Let $R$ be a ring. Recall that a module $M$ is called finitely presented if there is an exact sequence

$R^n \to R^m \to M \to 0$.

with $n,m \in \mathbb{N}$. A well known result states that any module $M$ is a filtered direct limit of finitely presented modules. A bit more generally, we can call a module $M$ 2-finitely presented if there is an exact sequence

$R^k \to R^n \to R^m \to M \to 0$

with $k,m,n \in \mathbb{N}$.

My question is: is any module the filtered direct limit of 2-finitely presented modules? what about higher order presentation?


1 Answer 1


Assume that a f.p. module $M$ is filtered direct limit of modules $(M_i)$.

Recall that this $M$ f.p. means that $\mathrm{Hom}(M,--)$ commutes with inductive limits. So the identity of $M$ can be lifted to $M\to M_i$ for $i$ large enough (say $i\ge i_0$), let $j_i$ be the lift $M\to M_i$, and let $N_i$ be the kernel of $M_i\to M$. Thus for $i\ge i_0$ we have $M_i=j_i(M)\times N_i$. But a direct factor of a module with your additional condition (I think it's called $FP_3$-Property) is still $FP_3$, so it's impossible if $M$ itself is not $FP_3$. So your expectation is hopeless: there is no way to "approximate" by modules with higher finiteness properties using direct limits.


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