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# homogDual -- Computes the dual of a pure homogeneous ideal

- Usage:
    `homogDual I`
- Inputs:
  - I, an instance of the type NCIdeal, or an NCQuotientRing or a Ring.
- Outputs:
  - an instance of the type NCIdeal, or an NCQuotientRing

## Description

The homogeneous dual of a pure ideal I in an NCPolynomialRing A is generated by the orthogonal complement to the generators of I under the natural pairing on the generating subspace of A and its linear dual. Though technically the dual ideal belongs to the tensor algebra on the dual space of generators, this method returns the dual ideal in the same NCPolynomialRing.

If the input is an NCQuotient ring, the method is applied to the defining ideal of the quotient and the corresponding quotient ring is returned.

Commonly used in conjunction with quadraticClosure.

    i1 : A = QQ{x,y,z}

    o1 = A

    o1 : NCPolynomialRing

    i2 : I = ncIdeal{x*z-z*x, y*z, x*y^2-y^2*x, x^3*y-y*x^3}

                                        2    2     3  3
    o2 = Two-sided ideal {-zx+xz, yz, -y x+xy , -yx +x y}

    o2 : NCIdeal

    i3 : J = quadraticClosure I

    o3 = Two-sided ideal {-zx+xz, yz}

    o3 : NCIdeal

    i4 : J' = homogDual J

                           2           2              2
    o4 = Two-sided ideal {x , yx, xy, y , zy, zx+xz, z }

    o4 : NCIdeal

## See also

- quadraticClosure -- Creates the subideal generated by quadratic elements of a given ideal

## Ways to use homogDual:

- `homogDual(NCIdeal)`
- `homogDual(NCQuotientRing)`
- `homogDual(Ring)`

## For the programmer

The object homogDual is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/NCAlgebra/NCAlgebraDoc.m2:2792:0.