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lyubeznikResolution -- create the Lyubeznik resolution of an ordered set of monomials.

Synopsis

Description

For a monomial ideal $M$ in a polynomial ring $S$, minimally generated by $L$, the Lyubeznik resolution is a resolution of $S/M$ determined by a total ordering of the minimal generators of $M$. It is the subcomplex of the Taylor resolution of $M$ induced on the rooted faces. If $L$ is used as input, the ordering is the order in which the monomials appear in $L$. If $M$ is used as the input, the ordering is obtained from $\operatorname{first} \operatorname{mingens} \operatorname{entries} M$. For more details on Lyubeznik resolutions and their construction, see Jeff Mermin Three Simplicial Resolutions, (English summary) Progress in commutative algebra 1, 127–141, de Gruyter, Berlin, 2012.

i1 : S = QQ[x,y];
 
i2 : M = monomialIdeal{x*y,x^2,y^3};

o2 : MonomialIdeal of S
 
i3 : F = lyubeznikResolution M;
 
i4 : F.dd

          1                    3
o4 = 0 : S  <---------------- S  : 1
               | xy x2 y3 |

          3                      2
     1 : S  <------------------ S  : 2
               {2} | -x -y2 |
               {2} | y  0   |
               {3} | 0  x   |

o4 : 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 $\mathrm{MonomialOrder}$ argument.

i5 : first entries mingens M

            2   3
o5 = {x*y, x , y }

o5 : List
 
i6 : F' = lyubeznikResolution({x^2,y^3,x*y});
 
i7 : F'.dd

          1                    3
o7 = 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   |

o7 : ChainComplexMap
 
i8 : F' = lyubeznikResolution(M,MonomialOrder=>{1,2,0});
 
i9 : F'.dd

          1                    3
o9 = 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   |

o9 : ChainComplexMap

See also

Ways to use lyubeznikResolution :

For the programmer

The object lyubeznikResolution is a method function with options.