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# buchbergerSimplicialComplex -- make the Buchberger complex of a monomial ideal

## Synopsis

• Usage:
buchbergerSimplicialComplex(L,R)
buchbergerSimplicialComplex(I,R)
• Inputs:
• I, ,
• L, a list, a minimal set of generators for a monomial ideal
• R, a ring, the ambient ring for the Buchberger complex
• Outputs:

## Description

If a monomial ideal is minimally generated by a list of monomials $L$, then the Buchberger complex is the simplicial complex whose vertices correspond to the monomials in $L$ and whose faces faces correspond subsets $F$ of $L$ for which no monomial in $L$ properly divides $\text{lcm} (F)$. When we say a monomial $m$ properly divides $\text{lcm} (F)$, we mean $m$ divides $\text{lcm} (F)$ and $(\text{lcm} (F))/m$ has the same support as $\text{lcm} (F)$.

The Buchberger complex is a generalization of the Buchberger graph, first introduced in Miller-Sturmfels Monomial Ideals and Planar Graphs as an important object of study for Gröbner bases. Oltaneau and Welker introduce the Buchberger complex in their paper The Buchberger Resolution.

 i1 : R = ZZ/101[x_0..x_4]; i2 : S = ZZ/101[a,b,c,d,e]; i3 : I = monomialIdeal(x_1^2, x_2^2, x_3^2, x_1*x_3, x_2*x_4); o3 : MonomialIdeal of R i4 : B1 = buchbergerSimplicialComplex(I,S) o4 = simplicialComplex | acde abcd | o4 : SimplicialComplex

The Buchberger complex supports a free resolution of $I$, called the Buchberger resolution of $I$.

 i5 : BRes = chainComplex(B1, Labels => first entries mingens I) 1 5 9 7 2 o5 = R <-- R <-- R <-- R <-- R 0 1 2 3 4 o5 : ChainComplex i6 : HH_0(BRes) o6 = cokernel | x_2x_4 x_3^2 x_1x_3 x_2^2 x_1^2 | 1 o6 : R-module, quotient of R i7 : all(1..dim B1+1, i -> prune HH_i(BRes) == 0) o7 = true i8 : BRes == buchbergerResolution(I) o8 = true

If the monomial ideal is square free, then the Buchberger complex is the simplex on $\#L$ vertices.

 i9 : L = {x_1*x_2, x_1*x_3*x_4, x_0*x_2*x_4}; i10 : B2 = buchbergerSimplicialComplex(L,S) o10 = simplicialComplex | abc | o10 : SimplicialComplex