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).

Algèbre commutative.– Fred Rohrer Jun 11 at 6:42and projective? – tj_ Jun 11 at 7:12lower boundon 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