The following question came up while trying to determine whether the extension problems in a spectral sequence are trivial.


Given a noetherian ring $R$ and a finitely generated $R$-module $M$ with a filtration $M=F\_0 \supset F\_1 \supset \ldots \supset F\_n \supset F\_{n+1}=0$ such that $M \cong \bigoplus\_{i=0}^n F\_i/F\_{i+1}$ (abstractly, but I do not want to assume that this isomorphism is induced by the maps in the filtration), is the filtration split in the sense that all the short exact sequences $0 \to F\_{i+1} \to F\_i \to F\_i/F\_{i+1} \to 0$ are split? 


I can prove this for $R=\mathbb{Z}$, but my proof does not seem to generalize well. In that case, one can work $p$-locally and then show that a short exact sequence $0 \to A \to B \to C \to 0$ of $p$-groups, 


$|\{x \in B|ord(x)\text{ divides }p^k\}|\geq |\{x \in A|ord(x)\text{ divides }p^k\}|\cdot|\{x \in C|ord(x)\text{ divides }p^k\}|$


, with equality iff the sequence is split, and then use that by the assumed isomorphisms, the number of these elements can never strictly increase.


Steven Landsburgs answer in [Do all exact sequences $0 \rightarrow A \rightarrow A \oplus B \rightarrow B \rightarrow 0$ split for finitely generated abelian groups?](https://mathoverflow.net/questions/167701/do-all-exact-sequences-0-rightarrow-a-rightarrow-a-oplus-b-rightarrow-b-ri) got me hoping that something like this might hold for finitely generated modules over noetherian rings, but a simple induction argument does not seem to work.


Does anyone know if this generalisation is true, and knows a proof?



Given a filtration of a finitely generated module over a noetherian ring that "looks" split, is it split?
Take the direct sum of the short exact sequences
$$0\to F\_{i+1}\to F\_i\to F\_i/F\_{i+1}\to0$$
for $0\leq i\leq n$.


This has the form 
$$0\to \bigoplus\_{i=1}^n F\_i\to \bigoplus\_{i=0}^n F\_i\to F\_0\to 0$$
and so splits by the linked answer of Steven Landsburg.


A direct summand of a split short exact sequence is split, so
$$0\to F\_{i+1}\to F\_i\to F\_i/F\_{i+1}\to0$$
is split for each $i$.


