Let $R$ be a commutative ring with unity If $$ 0\longrightarrow L \longrightarrow M \longrightarrow N \longrightarrow 0$$ is a short exact sequence of R modules. Now given two projective resolutions of $L$ and $N$; by Horseshoe lemma I can find a projective resolution of $M$.
My question is if I have $\textbf{finite}$ projective resolutions of both $M$ and $N$ then can I construct a projective resolution of $L$?
My attempt (or guess): Let $\textbf{P.}$ $\longrightarrow M$ and $\textbf{Q.}$ $\longrightarrow N$ be the projective resolutions.
Then for $f : M \longrightarrow N$ I have a chain map lift from $\textbf{P.}$ $\longrightarrow \textbf{Q.}$
So my guess is if I take the mapping cone $\textbf{P}_{n-1} \oplus \textbf{Q}_n$
It might give me a projective resolution for $L$, but I am unable to prove my claim.
Thanks in advance.