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

## Synopsis

• Usage:
cokernel f
• Inputs:
• f : A --> B, , , or
• Outputs:
• the object B/(image f)

## 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