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# coefficientRing(SimplicialComplex) -- get the coefficient ring of the underlying polynomial ring

## Synopsis

• Function: coefficientRing
• Usage:
coefficientRing Delta
• Inputs:
• Delta, ,
• Outputs:
• a ring, that is the coefficient ring of the polynomial ring that contains the defining Stanley–Reisner ideal

## Description

In this package, an abstract simplicial complex is represented as squarefree monomial ideal in a polynomial ring. This method returns the coefficient ring of this polynomial ring.

We construct the boundary of the $4$-sphere using three different polynomial rings.

 i1 : S = ZZ[a..e]; i2 : Δ = simplicialComplex {b*c*d*e, a*c*d*e, a*b*d*e, a*b*c*e, a*b*c*d} o2 = simplicialComplex | bcde acde abde abce abcd | o2 : SimplicialComplex i3 : ring Δ o3 = S o3 : PolynomialRing i4 : monomialIdeal Δ o4 = monomialIdeal(a*b*c*d*e) o4 : MonomialIdeal of S i5 : coefficientRing Δ o5 = ZZ o5 : Ring i6 : assert (ring Δ === S and coefficientRing Δ === ZZ and numgens ideal Δ === 1 )
 i7 : R = QQ[a..e]; i8 : Γ = simplicialComplex {b*c*d*e, a*c*d*e, a*b*d*e, a*b*c*e, a*b*c*d} o8 = simplicialComplex | bcde acde abde abce abcd | o8 : SimplicialComplex i9 : ring Γ o9 = R o9 : PolynomialRing i10 : monomialIdeal Γ o10 = monomialIdeal(a*b*c*d*e) o10 : MonomialIdeal of R i11 : coefficientRing Γ o11 = QQ o11 : Ring i12 : assert (ring Γ === R and coefficientRing Γ === QQ and numgens ideal Γ === 1)
 i13 : S' = ZZ/101[a..f]; i14 : Δ' = simplicialComplex {b*c*d*e, a*c*d*e, a*b*d*e, a*b*c*e, a*b*c*d} o14 = simplicialComplex | bcde acde abde abce abcd | o14 : SimplicialComplex i15 : ring Δ' o15 = S' o15 : PolynomialRing i16 : monomialIdeal Δ' o16 = monomialIdeal (a*b*c*d*e, f) o16 : MonomialIdeal of S' i17 : coefficientRing Δ' ZZ o17 = --- 101 o17 : QuotientRing i18 : assert (ring Δ' === S' and coefficientRing Δ' === ZZ/101 and numgens ideal Δ' === 2)

The Stanley–Reisner ideal is part of the defining data of an abstract simplicial complex, so this method does no computation.

Although an abstract simplicial complex can be represented by a Stanley–Reisner ideal in any polynomial ring with a sufficiently large number of variables, some operations in this package do depend of the choice of the polynomial ring (or its coefficient ring). For example, the chain complex of an abstract simplicial complex is, by default, constructed over the coefficient ring of its polynomial ring, and the dual of a simplicial complex (or monomial ideal) is dependent on the number of variables in the polynomial ideal.

 i19 : C = chainComplex Δ 1 5 10 10 5 o19 = ZZ <-- ZZ <-- ZZ <-- ZZ <-- ZZ -1 0 1 2 3 o19 : ChainComplex i20 : D = chainComplex Γ 1 5 10 10 5 o20 = QQ <-- QQ <-- QQ <-- QQ <-- QQ -1 0 1 2 3 o20 : ChainComplex i21 : C' = chainComplex Δ' ZZ 1 ZZ 5 ZZ 10 ZZ 10 ZZ 5 o21 = (---) <-- (---) <-- (---) <-- (---) <-- (---) 101 101 101 101 101 -1 0 1 2 3 o21 : ChainComplex i22 : assert (D == C ** QQ and C' == C ** (ZZ/101))