generators(Module) -- the generator matrix of a module

Synopsis

Function: generators
Usage:

generators M

gens M
Inputs:
- M, a module
Optional inputs:
- CoefficientRing => ..., default value null, unused option
Outputs:
- a matrix, the matrix of generators of M.

Description

Every module in Macaulay2 has, at least implicitly, a generator matrix and a matrix of relations, both of which are matrices between free modules. This function returns the generator matrix. The module is generated by (the images of) the columns of this matrix.

i1 : R = GF(8,Variable=>a)

o1 = R

o1 : GaloisField

i2 : f = R_0 ++ R_0^2 ++ R_0^3 ++ R_0^4

o2 = | a 0  0   0    |
     | 0 a2 0   0    |
     | 0 0  a+1 0    |
     | 0 0  0   a2+a |

             4      4
o2 : Matrix R  <-- R

i3 : generators image f

o3 = | a 0  0   0    |
     | 0 a2 0   0    |
     | 0 0  a+1 0    |
     | 0 0  0   a2+a |

             4      4
o3 : Matrix R  <-- R

i4 : generators cokernel f

o4 = | 1 0 0 0 |
     | 0 1 0 0 |
     | 0 0 1 0 |
     | 0 0 0 1 |

             4      4
o4 : Matrix R  <-- R

Caveat

This function returns a matrix with the given generators. This set of generators may not be minimal, or sorted in any particular order. Use trim(Module) or mingens(Module) instead.

Ways to use this method: