Macaulay2 » Documentation
Packages » DGAlgebras » Operations on DG Ideals » DGIdeal * DGIdeal
next | previous | forward | backward | up | index | toc

DGIdeal * DGIdeal -- Product of two DG ideals

Description

Preserves d-closure by the Leibniz rule: d(ab) = d(a) b \pm a d(b) shows d(IJ) \subset (dI) J + I (dJ) \subset IJ. When the ambient ring is commutative, multiplication is commutative.

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 = I * J

o6 = DGIdeal of Anat
     generators => | xy |

o6 : DGIdeal
 
i7 : isWellDefined K

o7 = true
 
i8 : (I * J) == (J * I)

o8 = true

The unit DGIdeal is a two-sided identity for * (as ideals).

i9 : U = dgIdeal(A, {1_Anat})

o9 = DGIdeal of Anat
     generators => | 1 |

o9 : DGIdeal
 
i10 : (U * I) == I

o10 = true

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.m2:2509:0.