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# scarfSimplicialComplex -- create the Scarf simplicial complex for a list of monomials

## Synopsis

• Usage:
scarfSimplicialComplex(L, R)
scarfSimplicialComplex(M, R)
• Inputs:
• L, a list, of monomials in a polynomial ring, that minimally generate a monomial ideal
• M, ,
• R, a ring, the ambient ring used to construct the Scarf simplicial complex.
• Outputs:

## Description

The Scarf simplicial complex is the simplicial complex that supports the Scarf complex of a monomial ideal. The Scarf complex does not need to be an acyclic simplicial complex. In fact, every simplicial complex that is not the boundary of a simplex is the Scarf complex for some monomial ideal, see Theorem 5.3 in Peeva, Irena; Velasco, Mauricio Frames and Degenerations of Monomial Resolutions. For more information on the Scarf simplicial complex and its construction, see Bayer, Dave; Peeva, Irena; Sturmfels, Bernd Monomial Resolutions. Math. Res. Lett. 5 (1998), no. 1-2, 31–46, or Jeff Mermin Three Simplicial Resolutions, (English summary) Progress in commutative algebra 1, 127–141, de Gruyter, Berlin, 2012.

 i1 : R = ZZ[a,b,c,d]; i2 : S = ZZ/17[x_0..x_3]; i3 : M = monomialIdeal(x_0*x_1,x_1*x_2,x_2*x_3) o3 = monomialIdeal (x x , x x , x x ) 0 1 1 2 2 3 o3 : MonomialIdeal of S i4 : D = scarfSimplicialComplex(M,R) o4 = simplicialComplex | bc ab | o4 : SimplicialComplex i5 : prune homology D o5 = -1 : 0 0 : 0 1 : 0 o5 : GradedModule i6 : M' = monomialIdeal(x_0*x_1,x_0*x_3,x_1*x_2,x_2*x_3) o6 = monomialIdeal (x x , x x , x x , x x ) 0 1 1 2 0 3 2 3 o6 : MonomialIdeal of S i7 : D' = scarfSimplicialComplex(M',R) o7 = simplicialComplex | cd bd ac ab | o7 : SimplicialComplex i8 : prune homology D' o8 = -1 : 0 0 : 0 1 1 : ZZ o8 : GradedModule