polyDiffMonomial -- Leibniz rule on a single monomial of a DG algebra

Usage:

b = polyDiffMonomial(A, m)
Inputs:
- A, an instance of the type DGAlgebra,
- m, a ring element, A monomial of A.natural (or the zero element).
Outputs:
- b, a ring element, The image d(m) in A.natural.

Description

polyDiffMonomial is the monomial kernel of the DG algebra differential. It splits m into its coefficient and its support powers and then applies the graded Leibniz rule variable by variable, keeping track of the sign produced when d passes an odd-degree generator. The routine handles the zero element without raising an exception, returning 0 in A.natural.

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(polyDiffMonomial(A, 0_(A.natural)) == 0_(A.natural))

i4 : gs = gens A.natural

o4 = {T   , T   }
       1,1   1,2

o4 : List

i5 : polyDiffMonomial(A, gs#0 * gs#1)

o5 = - y*T    + x*T
          1,1      1,2

o5 : R[T   ..T   ]
        1,1   1,2

On a two-variable Koszul generator product, the Leibniz rule gives d(T_1 T_2) = d(T_1) T_2 - T_1 d(T_2) = x T_2 - y T_1:

i6 : r = polyDiffMonomial(A, gs#0 * gs#1)

o6 = - y*T    + x*T
          1,1      1,2

o6 : R[T   ..T   ]
        1,1   1,2

i7 : assert(r == x * gs#1 - y * gs#0)

Ways to use polyDiffMonomial:

polyDiffMonomial(DGAlgebra,RingElement)

For the programmer

The object polyDiffMonomial is a method function.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/DGAlgebras/doc.m2:6955:0.

polyDiffMonomial -- Leibniz rule on a single monomial of a DG algebra

Description

See also

Ways to use polyDiffMonomial:

For the programmer