isWellDefined(DGModule) -- Check that a DG module has correct structure and that its differential squares to zero

Function: isWellDefined
Usage:

b = isWellDefined M
Inputs:
- M, an instance of the type DGModule,
Outputs:
- b, a Boolean value, True when M has all expected structural keys, its base DG algebra M.dgAlgebra is itself well-defined, the ring of M matches that of its DG algebra, each stored differential is a module element of M.natural, and d_M^2 = 0 on every generator.

Description

Performs a structural check (required keys and list-length agreement #M.diff == #M.Degrees), then verifies that M.dgAlgebra is a well-defined DG algebra, checks that each entry of M.diff is a vector or matrix over M.natural, and finally calls the internal isWellDefinedDifferential routine to verify d_M^2 = 0 up to the natural bound.

Diagnostic messages are emitted when debugLevel > 0.

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 : isWellDefined M

o6 = true

Ways to use this method:

isWellDefined(DGModule) -- Check that a DG module has correct structure and that its differential squares to zero

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/DGAlgebras/doc.m2:4986:0.

isWellDefined(DGModule) -- Check that a DG module has correct structure and that its differential squares to zero

Description

See also

Ways to use this method: