isHomogeneous(DGModule) -- Test whether the underlying graded module is homogeneous

i1 : R = ZZ/101[x, y]

o1 = R

o1 : PolynomialRing

i2 : A = koszulComplexDGA R

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

o2 : DGAlgebra

i3 : M = freeDGModule(A, {0, 2})

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

o3 : DGModule

i4 : isHomogeneous M

o4 = true

i5 : Anat = A.natural

o5 = Anat

o5 : PolynomialRing, 2 skew commutative variable(s)

i6 : M1 = freeDGModule(A, {0})

o6 = {Base ring => R               }
      DG algebra => Anat
                            1
      Natural module => Anat
      Generator degrees => {{0, 0}}
      Differentials on gens => {0}

o6 : DGModule

i7 : S = dgSubmodule(M1, matrix {{x_Anat, y_Anat}})

o7 = DGSubmodule of ambient DGModule
     Degrees  => {{0, 1}, {0, 1}}
                     2
     natural  => Anat
     inclusion => | x y |

o7 : DGSubmodule

i8 : isHomogeneous S

o8 = true

i9 : Q = M1 / S

o9 = DGQuotientModule Q = M / S
     Q.natural = cokernel | x y |
     Degrees   = {{0, 0}}

o9 : DGQuotientModule

i10 : isHomogeneous Q

o10 = true