isDGIdeal -- Test whether an ideal of A.natural is closed under the differential of A

Usage:

b = isDGIdeal(A, I)
Inputs:
- A, an instance of the type DGAlgebra,
- I, an ideal, An ideal of A.natural.
Outputs:
- b, a Boolean value, True iff every minimal generator g of I satisfies polyDifferential(A, g) % I == 0.

Description

Unlike dgIdeal (which d-saturates), isDGIdeal simply checks d-closure without modifying the input. Use it to validate that an externally-produced Ideal is already a DG ideal.

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 : isDGIdeal(A, ideal(x_Anat, y_Anat))

o4 = true

i5 : T = Anat_0   -- first Koszul variable T_{1,1}; d(T) = x

o5 = T
      1,1

o5 : Anat

i6 : isDGIdeal(A, ideal(T))

o6 = false

The second test returns false: the ideal (T) is not d-closed because d(T) = x is not in it. Passing the same seed to dgIdeal produces the d-closure.

i7 : J = dgIdeal(A, {T})

o7 = DGIdeal of Anat
     generators => | x T_(1,1) |

o7 : DGIdeal

i8 : isDGIdeal(A, J.natural)

o8 = true

Ways to use isDGIdeal:

isDGIdeal(DGAlgebra,Ideal)

For the programmer

The object isDGIdeal is a method function.

The source of this document is in DGAlgebras/doc.m2:2404:0.

isDGIdeal -- Test whether an ideal of A.natural is closed under the differential of A

Description

See also

Ways to use isDGIdeal:

For the programmer