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# addBaseChange -- changing the transition matrices of a toric vector bundle

- Usage:
    `F = addBaseChange(E,L)`
- Inputs:
  - E, a vector bundle on a toric variety (Kaneyama's description)
  - L, a list, with matrices over ZZ or QQ
- Outputs:
  - F, a vector bundle on a toric variety (Kaneyama's description)

## Description

addBaseChange replaces the transition matrices in E by the matrices in the List L. The matrices in L must be in GL($k$,ZZ) or GL($k$,QQ), where $k$ is the rank of the vector bundle T. The list has to contain one matrix for each maximal dimensional cone of the underlying fan over which E is defined. The fan can be recovered with fan(ToricVectorBundle). The vector bundle already has a list of pairs $(i,j)$ denoting the codim 1 intersections of two maximal cones with $i<j$ and they are ordered in lexicographic order. The matrices will be assigned to the pairs $(i,j)$ in that order. To see which codimension 1 cone corresponds to the pair $(i,j)$ use details(ToricVectorBundle). The matrix $A$ assigned to $(i,j)$ denotes the transition $(e_i^1,...,e_i^k) = (e_j^1,...,e_j^k)*A$. The matrices need not satisfy the regularity or the cocycle condition. These can be checked with regCheck and cocycleCheck.

    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 : F = addBaseChange(E,{matrix{{1,2},{0,1}},matrix{{1,0},{3,1}},matrix{{1,-2},{0,1}},matrix{{1,0},{-3,1}}})

    o3 = {dimension of the variety => 2 }
          number of affine charts => 4
          rank of the vector bundle => 2

    o3 : ToricVectorBundleKaneyama

    i4 : details F

    o4 = (HashTable{0 => (| 1 0 |, 0)  }, HashTable{(0, 1) => | 1 2 | })
                          | 0 1 |                             | 0 1 |
                    1 => (| 1 0  |, 0)              (0, 2) => | 1 0 |
                          | 0 -1 |                            | 3 1 |
                    2 => (| -1 0 |, 0)              (1, 3) => | 1 -2 |
                          | 0  1 |                            | 0 1  |
                    3 => (| -1 0  |, 0)             (2, 3) => | 1  0 |
                          | 0  -1 |                           | -3 1 |

    o4 : Sequence

    i5 : cocycleCheck F

    o5 = true

## See also

- addDegrees -- changing the degrees of a toric vector bundle
- regCheck -- checking the regularity condition for a toric vector bundle
- cocycleCheck -- checks if a toric vector bundle fulfills the cocycle condition

## Ways to use addBaseChange:

- `addBaseChange(ToricVectorBundleKaneyama,List)`

## For the programmer

The object addBaseChange is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/OldToricVectorBundles.m2:2225:0.