Abstract. We describe and investigate Arens-Michael envelopes of associative algebras and their homological properties. We also introduce and study analytic analogs of some classical ring-theoretic constructs: Ore extensions, Laurent extensions, and tensor algebras. For some finitely generated algebras, we explicitly describe their Arens-Michael envelopes as certain algebras of noncommutative power series, and we also show that the embeddings of such algebras in their Arens-Michael envelopes are homological epimorphisms (i.e., localizations in the sense of J. Taylor). For that purpose we introduce and study the concepts of relative homological epimorphism and relatively quasi-free algebra. The above results hold for multiparameter quantum affine spaces and quantum tori, quantum Weyl algebras, algebras of quantum (2 × 2)-matrices, and universal enveloping algebras of some Lie algebras of small dimensions.