# Generalization of finitely generated, finitely presented modules?

Let $R$ be a commutative ring and $M$ an $R$-module. The module $M$ is finitely generated iff there is an exact sequence $R^{k_0} \to M \to 0$. Similarly, $M$ is finitely presented iff there is an exact sequence $R^{k_1} \to R^{k_0} \to M \to 0$. It seems we could generalize this as follows: for $n \in \mathbb{Z}_{\ge 0}$ let us call $M$ a finitely $n$-presented module, if there is an exact sequence $R^{k_n} \to \dotsm \to R^{k_0} \to M \to 0$. So finitely generated = finitely $0$-presented, and finitely presented = finitely $1$-presented. You could also define finitely $\infty$-presented modules, which have a resolution with finite free modules. I would even try defining $M$ to be finitely $\omega$-presented, if it has a finite free resolution $0 \to R^{k_m} \to \dotsm \to R^{k_0} \to M \to 0$ (I think these have been studied a lot).

I haven't seen these notions defined before (except for the last one). Couldn't they be useful, or have they been used? I think both finitely generated and finitely presented modules are important, although if you are only interested in noetherian rings, there is no difference.

For example, there is a result saying that if $0 \to M' \to M \to M'' \to 0$ is exact, $M''$ is finitely presented and $M$ is finitely generated, then $M'$ is also finitely generated. You could generalize this: if $M$ is finitely $n$-presented and $M''$ is finitely $(n+1)$-presented, then $M'$ is finitely $n$-presented.

Also we could look at submodules: $M$ is noetherian iff every submodule of $M$ is a finitely generated. $M$ is coherent iff it is finitely generated, and every finitely generated submodule of $M$ is finitely presented. We could generalize this as follows: $M$ is $n$-coherent ($n \in \mathbb{Z}_{\ge 0}$) iff it is $(n-1)$-presented, and every finitely $(n-1)$-presented submodule of $M$ is finitely $n$-presented. So noetherian = $0$-coherent, and coherent = $1$-coherent. You could also define $R$ to be $n$-coherent iff it is an $n$-coherent $R$-module. The category of noetherian/coherent $R$-modules is abelian, so I guess the same should hold for the category of $n$-coherent $R$-modules.

• This is an interesting question. Do you have an idea how to define these notions in an arbitrary category? For f.g. and f.p. this has been done. Apr 10 '12 at 18:25

All these notions have been defined and studied long time ago. Serre called a module type $FL_n$ if it is finitely $n$-presented in your terminology. Type $FL_\infty$ and type $FL$ is used for finitely $\infty$-presented and finitely $\omega$-presented. They are studied a lot for group rings ($R$ does not need to be commutative) and show up in the definition of $G$-theory for general rings. Modules of type $FP_\infty$ are sometimes also called pseudo-coherent, a name/definition that goes back to SGA 6, I.2.9, see for Example 7.1.4 in Chuck Weibel's book on Algebraic K-theory.
There is a big literature about this at least for modules over group rings $R=\mathbb{Z}[G]$. The standard terminology is to say that $M$ has type $FP_n$ if it is finitely $n$-presented according to your definition. (Or maybe the indexing is shifted by one, I don't remember.) Peter Kropholler is one of the main authors so if you look for his papers you will get lots of references.