differential -- The list of generator differentials stored in a DG algebra or DG module

Usage:

d = differential A

d = differential M
Inputs:
- A, an instance of the type DGAlgebra,
- M, an instance of the type DGModule,
Outputs:
- d, a list, The stored list of differentials: for a DGAlgebra, one element of A.natural per DG generator; for a DGModule, one element of M.natural per natural generator.

Description

differential returns the raw list of generator differentials that was installed by setDiff. The i-th entry is the image of the i-th DG generator under d, and every other differential is computed from it by linearity and the graded Leibniz rule.

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 : differential A

o3 = map (R[T   ..T   ], R[T   ..T   ], {x, y, x, y})
             1,1   1,2      1,1   1,2

o3 : RingMap R[T   ..T   ] <-- R[T   ..T   ]
                1,1   1,2         1,1   1,2

i4 : assert(differential A === A.diff)

i5 : M = koszulComplexDGM R^1

o5 = {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}

o5 : DGModule

i6 : differential M

o6 = {0}

o6 : List

Ways to use differential:

differential(DGAlgebra)
differential(DGModule)

For the programmer

The object differential is a method function.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/DGAlgebras/doc.m2:7376:0.

differential -- The list of generator differentials stored in a DG algebra or DG module

Description

See also

Ways to use differential:

For the programmer