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# hullComplex -- gives the hull complex of a monomial ideal

## Synopsis

• Usage:
hullComplex I
hullComplex(t,I)
• Inputs:
• Outputs:
• , the hull complex of $I$ as described in Bayer-Sturmfels' Cellular Resolutions of Monomial Modules''

## Description

Given a monomial ideal $I$, this function returns the hull complex of that ideal. If an rational number $t$ is provided, this will set the base used in the exponents used to construct the polytope as described in Combinatorial Commutative Algebra.'' The resulting complex is only a resolution for $t\gg 0$. In particular $t > (n+1)!$ is sufficient where $n$ is the number of variables in the ring. If t is not provided, $(n+1)!+1$ will be used.

The example given below can be found as Example 4.23 in Miller-Sturmfels' Combinatorial Commutative Algebra.'' In this example, the resolution supported on the hull complex is minimal, but this is not always the case. We also see that for $t=3/2$ the resulting complex is no longer a resolution.

 i1 : S = QQ[x,y,z]; i2 : I = monomialIdeal (x^2*z, x*y*z, y^2*z, x^3*y^5, x^4*y^4, x^5*y^3); o2 : MonomialIdeal of S i3 : H = hullComplex I o3 = H o3 : CellComplex i4 : chainComplex H 1 6 7 2 o4 = S <-- S <-- S <-- S -1 0 1 2 o4 : ChainComplex i5 : cells(1,H)/cellLabel 2 4 4 5 3 5 4 3 5 4 5 2 o5 = {x*y z, x y z, x y z, x y , x y z, x y , x y*z} o5 : List i6 : cells(2,H)/cellLabel 4 5 5 4 o6 = {x y z, x y z} o6 : List i7 : isMinimal H o7 = true i8 : H2 = hullComplex (3/2,I) o8 = H2 o8 : CellComplex i9 : prune HH chainComplex H2 o9 = -1 : cokernel | y2z xyz x2z x3y5 x4y4 x5y3 | 0 : cokernel {4} | x 0 0 | {9} | 0 z x | 1 : 0 2 : 0 o9 : GradedModule