intersect(DGIdeal,DGIdeal) -- Intersection of two DG ideals

Function: intersect
Usage:

K = intersect(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 = intersect(I.natural, J.natural).

Description

Intersections preserve d-closure directly: if g lies in both I and J, then d(g) lies in both (by d-closure of each), hence in their intersection.

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})

o4 = DGIdeal of Anat
     generators => | x |

o4 : DGIdeal

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

o5 = DGIdeal of Anat
     generators => | y |

o5 : DGIdeal

i6 : K = intersect(I, J)

o6 = DGIdeal of Anat
     generators => | xy |

o6 : DGIdeal

i7 : isWellDefined K

o7 = true

Intersecting with the zero DGIdeal yields the zero DGIdeal.

i8 : Z = dgIdeal(A, {})

o8 = DGIdeal of Anat
     generators => 0

o8 : DGIdeal

i9 : ZI = intersect(Z, I)

o9 = DGIdeal of Anat
     generators => 0

o9 : DGIdeal

i10 : isWellDefined ZI

o10 = true

i11 : numgens ZI.natural == 0

o11 = true

Ways to use this method:

intersect(DGIdeal,DGIdeal) -- Intersection of two DG ideals

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/DGAlgebras/doc.m2:2588:0.

intersect(DGIdeal,DGIdeal) -- Intersection of two DG ideals

Description

See also

Ways to use this method: