Operations on DG Ideals -- Sum, product, power, intersection, colon, and other ideal-algebra operations on DG ideals

A DG ideal of a DG algebra A is a graded ideal of A.natural that is closed under the differential d of A. The DGIdeal type wraps such an ideal and exposes the standard M2 Ideal operations, each of which preserves d-closure and returns a new DGIdeal.

Constructor: dgIdeal takes a List, Matrix, or Ideal of elements of A.natural and iteratively d-saturates the ideal they generate until it is closed under d. Warning: if an input element is not a cycle, the d-saturation enlarges the ideal (possibly to the unit 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 : 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 : isWellDefined I

o6 = true

i7 : isDGIdeal(A, I.natural)

o7 = true

Ideal-algebra operations all preserve d-closure by elementary arguments (sum: d(I+J) \subset dI + dJ \subset I + J; product: by the Leibniz rule; intersection: direct; power: induction; colon: Leibniz + f J \subset I implies (df) J \subset I + f(dJ) \subset I). Concretely, they all return well-defined DGIdeals.

i8 : isWellDefined(I + J)

o8 = true

i9 : isWellDefined(I * J)

o9 = true

i10 : isWellDefined(I^3)

o10 = true

i11 : isWellDefined intersect(I, J)

o11 = true

i12 : isWellDefined(I : J)

o12 = true

Equality, containment, and reduction mirror the usual behavior on Ideal.

i13 : isSubset(I, I + J)

o13 = true

i14 : I != I + J

o14 = true

i15 : (x_Anat^2 % I) == 0

o15 = true

i16 : (y_Anat^2 % I) == y_Anat^2

o16 = true

Quotient DG algebras: A / I builds the DG algebra whose underlying ring is A.natural / I.natural, with differential descending from d (well-defined precisely because I is d-closed).

i17 : B = A / I

o17 = {Ring => R                 }
                             Anat
       Underlying algebra => ----
                               x
       Differential => {0, y}

o17 : DGAlgebra

i18 : class B

o18 = DGAlgebra

o18 : Type

i19 : isWellDefined B

o19 = true

Compatibility: any operation that takes an Ideal and lives on I.natural can be reached via I.natural, but the wrapped operations on DGIdeal return DGIdeals (so d-closure is certified on the result by running through dgIdeal once more).

Operations on DG Ideals -- Sum, product, power, intersection, colon, and other ideal-algebra operations on DG ideals

See also

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