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# lyubeznikSimplicialComplex -- create a simplicial complex supporting a Lyubeznik resolution of a monomial ideal

## Synopsis

• Usage:
lyubeznikSimplicialComplex(L,R)
lyubeznikSimplicialComplex(M,R)
• Inputs:
• L, a list, of monomials in a polynomial ring, that minimally generate a monomial ideal.
• M, ,
• R, , the ambient ring used in constructing the Lyubeznik simplicial complex.
• Optional inputs:
• MonomialOrder => a list, default value {}
• Outputs:

## Description

The Lyubeznik simplicial complex is the simplicial complex that supports the Lyubeznik resolution of an ordered set of monomials. This function is sensitive to the order in which the monomials in $L$ appear. If you are using a monomial ideal $M$ as your input, then the order of the monomials is given by $\operatorname{first} \operatorname{entries} \operatorname{mingens} M$.

 i1 : S = QQ[x,y]; i2 : R = QQ[a,b,c]; i3 : M = monomialIdeal{x*y,x^2,y^3}; o3 : MonomialIdeal of S i4 : D = lyubeznikSimplicialComplex(M,R) o4 = simplicialComplex | ac ab | o4 : SimplicialComplex

The lyubeznik resolution of $M$ is the homogenization of $D$ by $M$ (See chainComplex(SimplicialComplex,Labels=>...)).

 i5 : L = lyubeznikResolution(M); i6 : L.dd 1 3 o6 = 0 : S <---------------- S : 1 | xy x2 y3 | 3 2 1 : S <------------------ S : 2 {2} | -x -y2 | {2} | y 0 | {3} | 0 x | o6 : ChainComplexMap i7 : L' = chainComplex(D,Labels=>(first entries mingens M)); i8 : L'.dd 1 3 o8 = 0 : S <---------------- S : 1 | xy x2 y3 | 3 2 1 : S <------------------ S : 2 {2} | -x -y2 | {2} | y 0 | {3} | 0 x | o8 : ChainComplexMap

Changing the order of the generators may change the output. We can do this by manually entering the permuted list of generators, or by using the optional $\operatorname{MonomialOrder}$ argument.

 i9 : first entries mingens M 2 3 o9 = {x*y, x , y } o9 : List i10 : D' = lyubeznikSimplicialComplex(M,R,MonomialOrder=>{1,2,0}) o10 = simplicialComplex | abc | o10 : SimplicialComplex i11 : D' = lyubeznikSimplicialComplex({x^2,y^3,x*y},R) o11 = simplicialComplex | abc | o11 : SimplicialComplex i12 : (lyubeznikResolution(M,MonomialOrder=>{1,2,0})).dd 1 3 o12 = 0 : S <---------------- S : 1 | x2 y3 xy | 3 3 1 : S <--------------------- S : 2 {2} | -y3 -y 0 | {3} | x2 0 -x | {2} | 0 x y2 | 3 1 2 : S <--------------- S : 3 {5} | 1 | {3} | -y2 | {4} | x | o12 : ChainComplexMap