What's the intuition behind a split exact sequence?

I've learned the following definition for a split exact sequence:

A short exact sequence of $R$-module homomorphisms $0\to A \stackrel{f}{\to} B \stackrel{g}{\to} C \to 0$ is split if there exists a homomorphism $\alpha:B\to A$ such that $\alpha\circ f = 1$ or a homomorphism $\beta:C\to B$ such that $g\circ \beta = 1$.

I can never remember whether the definition requires $\alpha\circ f = 1$ or $f \circ \alpha = 1$, and similarly for the other mapping, $\beta$. I suspect that my inability to remember this stems from an underlying lack of understanding about what split exact sequences are for, what a sequence being split really tells us about the modules and morphisms involved, and what concept split exact sequences are meant to capture or generalize.

Is there a good way to remember the directions of composition for these definitions?

What is the intuition, utility, and underlying meaning behind a split exact sequence?

The direction of the maps for splitting is always in the opposite direction of the original maps. So you should think of "splitting" as being able to reverse the sequence to obtain $0\to C\to B\to A\to 0$. In terms of the order, you always end up with identity maps on the end. So to split is to have $A\to B\to A$ being the identity on $A$, and to have $C\to B\to C$ being the identity on $C$. In particular, your example would have $\alpha\circ f=\mathbb{1}_A$.

Intuitively, at least in the finitely generated case, $B$ is "bigger" than $A$ and $C$, so we cannot find an injection from $B$ to $A$ or a surjection from $C$ to $B$, so we cannot hope for the maps to be composed in the opposite order to obtain the identity on $B$.

The way I think of splitting is that a split exact sequence $0\to A\to B\to C\to 0$ is one for which there is a (possibly non-canonical) isomorphism $B\cong A\oplus C$, and for which $f$ and $g$ look like the "natural" maps into and from a direct sum, i.e. inclusion and projection. Thus the sequence looks like $$ 0\to A\to A\oplus C\to C\to 0. $$ In particular, the sections now look like inclusion into one factor, $c\to (0,c)$, or projection onto another factor, $(a,c)\to a$.

When you have a short exact sequence

$$ 0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0 $$

$B$ canonically has a submodule $\mathrm{im}(f) \cong A$ and a quotient module $\mathrm{coim}(f) \cong C$. When the sequence is split, we can actually write $B$ as a direct sum of these two modules, which lets us decompose the given short exact sequence into a direct sum of two short exact sequences:

$$ \begin{matrix} 0 \to A \xrightarrow{f} &B& \xrightarrow{g} C \to 0 \\ & \cong \\ 0 \to A \xrightarrow{f} &\mathrm{im}(f)& \to0 \to 0 \\ & \oplus \\ 0 \to 0 \to & \mathrm{coim}(g) & \xrightarrow{g} C \to 0 \end{matrix}$$

I imagine this is where the term split comes from.

A splitting $\beta$ of the epimorphism $g$ is meant to be the monic map expressing how $\mathrm{coim}(g)$ is being made into a submodule of $B$ in a way compatible with $g$. So $g \circ \beta = 1_C$.

Similarly for the splitting $\alpha$ of the monomorphism $f$.

Having a pair of morphisms

$$ Y \xrightarrow{u} X \xrightarrow{v} Y $$ with $v \circ u = 1_Y$ is a significant phenomenon going by lots of names coming from lots of mathematical settings. For example:

• $v$ is a left inverse of $u$
• $u$ is a right inverse of $v$
• $u$ is a split monomorphism, with $v$ its splitting
• $v$ is a split epimorphism, with $u$ its splitting
• $u$ is a section of $v$
• We say that $Y$ is a retract of $X$, given by the retraction $(u,v)$
• $u \circ v$ is a split idempotent, with the splitting given by $(u,v)$