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diff(DGAlgebra,RingElement) -- Apply the differential of a DGAlgebra to a ring element

Description

Returns the image of a under the differential of A. The element a must be a ring element of the underlying graded-commutative algebra A.natural; products, sums, and polynomial expressions in the DG generators are all valid inputs. The differential is applied by the Leibniz rule, so homogeneous components are processed independently.

i1 : R = ZZ/101[x,y,z]

o1 = R

o1 : PolynomialRing
 
i2 : A = koszulComplexDGA R

o2 = {Ring => R                          }
      Underlying algebra => R[T   ..T   ]
                               1,1   1,3
      Differential => {x, y, z}

o2 : DGAlgebra
 
i3 : diff(A, T_(1,1) * T_(1,2))

o3 = - y*T    + x*T
          1,1      1,2

o3 : R[T   ..T   ]
        1,1   1,3
 
i4 : diff(A, diff(A, T_(1,1) * T_(1,2) * T_(1,3)))

o4 = 0

o4 : R[T   ..T   ]
        1,1   1,3

When a is a cycle, the result is zero:

i5 : z = x*T_(1,2) - y*T_(1,1)

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

o5 : R[T   ..T   ]
        1,1   1,3
 
i6 : diff(A, z)

o6 = 0

o6 : R[T   ..T   ]
        1,1   1,3

To solve the equation diff(A, b) == z for a boundary z, use getBoundaryPreimage.

See also

Ways to use this method:

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/DGAlgebras/doc.m2:8767:0.