Let $R$ be a ring (not necessarily commutative, but with a unit). Recall that an $R$-module $M$ is of type $FP_n$ if $M$ has a partial projective resolution of length $n$ whose terms are all finitely generated (thus $FP_0$ means finitely generated and $FP_1$ means finitely presented). Consider an extension of modules

$$0 \longrightarrow M' \longrightarrow M \longrightarrow M'' \longrightarrow 0.$$

Assume that $M$ is of type $FP_n$ and that $M''$ is of type $FP_m$. What can we say about $M'$? I suppose that I am also interested in the other possibilities (where finiteness properties of $M'$ and $M$, or of $M'$ and $M''$, are given), but the one I indicated above is the most relevant one for what I am doing. Any references for these kinds of finiteness conditions in homological algebra are also welcome (the only one I know is Brown's book on group cohomology, and of course it focuses on examples coming from group cohomology).

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    You might find some results about this in Exercice I.2.6 in Bourbaki's Algèbre commutative. – Fred Rohrer Jun 11 at 6:42
  • Doesn't $FP_0$ mean finitely generated and projective ? – tj_ Jun 11 at 7:12
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    @tj_ : No, $M\ FP_0$ means only that there is a surjective homomorphism $R^n\twoheadrightarrow M$. – abx Jun 11 at 7:39
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    The exercise mentioned by Fred Rohrer gives a lower bound on the possible length $r$ of a finite presentation of $M'$, namely $r\geq \min(n,m-1)$. Of course you cannot hope for an upper bound: take for $M'$ an ideal $\mathfrak{a}$ of $R$ which is not finitely generated, $M=R$ and $M''=R/\mathfrak{a}$. – abx Jun 11 at 7:41
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    @abx: Thanks, I mixed the notion with the projective dimension. – tj_ Jun 11 at 9:12

Let $M'$ resp. $M''$ be of type $FP_{m'}$ resp. $FP_{m''}$.

Then $M$ is of type $FP_m$ for $m = \min(m', m'')$.

Proof: Let $P' \to M' \to 0$ be a projective resolution with the modules of $P'$ finitely generated up to degree $m'$. Let $P''$ be resp. for $M''$. By the Horseshoe Lemma there is a projective resolution $P \to M\to 0$ such that $P_i = P'_i \oplus P''_i$ in each degree $i\ge 0$ (Weibel, Homological Algebra, 2.2.8). In particular, $P_i$ is finitely generated for $i \le \min(m', m'')$. q.e.d.

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