isMinimal -- check if a labeled cell complex supports a minimal resolution

Synopsis

Usage:

isMinimal C
Inputs:
- C, a cell complex,
Outputs:
- a Boolean value, true if the cellular resolution supported on C is minimal false otherwise

Description

This determines whether the cell complex C supports a minimal free resolution of the monomialIdeal I generated by the labels of the vertices of C. Note: we assume the cell complex is free.

i1 : R = QQ[x,y,z];

i2 : v1 = newCell({},x^2*y);

i3 : v2 = newCell({},y*z);

i4 : v3 = newCell({},z^3);

i5 : e12 = newCell({v1,v2});

i6 : e13 = newCell({v1,v3});

i7 : e23 = newCell({v2,v3});

i8 : f123 = newCell({e12,e13,e23});

i9 : C = cellComplex(R,{e12,e23});

i10 : isMinimal C

o10 = true

i11 : D = cellComplex(R,{f123});

i12 : isMinimal D

o12 = false

Caveat

This function makes no attempt to check that the cell complex gives a resolution or that the complex is a complex of free modules.

Ways to use isMinimal:

isMinimal(CellComplex)

For the programmer