coimage -- coimage of a map

Synopsis

Usage:
coimage f
Inputs:
- f, a ring map, a matrix, a chain complex map, or a graded module map, from A to B
Outputs:
- The object A/(kernel f)

Description

In each of these cases, the coimage is isomorphic to the image, but the coimage is presented as a quotient object of the source of the map, whereas the image is presented as a subobject of the target of the map. For rings, we can represent quotient rings, but not subrings, in Macaulay2. Hence, for ring maps, we can compute the coimage, but not the image.

The isomorphism between coimage and image is not always obvious, as the following example shows.

i1 : R = QQ[a..d];

i2 : M = matrix{{a^3,b^3-c^3,a*b*c,a*(b^2-c^2)}}

o2 = | a3 b3-c3 abc ab2-ac2 |

             1       4
o2 : Matrix R  <--- R

i3 : image M

o3 = image | a3 b3-c3 abc ab2-ac2 |

                             1
o3 : R-module, submodule of R

i4 : coimage M

o4 = cokernel {3} | 0    -bc 0     -b2+c2 |
              {3} | a    0   0     0      |
              {3} | b-c  a2  b2-c2 0      |
              {3} | -b-c 0   -bc   a2     |

                            4
o4 : R-module, quotient of R

i5 : kernel M

o5 = image {3} | 0    -bc 0     -b2+c2 |
           {3} | a    0   0     0      |
           {3} | b-c  a2  b2-c2 0      |
           {3} | -b-c 0   -bc   a2     |

                             4
o5 : R-module, submodule of R

Ways to use `coimage` :

coimage(ChainComplexMap)
coimage(GradedModuleMap)
coimage(Matrix)
coimage(RingMap)

coimage -- coimage of a map

Synopsis

Description

See also

Ways to use coimage :

Ways to use `coimage` :