# idealToComplex -- The complex associated to a reduced monomial ideal.

## Synopsis

• Usage:
idealToComplex(I)
idealToComplex(I,C)
• Inputs:
• Outputs:
• ,

## Description

The Stanley-Complex associated to a reduced monomial ideal.

This is a complex of faces of either C, or if C is not given, a complex of faces of simplex ring I.

 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