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# cokernel(ToricVectorBundleKlyachko,Matrix) -- the cokernel of a morphism to a vector bundle

- Function: cokernel
- Usage:
    `E1 = coker(E,M)`
- Inputs:
  - E, a vector bundle on a toric variety (Klyachko's description)
  - M, a matrix, over ZZ or QQ
- Outputs:
  - E1, a vector bundle on a toric variety (Klyachko's description)

## Description

M must be a matrix over ZZ or QQ where the target space is the space of the bundle, i.e., the matrix must have $k$ rows if the bundle has rank $k$. Then the new bundle is given on each ray $\rho$ by the following filtration of coker(E,M)${}^\rho = ( E^{\rho} ) / $im(M) :

coker(E,M)${}^\rho(i) := E^{\rho}(i) / ( E^{\rho}(i) \cap $ im(M) ).

    i1 : E = tangentBundle hirzebruchFan 2

    o1 = {dimension of the variety => 2 }
          number of affine charts => 4
          number of rays => 4
          rank of the vector bundle => 2

    o1 : ToricVectorBundleKlyachko

    i2 : E = E ** E

    o2 = {dimension of the variety => 2 }
          number of affine charts => 4
          number of rays => 4
          rank of the vector bundle => 4

    o2 : ToricVectorBundleKlyachko

    i3 : M = matrix {{1,0},{0,1},{1,0},{0,1/1}}

    o3 = | 1 0 |
         | 0 1 |
         | 1 0 |
         | 0 1 |

                  4       2
    o3 : Matrix QQ  <-- QQ

    i4 : E1 = coker(E,M)

    o4 = {dimension of the variety => 2 }
          number of affine charts => 4
          number of rays => 4
          rank of the vector bundle => 2

    o4 : ToricVectorBundleKlyachko

    i5 : details E1

    o5 = HashTable{| -1 | => (| -1/2 1/2 |, | -2 -1 |)}
                   | 2  |     | 1    0   |
                   | 0  | => (| 0 1 |, | -2 -1 |)
                   | -1 |     | 1 0 |
                   | 0 | => (| 0 1 |, | -2 -1 |)
                   | 1 |     | 1 0 |
                   | 1 | => (| 1 0 |, | -2 -1 |)
                   | 0 |     | 0 1 |

    o5 : HashTable

## See also

- image(ToricVectorBundleKlyachko,Matrix) -- the image of a vector bundle under a morphism
- kernel(ToricVectorBundleKlyachko,Matrix) -- the kernel of a morphism to a vector bundle

## Ways to use this method:

- cokernel(ToricVectorBundleKlyachko,Matrix) -- the cokernel of a morphism to a vector bundle

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/OldToricVectorBundles.m2:2633:0.