I am wondering if there is a link between a projective resolution of an R-module M and its composition serie. If we have M of finite length can we conclude something ? If we know one of the composition series can we construct a projective resolution ?
Thank you really much.
The answer is very much no. For instance, two non-isomorphic modules with the same composition factors can have very different projective resolutions.
Here's an example: take $k$ a field, and $R = k[x]/(x^2)$. Let $S = k[x]/(x)$ be the simple $R$-module. Then the $R$-modules $R$ and $S^2$ have the same composition factors, but their projective resolutions are $$ 0 \to R \xrightarrow{1}R \to 0 $$ and $$ \ldots \xrightarrow{x} R\xrightarrow{x} R\xrightarrow{x} R \to S \to 0, $$ respectively. Thus $R$ has projective dimension $0$, while $S$ has infinite projective dimension.