“Short exact sequences”, longer than classical one

Question

A standard short exact sequence is a complex $$0\to A\to B\to C \to 0$$ Based on this concept we get a lot of new concept in homological algebras. For example left exactness, right exactness, derived functor, etc. And we get some applications in other areas e.g algebraic topology.

Now let'us consider an exact sequence with more than 3 terms. For example 4 object or more as follows:

$$0\to A\to B\to C\to D \to 0$$

Do we get some thing new with these new concept of short exact sequence? Can we find some non trivial applications in homological algebra or algebraic topology?

Note that in this setting some concepts as left or right exactness have the obvious modifications.

Answer 1

Length 4 exact sequences are pretty boring because, in normal abelian categories, they correspond to ordinary morphisms. We can extend $f:X\to Y$ to an exact sequence $$ 0 \xrightarrow{} \operatorname{ker}f \to X \xrightarrow{f} Y \to \operatorname{coker}f \to 0. $$ Similarly, every exact sequence of length four will arise this way. Passing to length 5 sequences isn't going to be terribly interesting either because the sequences now look like this $$ 0 \xrightarrow{} \operatorname{ker}f \to X \xrightarrow{f} Y \xrightarrow{g} Z \to\operatorname{coker}g \to 0. $$ There's nothing new going on here. In fact, as a different poster has mentioned all these longer exact sequences can be produced by splicing together short exact sequences. The natural way to generalise left exactness, right exactness and derived functors turns out to involve weakening the notion of a abelian category, to obtain an $n$-abelian category. This is called Higher Homological Algebra and you can read more about it in this paper of Gustavo Jasso.

Answer 2

The short exact sequence is more fundamental, because every longer chain complex can be obtained by "splicing" SES.

In your example, we have

$$0 \to A \to_f B \to \operatorname{coker} f \to 0$$ and $$ 0 \to \ker h \to C \to_h D \to 0$$

Now let $g:B \to C$ be the middle map in your exact sequence. Note however that $\operatorname{coker} f=B/\ker g \cong\operatorname{im} g$ while $\ker h = \operatorname{im} g$ as well by exactness. Thus, we can use these identifications to "splice" our SES back together. See here for mode details and here for the "other direction."

One possible generalization is the notion of SES in an abelian category, which for example gives SES of chain complexes $0 \to A_\bullet \to B_\bullet \to C_\bullet \to 0$. This also gives a LES in homology by the snake lemma.

Answer 3

This should probably just be a comment, but note that, restricting attention to complexes of finite-dimensional vector spaces, a short exact sequence $$0 \to A \to B \to C \to 0$$ satisfies the dimensional relation $$ \dim(A) +\dim(C)= \dim(B).$$ In the case of a four-term sequence $$0 \to A \to B \to C \to D \to 0,$$ we get $$\dim(A) + \dim(C) = \dim(B) +\dim(D).$$ This keeps working for exact sequences of any finite length $$0 \to A_0 \to A_1 \to \cdots \to A_n \to 0.$$ You would usually write the associated dimensional condition as $$\sum_k (-1)^k \dim(A_k)=0$$ and say something like "the Euler characteristic of the exact complex is zero".