  Let $R$ be any ring with identity. We show that the homotopy category of all
acyclic chain complexes of pure-projective $R$-modules is a compactly generated
triangulated category. We do this by constructing abelian model structures that
put this homotopy category into a recollement with two other compactly
generated triangulated categories: The usual derived category of $R$ and the
pure derived category of $R$. This also gives a new model for the derived
category.
