How do you split a long exact sequence into short exact sequences?

Question

How does one split up a long exact sequence into short exact sequences?

Say you have some longs exact sequences of modules $$ 0\longrightarrow M_1\stackrel{\phi_1}{\longrightarrow}M_2\stackrel{\phi_2}{\longrightarrow}M_3\stackrel{\phi_3}{\longrightarrow}M_4\stackrel{\phi_4}{\longrightarrow}\cdots $$ I've read it's possible to split this into short exact sequences. What exactly does that mean? Would it be written as short exact sequences, one appended to another like $$ 0\longrightarrow N_1\longrightarrow M_1\longrightarrow N'_1\longrightarrow 0\longrightarrow N_2\longrightarrow M_2\longrightarrow N'_2\longrightarrow 0 \longrightarrow\cdots? $$ If so, how does this work? Merci.

Answer 1

You can think of the long exact sequence $$0\longrightarrow M_1\stackrel{\phi_1}{\longrightarrow}M_2\stackrel{\phi_2}{\longrightarrow}M_3\stackrel{\phi_3}{\longrightarrow}M_4\stackrel{\phi_4}{\longrightarrow}\cdots$$ as a collection of short exact sequences $$0\longrightarrow M_1\stackrel{\phi_1}{\longrightarrow}M_2\stackrel{\phi_2}{\longrightarrow}\mathrm{Image}(\phi_2)\longrightarrow 0$$ $$0\longrightarrow\mathrm{Coker}(\phi_2)\stackrel{\phi_3}{\longrightarrow} M_4\stackrel{\phi_4}{\longrightarrow}\mathrm{Image}(\phi_4)\longrightarrow 0$$ $$\vdots$$ where each sequence after the first begins with the relevant cokernel (well, so does the first, but this is just $M_1$) and ends with the relevant image. I have abused notation here by writing $\phi_n$ for the maps from the cokernel which where originally from the corresponding module; this is not a serious issue because exactness of the original sequence ensures that the natural maps (defined by sending an equivalence class to the image of a representative) will be well-defined. One could write this as a single long chain like you proposed, but I prefer not to.

Answer 2

Just to expand on @AlexBecker's answer above.

Let's say we have a long exact sequence of $A$-modules $$\dots \to M_{i-1} \xrightarrow{f_i} M_i \xrightarrow{f_{i+1}} M_{i+1} \dots$$

If we let $N_i = \operatorname{Im}(f_i) = \operatorname{ker}(f_{i+1})$ for each $i$, then we obtain (for each $i$) a short exact sequence $$0 \to N_i \xrightarrow{\iota} M_i \xrightarrow{\pi} M_i / N_i \to 0 \ \ \ \ \ \ \ \ \ (*)$$ where $\iota : N_i \to M_i$ is the inclusion map and $\pi : M_i \to M_i/N_i$ is the canonical epimorphism. It shouldn't be too hard to prove that the above sequence is exact.

Now since $N_{i+1} = \operatorname{Im}(f_{i+1})$, by the first isomorphism theorem we have that $M_i/\operatorname{ker}(f_{i+1}) \cong \operatorname{Im}(f_{i+1}) \iff M_i/N_i \cong N_{i+1}$. If we let $f : M_i/N_i \to N_{i+1}$ be this isomorphism then letting $\pi' = f \circ \pi$ we obtain a short exact sequence

$$0 \to N_i \xrightarrow{\iota} M_i \xrightarrow{\pi'} N_{i+1} \to 0 \ \ \ \ \ \ \ \ \ (**)$$

which we can then rewrite by definition of the $N_i$'s as

$$0 \to \operatorname{Im}(f_i) \xrightarrow{\iota} M_i \xrightarrow{\pi'} \operatorname{Im}(f_{i+1}) \to 0 \ \ \ \ \ \ \ \ \ (**)$$

Now here's a Lemma that we will use below

Lemma: If we have $A$-modules $M, M', M''$, then $0 \to M' \xrightarrow{f} M \xrightarrow{g} M'' \to 0$ is exact $\iff$ $f$ is injective, $g$ is surjective and $g$ induces an isomorphism of $\operatorname{Coker}(f)$ onto $M''$

Replacing $\pi'$ in the exact sequence $(**)$ by $\pi''$ which is the isomorphism induced by $\pi'$ of $\operatorname{Coker}(\iota)$ onto $N_{i+1}$ we obtain the following short exact sequence

$$0 \to \operatorname{ker}(f_{i+1}) \xrightarrow{\iota} M_i \xrightarrow{\pi''} \operatorname{Coker}(\iota) \to 0 \ \ \ \ \ \ \ \ \ (***)$$

So these are three ways (up to isomorphism of a term in the sequence) of splitting a long exact sequence into a short exact sequence.

Answer 3

“Splitting long exact sequence into short exact sequences” means long exact sequence can be constructed from collection of short exact sequences. To be precise, $$\cdots \longrightarrow M_{i-1}\stackrel{f_i}{\longrightarrow}M_{i}\stackrel{f_{i+1}}{\longrightarrow}M_{i+1}\stackrel{}{\longrightarrow}\cdots$$ is long exact sequence if and only if $\exists$ $A$-modules $N_i$ such that
$$\vdots \\ 0 \to N_{i-1} \xrightarrow{g_{i-1}} M_{i-1} \xrightarrow{h_{i-1}} N_{i} \to 0\\ 0 \to N_{i} \xrightarrow{g_i} M_i \xrightarrow{h_i} N_{i+1} \to 0 \\ 0 \to N_{i+1} \xrightarrow{g_{i+1}} M_{i+1} \xrightarrow{h_{i+1}} N_{i+2} \to 0 \\ \vdots$$ are short exact sequences.

Proof: $(\Rightarrow)$ Take $N_i=\text{Im}(f_i)=\text{Ker}(f_{i+1})$ and $g_i=\iota$ (incusion map) and $h_i=f_{i+1}$.

$(\Leftarrow)$ Let $f_i=g_ih_{i-1}$ and $f_{i+1}=g_{i+1}h_i$. We show $M_{i-1} \xrightarrow{f_i} M_{i} \xrightarrow{f_{i+1}} M_{i}$ is exact. Since $h_{i-1}$ is surjective, $\text{Im}(f_i)=\text{Im}(g_i)$. Since $g_{i+1}$ is injective, $\text{Ker}(f_{i+1})=\text{Ker}(h_i)$. Therefore $\text{Im}(f_i)=\text{Im}(g_i)=\text{Ker}(h_i)= \text{Ker}(f_{i+1})$ and $M_{i-1} \xrightarrow{f_i} M_{i} \xrightarrow{f_{i+1}} M_{i}$ is exact.