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# complexToIdeal -- The monomial ideal associated to a complex.

- Usage:
    `complexToIdeal(C)`
- Inputs:
  - C, an embedded complex,
- Optional inputs:
  - embedded => ..., default value null, Use only faces of the embedding complex.
- Outputs:
  - an ideal,

## Description

The Stanley-Reisner ideal associated to a complex, i.e., the ideal of non-faces of C.

If the option embedded=>true is given, then only Faces of embeddingComplex C are used.

    i1 : R=QQ[x_0..x_4]

    o1 = R

    o1 : PolynomialRing

    i2 : addCokerGrading(R)

    o2 = | -1 -1 -1 -1 |
         | 1  0  0  0  |
         | 0  1  0  0  |
         | 0  0  1  0  |
         | 0  0  0  1  |

                  5       4
    o2 : Matrix ZZ  <-- ZZ

    i3 : C0=simplex(R)

    o3 = 4: x x x x x
             0 1 2 3 4

    o3 : complex of dim 4 embedded in dim 4 (printing facets)
         equidimensional, simplicial, F-vector {1, 5, 10, 10, 5, 1}, Euler = 0

    i4 : I=ideal(x_0*x_1,x_1*x_2,x_2*x_3,x_3*x_4,x_4*x_0)

    o4 = ideal (x x , x x , x x , x x , x x )
                 0 1   1 2   2 3   3 4   0 4

    o4 : Ideal of R

    i5 : C=idealToComplex(I)

    o5 = 1: x x  x x  x x  x x  x x
             0 2  0 3  1 3  1 4  2 4

    o5 : complex of dim 1 embedded in dim 4 (printing facets)
         equidimensional, simplicial, F-vector {1, 5, 5, 0, 0, 0}, Euler = -1

    i6 : embeddingComplex C

    o6 = 4: x x x x x
             0 1 2 3 4

    o6 : complex of dim 4 embedded in dim 4 (printing facets)
         equidimensional, simplicial

    i7 : idealToComplex(I,C0)

    o7 = 1: x x  x x  x x  x x  x x
             0 2  0 3  1 3  1 4  2 4

    o7 : complex of dim 1 embedded in dim 4 (printing facets)
         equidimensional, simplicial, F-vector {1, 5, 5, 0, 0, 0}, Euler = -1

    i8 : complexToIdeal(C)

    o8 = ideal (x x , x x , x x , x x , x x )
                 0 1   1 2   2 3   0 4   3 4

    o8 : Ideal of R

    i9 : cC=idealToCoComplex(I,C0)

    o9 = 2: x x x  x x x  x x x  x x x  x x x
             0 1 3  0 2 3  0 2 4  1 2 4  1 3 4

    o9 : co-complex of dim 2 embedded in dim 4 (printing facets)
         equidimensional, simplicial, F-vector {0, 0, 0, 5, 5, 1}, Euler = 1

    i10 : cC==complement C

    o10 = true

    i11 : I==coComplexToIdeal(cC)

    o11 = true

    i12 : dualize cC

    o12 = 1: v v  v v  v v  v v  v v
              0 2  0 3  1 3  1 4  2 4

    o12 : complex of dim 1 embedded in dim 4 (printing facets)
          equidimensional, simplicial, F-vector {1, 5, 5, 0, 0, 0}, Euler = -1

## See also

- coComplexToIdeal -- The monomial ideal associated to a CoComplex.
- idealToComplex -- The complex associated to a reduced monomial ideal.
- idealToCoComplex -- The co-complex associated to a reduced monomial ideal.
- embeddingComplex -- The embedding complex of a complex or co-complex.

## Ways to use complexToIdeal:

- `complexToIdeal(Complex)`

## For the programmer

The object complexToIdeal is a method function with options.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/SRdeformations.m2:6504:0.