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

Synopsis

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 complex(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 : ComplexMap
i7 : L' = complex(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 : ComplexMap

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 : ComplexMap

See also

Ways to use lyubeznikSimplicialComplex:

For the programmer

The object lyubeznikSimplicialComplex is a method function with options.