isAcyclic(DGQuotientModule) -- Determine whether a DG quotient module has vanishing positive-degree homology

i1 : R = ZZ/101[x]

o1 = R

o1 : PolynomialRing

i2 : A = koszulComplexDGA R

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

o2 : DGAlgebra

i3 : M = freeDGModule(A, {0, 1})

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

o3 : DGModule

i4 : natGens = apply(rank M.natural, i -> (M.natural)_i)

o4 = {| 1 |, | 0 |}
      | 0 |  | 1 |

o4 : List

i5 : setDiff(M, {0, x * natGens#0})

o5 = {Base ring => R                       }
      DG algebra => R[T   ]
                       1,1
                                 2
      Natural module => (R[T   ])
                            1,1
      Generator degrees => {{0, 0}, {1, 0}}
      Differentials on gens => {0, | x |}
                                   | 0 |

o5 : DGModule

i6 : Sfull = dgSubmodule(M, id_(M.natural))

o6 = DGSubmodule of ambient DGModule
     Degrees  => {{0, 0}, {1, 0}}
                          2
     natural  => (R[T   ])
                     1,1
     inclusion => {0, 0} | 1 0 |
                  {1, 0} | 0 1 |

o6 : DGSubmodule

i7 : Q0 = M / Sfull

o7 = DGQuotientModule Q = M / S
     Q.natural = cokernel {0, 0} | 1 0 |
                          {1, 0} | 0 1 |
     Degrees   = {{0, 0}, {1, 0}}

o7 : DGQuotientModule

i8 : isAcyclic Q0

o8 = true