DGIdeal : DGIdeal -- Ideal quotient (colon) of two DG ideals

Operator: :
Usage:

K = I : J
Inputs:
- I, an instance of the type DGIdeal,
- J, an instance of the type DGIdeal, With I.dgAlgebra === J.dgAlgebra.
Outputs:
- K, an instance of the type DGIdeal, The DGIdeal with K.natural = I.natural : J.natural — i.e. { f \in A.natural | f J \subset I }.

Description

Preserves d-closure: if f J \subset I then by Leibniz d(f) J = d(fJ) - f d(J) \subset dI + f J \subset I + I = I, so d(f) \in I : J.

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

o1 = R

o1 : PolynomialRing

i2 : A = koszulComplexDGA R

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

o2 : DGAlgebra

i3 : Anat = A.natural

o3 = Anat

o3 : PolynomialRing, 2 skew commutative variable(s)

i4 : I = dgIdeal(A, {x_Anat * y_Anat})

o4 = DGIdeal of Anat
     generators => | xy |

o4 : DGIdeal

i5 : J = dgIdeal(A, {y_Anat})

o5 = DGIdeal of Anat
     generators => | y |

o5 : DGIdeal

i6 : K = I : J

o6 = DGIdeal of Anat
     generators => | x |

o6 : DGIdeal

i7 : isWellDefined K

o7 = true

Ways to use this method:

DGIdeal : DGIdeal -- Ideal quotient (colon) of two DG ideals

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/DGAlgebras/doc.m2:2621:0.

DGIdeal : DGIdeal -- Ideal quotient (colon) of two DG ideals

Description

See also

Ways to use this method: