substitute(SimplicialComplex,PolynomialRing) -- change the ring of a simplicial complex

Synopsis

Function: substitute
Usage:

substitute(Delta, R)
Inputs:
- Delta, an abstract simplicial complex,
- R, a polynomial ring,
Outputs:
- an abstract simplicial complex,

Description

Given a polynomial ring $R$, with enough variables, we can create a simplicial complex identical to $\Delta$, defined over the ring $R$.

i1 : S = ZZ/23[x,y,z,w];

i2 : Δ = simplexComplex(3,S)

o2 = simplicialComplex | xyzw |

o2 : SimplicialComplex

i3 : R = ZZ/101[a,b,c,d,e];

i4 : Γ = substitute(Δ, R)

o4 = simplicialComplex | abcd |

o4 : SimplicialComplex

This method is a works by applying substitute(RingElement,Ring) to the facets of $\Delta$.

i5 : code(substitute, SimplicialComplex, PolynomialRing)

o5 = -- code for method: substitute(SimplicialComplex,PolynomialRing)
     /usr/local/share/Macaulay2/
     SimplicialComplexes/Code.m2:671:77-679:71: --source code:
     substitute(SimplicialComplex, PolynomialRing) := SimplicialComplex => (D, R) -> (
         if ideal D === ideal(1_(ring D))
         then (
             I := sub(ideal D, R);
             simplicialComplex monomialIdeal I
             )
         else (
             n := numgens ring D;
             simplicialComplex for F in facets D list sub(F, (vars R)_{0..n-1})
             )
         )

Ways to use this method: