# Homological criteria for finite generation and finite presentation of modules?

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:

1. If M is finitely generated then N is finitely generated. (Trivial.)
2. If L and N are finitely generated then M is finitely generated. (Trivial.)
3. If N is finitely presented, and M is finitely generated, then L is finitely generated. (Theorem 2.6)
4. If M is finitely presented, and L is finitely generated, then N is finitely presented. (Exercise 2.5b)
5. 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.

• If $A$ is a non-negatively graded and connected over a field $k$, and your modules are graded with bounded below graduations, then you can use $H_i (M)=Tor^A(k,M)$: a module is $FL_n$ if $H_i(M)$ is finite dimensional for $i \leq n$. If you want zero cohomology groups, you can notice that $H_i$ takes values in graded vector spaces, and compose it with the functor which localizes at the Serre subcategory of finite dimensional graded vector spaces —call $H'$ the composition; now a module is in $FL_n$ if $H_i'(M)=0$ for $i\leq n$. Apr 14, 2014 at 5:30

I suggest you go and look at the literature. In particular the material in SGA 6 concerning pseudo-coherent complexes. Namely, what you call an $FL_n$ module is a $(-n)$-pseudo-coherent module, see Section Tag 064N. The definition applies to objects of the derived category of modules, and there one has a more general result for distinguished triangles.