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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.

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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. – Martin Brandenburg Apr 10 '12 at 18:25
up vote 7 down vote accepted

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.

A good starting point might be K.S. Brown's book "Cohomology of groups", Chapter VIII is about finiteness conditions.

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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.

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