Let $R$ be a commutative Noetherian ring. An acyclic complex $P$ of projective $R$-modules is called totally acyclic if for every projective $R$-module $Q$, the complex Hom$_R(P, Q)$ is also acyclic.
My question is: If $P$ is a totally acyclic complex of projective $R$-modules, $\mathfrak p$ is a prime ideal of $R$, then is the acyclic complex $P_{\mathfrak p}$ of projective $R_{\mathfrak p}$-modules also totally acyclic ?
I can see this if $P$ is a complex of finitely generated projective $R$-modules, but otherwise, I have no idea ...
