientRing

i2 : A = koszulComplexDGA R

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

o2 : DGAlgebra

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

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

o3 : DGModule

i4 : idM = identityDGModuleMap Mdg

                         2
o4 = {Source => (R[T   ])      }
                    1,1
                         2
      Target => (R[T   ])
                    1,1
      Natural => {0, 0} | 1 0 |
                 {2, 2} | 0 1 |

o4 : DGModuleMap

i5 : zM  = zeroDGModuleMap(Mdg, Mdg)

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

o5 : DGModuleMap

i6 : idM + zM == idM

o6 = true

i7 : zM * idM == zM

o7 = true

i8 : idM * zM == zM

o8 = true

i9 : Q = cokernel zM

o9 = DGQuotientModule Q = M / S
                          2
     Q.natural = (R[T   ])
                     1,1
     Degrees   = {{0, 0}, {2, 2}}

o9 : DGQuotientModule

i10 : rank Q.natural == rank Mdg.natural

o10 = true

i11 : K = kernel zM

o11 = DGSubmodule of ambient DGModule
      Degrees  => {{0, 0}, {2, 2}}
                           2
      natural  => (R[T   ])
                      1,1
      inclusion => {0, 0} | 1 0 |
                   {2, 2} | 0 1 |

o11 : DGSubmodule

i12 : instance(K, DGSubmodule)

o12 = true