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

Synopsis

• Usage:
lyubeznikResolution L
lyubeznikResolution M
• Inputs:
• L, a list, of monomials in a polynomial ring, that minimally generate a monomial ideal.
• M, ,
• Optional inputs:
• MonomialOrder => a list, default value {}
• Outputs:
• , the Lyubeznik resolution of $S/M$.

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