underlyingAlgebra -- The graded-commutative algebra carrying the DG structure

Usage:

N = underlyingAlgebra A

N = underlyingAlgebra M
Inputs:
- A, an instance of the type DGAlgebra,
- M, an instance of the type DGModule,
Outputs:
- N, a ring, The graded-commutative ring A.natural (the DG algebra's own ring) or, for a DG module, the graded free module M.natural (as a Module over A.natural).

Description

For a DGAlgebra, underlyingAlgebra returns the graded-commutative ring A.natural -- the ring whose elements one manipulates when writing polynomials in the DG generators. For a DGModule it returns the underlying A.natural-module M.natural on which the differential acts. In both cases the result is what you typically use before referring to DG generators by name.

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 : assert(underlyingAlgebra A === A.natural)

i4 : numgens underlyingAlgebra A

o4 = 2

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 : assert(underlyingAlgebra M === M.natural)

Ways to use underlyingAlgebra:

underlyingAlgebra(DGAlgebra)
underlyingAlgebra(DGModule)

For the programmer

The object underlyingAlgebra is a method function.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/DGAlgebras/doc.m2:7335:0.

underlyingAlgebra -- The graded-commutative algebra carrying the DG structure

Description

See also

Ways to use underlyingAlgebra:

For the programmer