How is the cokernel of a module homomorphism defined?

Conditions

• $f:A\to B$ is a homomorphism of $R$-modules.
• $\operatorname{im}(f)$ is a submodule of $B$.

Reasoning, step by step

1. Identify the codomain $B$ of the homomorphism $f$.
2. Determine the image of $f$, denoted $\operatorname{im}(f)$, which is the set of all elements in $B$ that are mapped to by some element in $A$.
3. Form the quotient module $B/\operatorname{im}(f)$.
4. This quotient is the cokernel of $f$.
5. Symbolically, $\operatorname{coker}(f) = B/\operatorname{im}(f)$.
6. In the specific context of the video, for $i:X\to Z$, $\operatorname{coker}(i) = Z/\operatorname{im}(i)$.
7. The video explicitly states this definition to prepare for the proof of the lemma.

Example

The video states: 'For a generic homomorphism $f:A\to B$, the board writes $\operatorname{coker} f = B/\operatorname{im} A$. This prepares the later identification of $Z/\operatorname{im} i$ with $\operatorname{coker}(i)$.'

Common misconceptions

• Confusing the cokernel with the kernel. The kernel is a submodule of the domain ($A$), while the cokernel is a quotient of the codomain ($B$).
• Thinking the cokernel is the image itself. The cokernel is the quotient of the codomain by the image.
• Assuming the cokernel is always zero. It is zero if and only if the map is surjective.

Watch the explanation

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