isWellDefinedDifferential -- Semantic check that d^2 = 0 for a DG algebra or DG module

Usage:

b = isWellDefinedDifferential A

b = isWellDefinedDifferential M
Inputs:
- A, an instance of the type DGAlgebra, A DG algebra.
- M, an instance of the type DGModule, A DG module.
Outputs:
- b, a Boolean value, true when the composition d_{n-1} * d_n is zero for every relevant n.

Description

The semantic half of the well-definedness check: it verifies that the differential squares to zero on every generator. The structural half (isValidDGAlgebra, and the corresponding isValidDGModule for modules) is called first as a pre-condition.

This is the primitive called internally by isWellDefined(DGAlgebra) and the user-facing isWellDefined(DGModule). It is exported for low-level use when only the d^2 = 0 half of well-definedness needs to be checked.

i1 : R = QQ[x, y] / ideal(x^2, y^2)

o1 = R

o1 : QuotientRing

i2 : A = koszulComplexDGA R

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

o2 : DGAlgebra

i3 : isWellDefinedDifferential A

o3 = true

i4 : M = freeDGModule(A, {0})

o4 = {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}

o4 : DGModule

i5 : isWellDefinedDifferential M

o5 = true

Ways to use isWellDefinedDifferential:

isWellDefinedDifferential(DGAlgebra)
isWellDefinedDifferential(DGModule)

For the programmer

The object isWellDefinedDifferential is a method function.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/DGAlgebras/doc.m2:6797:0.

isWellDefinedDifferential -- Semantic check that d^2 = 0 for a DG algebra or DG module

Description

See also

Ways to use isWellDefinedDifferential:

For the programmer