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natural -- The underlying graded-commutative ring of a DGAlgebra or DGModule

Description

For a DGAlgebra A, the key A.natural holds the underlying graded-commutative polynomial ring of A with the differential forgotten. This is the ambient ring in which algebra generators live, in which cycles and boundaries are expressed, and against which differentials are compared. It is always a polynomial ring over A.ring (with the skew-commutative flag set on odd-hom-degree generators), never a quotient.

i1 : R = ZZ/101[a, b, c, d]

o1 = R

o1 : PolynomialRing
 
i2 : A = koszulComplexDGA R

o2 = {Ring => R                          }
      Underlying algebra => R[T   ..T   ]
                               1,1   1,4
      Differential => {a, b, c, d}

o2 : DGAlgebra
 
i3 : A.natural

o3 = R[T   ..T   ]
        1,1   1,4

o3 : PolynomialRing, 4 skew commutative variable(s)
 
i4 : A.ring === R

o4 = true
 
i5 : gens A.natural

o5 = {T   , T   , T   , T   }
       1,1   1,2   1,3   1,4

o5 : List

For a DGModule M, the same key holds the underlying free A.natural-module with the differential forgotten. The differential is recorded separately in M.diff and the multi-degrees of the natural generators in M.Degrees:

i6 : B = freeDGModule(A, {0, 1, 2})

o6 = {Base ring => R                               }
      DG algebra => R[T   ..T   ]
                       1,1   1,4
                                       3
      Natural module => (R[T   ..T   ])
                            1,1   1,4
      Generator degrees => {{0, 0}, {1, 0}, {2, 0}}
      Differentials on gens => {0, 0, 0}

o6 : DGModule
 
i7 : B.natural

                    3
o7 = (R[T   ..T   ])
         1,1   1,4

o7 : R[T   ..T   ]-module, free, degrees {{0, 0}, {1, 0}, {2, 0}}
        1,1   1,4
 
i8 : rank B.natural

o8 = 3

See also

For the programmer

The object natural is a symbol.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/DGAlgebras/doc.m2:1198:0.