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DGSubmodule == DGSubmodule -- Equality and containment of DG submodules

Description

Equality requires the ambients to be the same object (===) and the images of the inclusions to agree. Containment further requires the DGSubmodules share an ambient and tests the image inclusions.

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}})

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

o5 : DGSubmodule
 
i6 : U = dgSubmodule(M, matrix {{x_Anat, y_Anat}})

o6 = DGSubmodule of ambient DGModule
     Degrees  => {{0, 1}, {0, 1}}
                     2
     natural  => Anat
     inclusion => | x y |

o6 : DGSubmodule
 
i7 : isSubset(S, U)

o7 = true
 
i8 : not isSubset(U, S)

o8 = true
 
i9 : S == S

o9 = true

Two DGSubmodules built from distinct ambient DGModules are never equal, even if the ambient modules are abstractly isomorphic.

i10 : M' = freeDGModule(A, {0})

o10 = {Base ring => R               }
       DG algebra => Anat
                             1
       Natural module => Anat
       Generator degrees => {{0, 0}}
       Differentials on gens => {0}

o10 : DGModule
 
i11 : S' = dgSubmodule(M', matrix {{x_Anat}})

o11 = DGSubmodule of ambient DGModule
      Degrees  => {{0, 1}}
                      1
      natural  => Anat
      inclusion => | x |

o11 : DGSubmodule
 
i12 : not (S == S')

o12 = true

See also

Ways to use this method:

The source of this document is in DGAlgebras/doc.m2:3873:0.