isWellDefined(DGSubmodule) -- Check that a DG submodule is d-closed in its ambient DG module

Function: isWellDefined
Usage:

b = isWellDefined S
Inputs:
- S, an instance of the type DGSubmodule,
Outputs:
- b, a Boolean value, True when S has all expected structural keys, its ambient is a well-defined DG module, and the column span of S.inclusion.natural in S.ambient.natural is stable under d_{S.ambient}.

Description

After a structural key check, verifies that S.ambient is a well-defined DG module, then checks d-closure: for every column v of the inclusion matrix, d_M(v) lies in the column span of the inclusion matrix modulo further columns. This is the one nontrivial condition that distinguishes a DG submodule from a general A.natural-submodule of M.natural.

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 : S = dgSubmodule(M, (id_(M.natural))_{1})

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

o6 : DGSubmodule

i7 : isWellDefined S

o7 = true

Ways to use this method:

isWellDefined(DGSubmodule) -- Check that a DG submodule is d-closed in its ambient DG module

The source of this document is in DGAlgebras/doc.m2:5026:0.

isWellDefined(DGSubmodule) -- Check that a DG submodule is d-closed in its ambient DG module

Description

See also

Ways to use this method: