What does it mean for a sequence of modules to be exact in this video?

Conditions

• The sequence consists of module homomorphisms.
• Exactness is checked at each internal object (not the ends, unless specified by zeros).

Reasoning, step by step

1. Identify a middle object in the sequence, e.g., $Z$ in $X \xrightarrow{i} Z \xrightarrow{p} Y$.
2. Identify the map coming into $Z$, which is $i:X\to Z$.
3. Identify the map going out of $Z$, which is $p:Z\to Y$.
4. Compute the image of the incoming map, $\operatorname{im}(i)$.
5. Compute the kernel of the outgoing map, $\ker(p)$.
6. The sequence is exact at $Z$ if $\operatorname{im}(i) = \ker(p)$.
7. The video states: 'remember exact sequence means that image same as kernel'.
8. This definition is applied to both parts of Lemma 2.3.

Example

The video states: 'Before proving anything, the speaker recalls the definition being used: exactness means that at each middle object, the image of the incoming map equals the kernel of the outgoing map. For the displayed sequence, the relevant equality is $\operatorname{im}(i)=\ker(p)$.'

Common misconceptions

• Thinking exactness means the maps are isomorphisms.
• Confusing image and kernel.
• Assuming exactness only applies to short exact sequences (0->A->B->C->0); it applies to any sequence at any internal object.

Watch the explanation

Explore next

• What does it mean for a sequence of modules to be exact in this video?
• How is exactness used in the proof of Lemma 2.3?
• What is the difference between exactness at the left and right ends?