Exact Sequence

Definition

A sequence of groups and morphisms $\cdots \to A_{n+1}\stackrel{f_{n+1}}{\to }{A}_n\stackrel{f_n}{\to }{A}_{n-1}\to \cdots$

is exact at $A_n$ if the image of the incoming morphism is equal to the kernel of the outgoing morphism:

$\text{im}(f_{n+1})=\text{ker}(f_n)$

A sequence is called an exact sequence if it is exact at every object (except the ends of a finite sequence).

Short Exact Sequence

A short exact sequence (SES) is an exact sequence of the form:

$0\to A\stackrel{i}{\to }B\stackrel{p}{\to }C\to 0$

This structure implies:

1. $i$ is injective ($\text{ker}(i)=0$).
2. $p$ is surjective ($\text{im}(p)=C$).
3. $\text{im}(i)=\text{ker}(p)$, meaning $A\cong \text{im}(i)$ is a normal subgroup of $B$ and $B/A\cong C$.

Long Exact Sequence

In homological algebra, a long exact sequence often arises from a short exact sequence of chain complexes. Given a SES of chain complexes $0\to \mathcal{A}\to \mathcal{B}\to \mathcal{C}\to 0$, there exists a long exact sequence in homology:

$\cdots \to H_n(A)\to H_n(B)\to H_n(C)\stackrel{\partial }{\to }{H}_{n-1}(A)\to \cdots$

where $\partial$ is the connecting homomorphism.

Categorical Generalization

In an abelian category, a sequence is exact if the kernel of $f_n$ is the image of $f_{n+1}$. This is defined using the factorization of a morphism through its coimage, which in an abelian category is isomorphic to its image.

Properties

• A sequence $0\to A\stackrel{f}{\to }B$ is exact if and only if $f$ is a monomorphism.
• A sequence $A\stackrel{g}{\to }B\to 0$ is exact if and only if $g$ is an epimorphism.
• The sequence $0\to A\stackrel{f}{\to }B\to 0$ is exact if and only if $f$ is an isomorphism.

References

1. Riehl, E. (2016). Category Theory in Context. Dover Publications.
2. Mac Lane, S. (1998). Categories for the Working Mathematician. Springer.
3. Aluffi, P. (2009). Algebra: Chapter 0. American Mathematical Society.

Would you like me to demonstrate the Splitting Lemma for short exact sequences?