# cocycleCheck -- checks if a toric vector bundle fulfills the cocycle condition

## Synopsis

• Usage:
b = cocycleCheck E
• Inputs:
• Outputs:
• b, , whether E satisfies the cocyle condition

## Description

The transition matrices in E define an equivariant toric vector bundle if they satisfy the cocycle condition. I.e. in this implementation of complete fans this means that for every codimension 2 cone of the fan the cycle of transition matrices of codimension 1 cones containing the codimension 2 cone gives the identity when multiplied.

 i1 : E = toricVectorBundle(2,pp1ProductFan 2,"Type" => "Kaneyama") o1 = {dimension of the variety => 2 } number of affine charts => 4 rank of the vector bundle => 2 o1 : ToricVectorBundleKaneyama i2 : details E o2 = (HashTable{0 => (| 1 0 |, 0) }, HashTable{(0, 1) => | 1 0 |}) | 0 1 | | 0 1 | 1 => (| 1 0 |, 0) (0, 2) => | 1 0 | | 0 -1 | | 0 1 | 2 => (| -1 0 |, 0) (1, 3) => | 1 0 | | 0 1 | | 0 1 | 3 => (| -1 0 |, 0) (2, 3) => | 1 0 | | 0 -1 | | 0 1 | o2 : Sequence i3 : A = matrix{{1,2},{0,1}}; 2 2 o3 : Matrix ZZ <--- ZZ i4 : B = matrix{{1,0},{3,1}}; 2 2 o4 : Matrix ZZ <--- ZZ i5 : C = matrix{{1,-2},{0,1}}; 2 2 o5 : Matrix ZZ <--- ZZ i6 : E1 = addBaseChange(E,{A,B,C,matrix{{1,0},{0,1}}}) o6 = {dimension of the variety => 2 } number of affine charts => 4 rank of the vector bundle => 2 o6 : ToricVectorBundleKaneyama i7 : cocycleCheck E1 o7 = false i8 : D = inverse(B)*A*C o8 = | 1 0 | | -3 1 | 2 2 o8 : Matrix ZZ <--- ZZ i9 : E1 = addBaseChange(E,{A,B,C,D}) o9 = {dimension of the variety => 2 } number of affine charts => 4 rank of the vector bundle => 2 o9 : ToricVectorBundleKaneyama i10 : cocycleCheck E1 o10 = true