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

Synopsis

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

See also

Ways to use scarfSimplicialComplex :

For the programmer

The object scarfSimplicialComplex is a method function.