Under mild assumptions, we construct the two Matlis additive category equivalences for an associative ring epimorphism u : R −→ U . Assuming that the ring epimorphism is homological of flat/projective dimension 1, we discuss the abelian categories of u-comodules and u-contramodules and construct the recollement of unbounded derived categories of R-modules, U -modules, and complexes of R-modules with u-co/contramodule cohomology. Further assumptions allow to describe the third category in the recollement as the unbounded derived category of the abelian categories of u-comodules and u-contramodules. For commutative rings, we also prove that any homological epimorphism of projective dimension 1 is flat. Injectivity of the map u is not required.The following result can be also found in [5, Corollary 4.4].Corollary 7.3. Let u : R −→ U be a homological ring epimorphism. Assume that fd U R ≤ 1 and pd R U ≤ 1. Suppose further that (U/R) ⊗ R J = 0 for all injective left R-modules J and Hom R (U/R, F ) = 0 for all projective left R-modules F . Then for every conventional derived category symbol ⋆ = b, +, −, or ∅, there is a triangulated equivalence between the derived categories of the abelian categories R-mod u-co and R-mod u-ctra of left u-comodules and left u-contramodules,Proof. According to Corollary 6.2 and Theorem 7.1(a-b), we have a chain of triangulated equivalences D ⋆ (R-mod u-co ) ∼ = D ⋆ u-co (R-mod) ∼ = D ⋆ u-ctra (R-mod) ∼ = D ⋆ (R-mod u-ctra ).