DGIdeal -- The class of d-closed graded ideals of a DG algebra

Description

A DGIdeal wraps a graded ideal of A.natural that is closed under the differential d of A. It records the ambient DG algebra, the underlying Ideal, and a chosen set of generators. Create one via dgIdeal.

The standard ideal-algebra operations (DGIdeal + DGIdeal, DGIdeal * DGIdeal, DGIdeal ^ ZZ, intersect(DGIdeal,DGIdeal), DGIdeal : DGIdeal) all preserve d-closure and return new DGIdeals. See Operations on DG Ideals for a guided tour.

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 : class I

o5 = DGIdeal

o5 : Type

i6 : ambient I === A

o6 = true

i7 : ring I === R

o7 = true

i8 : numgens I

o8 = 1

Functions and methods returning an object of class DGIdeal:

dgIdeal -- Construct a DG ideal from a list, matrix, or ideal of generators
intersect(DGIdeal,DGIdeal) -- Intersection of two DG ideals

Methods that use an object of class DGIdeal:

ambient(DGIdeal) -- see ambient -- Accessors for the ambient object, inclusion, projection, and killed submodule
DGAlgebra / DGIdeal -- Quotient DG algebra by a DG ideal
DGIdeal * DGIdeal -- Product of two DG ideals
DGIdeal * DGModule (missing documentation)
DGIdeal * DGSubmodule (missing documentation)
DGIdeal ** Ring -- Base change of a DG ideal along a ring map
DGIdeal + DGIdeal -- Sum of two DG ideals
DGIdeal : DGIdeal -- Ideal quotient (colon) of two DG ideals
DGIdeal == DGIdeal -- Equality and containment of DG ideals
isSubset(DGIdeal,DGIdeal) -- see DGIdeal == DGIdeal -- Equality and containment of DG ideals
DGIdeal ^ ZZ -- Nonnegative integer power of a DG ideal
ideal(DGIdeal) (missing documentation)
isHomogeneous(DGIdeal) -- Predicates on a DG ideal
isZero(DGIdeal) -- see isHomogeneous(DGIdeal) -- Predicates on a DG ideal
isWellDefined(DGIdeal) -- Verify that a DGIdeal has correct structure and is d-closed
generators(DGIdeal) -- see mingens(DGIdeal) -- Basic accessors on a DG ideal
mingens(DGIdeal) -- Basic accessors on a DG ideal
module(DGIdeal) -- see mingens(DGIdeal) -- Basic accessors on a DG ideal
numgens(DGIdeal) -- see mingens(DGIdeal) -- Basic accessors on a DG ideal
ring(DGIdeal) -- see mingens(DGIdeal) -- Basic accessors on a DG ideal
net(DGIdeal) (missing documentation)
minimalPresentation(DGIdeal) -- see prune(DGIdeal) -- Trim a DG ideal down to a minimal generating set
prune(DGIdeal) -- Trim a DG ideal down to a minimal generating set
RingElement % DGIdeal -- Reduce a ring element modulo a DG ideal
ZZ % DGIdeal -- see RingElement % DGIdeal -- Reduce a ring element modulo a DG ideal
toComplex(DGIdeal) -- Export a DG ideal to a Complex (not yet implemented)

For the programmer

The object DGIdeal is a type, with ancestor classes MutableHashTable < HashTable < Thing.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/DGAlgebras/doc.m2:2287:0.

DGIdeal -- The class of d-closed graded ideals of a DG algebra

Description

See also

Functions and methods returning an object of class DGIdeal:

Methods that use an object of class DGIdeal:

For the programmer