next | previous | forward | backward | up | top | index | toc | Macaulay2 web site
Macaulay2Doc :: cokernel

cokernel -- cokernel of a map of modules, graded modules, or chaincomplexes

Synopsis

Description

coker is a synonym for cokernel.

The generators of the cokernel are provided by the generators of the target of f. In other words, cover target f and cover cokernel f are equal.

An argument f that is a RingElement is interpreted as a one by one matrix.

i1 : R = ZZ[a..d];
i2 : M = cokernel matrix{{2*a-b,3*c-5*d,a^2-b-3}}

o2 = cokernel | 2a-b 3c-5d a2-b-3 |

                            1
o2 : R-module, quotient of R
If f is a matrix, and the target of f is a submodule, the resulting module will be a subquotient module.
i3 : f = map(a*M, M, a^3+a^2*b)

o3 = {1} | a+10b+18 |

o3 : Matrix
i4 : (target f,source f)

o4 = (subquotient (| a |, | 2a-b 3c-5d a2-b-3 |), cokernel | 2a-b 3c-5d
     ------------------------------------------------------------------------
     a2-b-3 |)

o4 : Sequence
i5 : N = cokernel f

o5 = subquotient (| a |, | a2+10ab+18a 2a-b 3c-5d a2-b-3 |)

                               1
o5 : R-module, subquotient of R
i6 : minimalPresentation N

o6 = cokernel | 81 27d 3c-5d 3b-18 a+b-9 9d2 bd-6d b2-b-30 3d3 d4 |

                            1
o6 : R-module, quotient of R

See also

Ways to use cokernel :