How is $\operatorname{coker}(i)\simeq Y$ proved from the exact sequence?

Conditions

• $X \xrightarrow{i} Z \xrightarrow{p} Y \to 0$ is an exact sequence of $R$-modules.
• $R$ is a commutative ring.

Reasoning, step by step

1. Start with the exact sequence $X \xrightarrow{i} Z \xrightarrow{p} Y \to 0$.
2. Use exactness at $Z$ to establish $\operatorname{im}(i) = \ker(p)$.
3. Use exactness at $Y$ (from $Y\to 0$) to establish $\operatorname{im}(p) = Y$.
4. Apply the First Isomorphism Theorem to the map $p:Z\to Y$, yielding $Z/\ker(p) \simeq \operatorname{im}(p)$.
5. Substitute $\operatorname{im}(p) = Y$ into the isomorphism to get $Z/\ker(p) \simeq Y$.
6. Substitute $\ker(p) = \operatorname{im}(i)$ into the denominator to get $Z/\operatorname{im}(i) \simeq Y$.
7. Recognize $Z/\operatorname{im}(i)$ as $\operatorname{coker}(i)$ by definition.
8. Conclude $\operatorname{coker}(i) \simeq Y$.
9. The video explicitly walks through these substitution steps on the whiteboard.

Example

The video states: 'The board successively writes $\operatorname{im} i=\ker p$, $X/\ker i\simeq \operatorname{im} i$, $Z/\ker p\simeq \operatorname{im} p$, $\operatorname{im} p=Y$, and $Z/\operatorname{im} i\simeq Y$. ... Finally, replace $\ker p$ by $\operatorname{im} i$ using the earlier exactness relation $\operatorname{im} i=\ker p$. The quotient becomes $Z/\operatorname{im} i \simeq Y$. By the definition written at the start, $Z/\operatorname{im} i$ is exactly $\operatorname{coker}(i)$, so the desired conclusion $\operatorname{coker}(i)\simeq Y$ follows.'

Common misconceptions

• Forgetting to use the exactness at $Y$ to conclude $\operatorname{im}(p)=Y$.
• Confusing the First Isomorphism Theorem with the definition of cokernel.
• Assuming the proof requires the map $i$ to be injective (which is not given in this part of the lemma).

Watch the explanation

Explore next

• How is $\operatorname{coker}(i)\simeq Y$ proved from the exact sequence?
• What role does the First Isomorphism Theorem play in this proof?
• Why is exactness at $Y$ necessary?