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super(DGSubmodule) -- Ambient DGModule, cover, and relation matrix of a DG sub- or quotient module

Description

These mirror the analogous M2 Module accessors.

super S (for a DGSubmodule S) and super Q, cover Q (for a DGQuotientModule Q) all return the ambient DGModule M from which the sub- or quotient was built.

relations Q returns the inclusion matrix of the killed submodule, encoding the cokernel presentation of Q.

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})

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

o3 : DGModule
 
i4 : Anat = A.natural

o4 = Anat

o4 : PolynomialRing, 2 skew commutative variable(s)
 
i5 : S = dgSubmodule(M, matrix {{x_Anat, y_Anat}})

o5 = DGSubmodule of ambient DGModule
     Degrees  => {{0, 1}, {0, 1}}
                     2
     natural  => Anat
     inclusion => | x y |

o5 : DGSubmodule
 
i6 : super S === M

o6 = true
 
i7 : Q = M / S

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

o7 : DGQuotientModule
 
i8 : super Q === M

o8 = true
 
i9 : cover Q === M

o9 = true
 
i10 : numcols relations Q == 2

o10 = true

See also

Ways to use this method:

  • super(DGQuotientModule)
  • super(DGSubmodule) -- Ambient DGModule, cover, and relation matrix of a DG sub- or quotient module

The source of this document is in DGAlgebras/doc.m2:3694:0.