(I'm new here; if I'm doing something wrong please help me out.)

In short, my question is: There are some results on the behavior of finite generation and finite presentation in exact sequences (of modules over an arbitrary commutative ring A), which feel like they come from a long exact sequence in homology. Is there some homology theory or something that explains this?

OK, first some motivation. Here are some facts (which I ran across in Matsumura's CRT).

Suppose

$0 \rightarrow L \rightarrow M \rightarrow N \rightarrow 0$

is an exact sequence of A-modules, A a commutative ring. Then we have:

- If M is finitely generated then N is finitely generated. (Trivial.)
- If L and N are finitely generated then M is finitely generated. (Trivial.)
- If N is finitely presented, and M is finitely generated, then L is finitely generated. (Theorem 2.6)
- If M is finitely presented, and L is finitely generated, then N is finitely presented. (Exercise 2.5b)
- If L and N are finitely presented then so is M. (Exercise 2.5a)

We can generalize these results as follows. (Also, there's another post where part of the generalized result was discussed.) Generalization of finitely generated, finitely presented modules?

Say a module $M$ is of type $FL_n$ if there is an exact sequence

$A^{k_n} \rightarrow \cdots \rightarrow A^{k_0} \rightarrow M \rightarrow 0$.

Note that $FL_0$ means finitely generated, and $FL_1$ means finitely presented.

Let $L$, $M$, $N$ be in an exact sequence as above, and consider the sequence of statements:

..., $L$ is $FL_1$, $M$ is $FL_1$, $N$ is $FL_1$, $L$ is $FL_0$, $M$ is $FL_0$, $N$ is $FL_0$, (true).

The general fact is that any statement is this sequence is implied by its two neighbors. (For instance, if $M$ is $FL_1$ and $L$ is $FL_0$ then $N$ is $FL_1$.) This can be proven by induction and the snake lemma (unless I made a mistake!). The five results above are special cases of this general fact.

Now for my question: Is this general fact a consequence of a long exact sequence in homology? If there were some homology theory such that $H_i(M)$ could detect whether $M$ was of type $FL_i$, the fact would follow immediately. (Maybe $M$ is of type $FL_i$ if and only if $H_i(M)$ is finitely generated, for instance.) Is there such a theory?

One last remark: this post (finitely presented implies always finitely presented, Does "finitely presented" mean "always finitely presented"? (Answered: Yes!)) also makes me feel like there might be something interesting going on, even though I can't say precisely what.