I am trying to construct the projective resolution in the category of chain complexes of
$\dots \to 0 \to M \to 0 \to \dots$
It seems like it should be possible to do this in terms of the projective resolution of $M$ but I am completely stuck.
I know a projective chain complex is split exact and formed by projectives, so if we think of the resolution as a half plane double complex, the column with $M$ must be a projective resolution of $M$.
I was trying to use the trick of $0 \to P \to P \to 0$ is a projective complex whenever $P$ is projective, but if I put that on top of our complex we don't necessarily get exactness.
Best Answer
If $$\dots\to P_2\to P_1\to P_0 \to M\to0$$ is a projective resolution of $M$ as a module, then $\dots\to0\to M\to0\to\dots$ has a resolution (by projective chain complexes) in the category of chain complexes of the following form (I'll let you figure out the differentials):
$\require{AMScd}$ \begin{CD} @.\vdots@.\vdots@.\vdots@.\vdots@.\vdots@.\vdots@.\\ @.@VVV@VVV@VVV@VVV@VVV@VVV\\ \cdots@>>>0@>>>P_2@>>>P_2\oplus P_1@>>>P_1\oplus P_0@>>> P_0@>>>0@>>>\cdots\\ @.@VVV@VVV@VVV@VVV@VVV@VVV\\ \cdots@>>>0@>>> P_1@>>>P_1\oplus P_0@>>>P_0@>>>0@>>>0@>>>\cdots\\ @.@VVV@VVV@VVV@VVV@VVV@VVV\\ \cdots@>>>0@>>>P_0@>>>P_0@>>>0@>>>0@>>>0@>>>\cdots\\ @.@VVV@VVV@VVV@VVV@VVV@VVV\\ \cdots@>>>0@>>>M@>>>0@>>>0@>>>0@>>>0@>>>\cdots\\ @.@VVV@VVV@VVV@VVV@VVV@VVV\\ @[email protected]@[email protected]@[email protected] \end{CD}