koszulComplexDGA(List) -- The Koszul complex on a list of ring elements, as a DG algebra

• Function: koszulComplexDGA

• Usage:

  A = koszulComplexDGA diffList

• Inputs:
  • diffList, a list, A nonempty list of RingElement s, all lying in a common ring R. These become the images of the Koszul generators under the differential.
• Optional inputs:
  • Variable => ..., default value "T", The Koszul complex on the variables of a ring, as a DG algebra
• Outputs:
  • A, an instance of the type DGAlgebra, The Koszul complex over R on the sequence diffList.

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

o1 = R

o1 : PolynomialRing

i2 : A = koszulComplexDGA({x^2, y*z})

o2 = {Ring => R                          }
      Underlying algebra => R[T   ..T   ]
                               1,1   1,2
                        2
      Differential => {x , y*z}

o2 : DGAlgebra

i3 : gens A.natural

o3 = {T   , T   }
       1,1   1,2

o3 : List

i4 : flatten entries matrix A.diff

       2
o4 = {x , y*z, x, y, z}

o4 : List

i5 : apply(3, i -> numgens prune HH_i toComplex A)

o5 = {1, 0, 0}

o5 : List