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We develop the basic constructions of homological algebra in the(appropriately defined) unbounded derived categories of modules over algebrasover coalgebras over noncommutative rings (which we call semialgebras overcorings). We define double-sided derived functors SemiTor and SemiExt of thefunctors of semitensor product and semihomomorphisms, and construct anequivalence between the exotic derived categories of semimodules andsemicontramodules.  Certain (co)flatness and/or (co)projectivity conditions have to be imposed onthe coring and semialgebra to make the module categories abelian (and thecotensor product associative). Besides, for a number of technical reasons wemostly have to assume that the basic ring has a finite homological dimension(no such assumptions about the coring and semialgebra are made).  In the final sections we construct model category structures on thecategories of complexes of semi(contra)modules, and develop relativenonhomogeneous Koszul duality theory for filtered semialgebras andquasi-differential corings.  Our motivating examples come from the semi-infinite cohomology theory.Comparison with the semi-infinite (co)homology of Tate Lie algebras and gradedassociative algebras is established in appendices, and the semi-infinitehomology of a locally compact topological group relative to an open profinitesubgroup is defined. An application to the correspondence between complexes ofrepresentations of an infinite-dimensional Lie algebra on the complementarycentral charge levels ($c$ and $26-c$ for the Virasoro) is worked out.

Homological algebra of semimodules and semicontramodules: Semi-infinite homological algebra of associative algebraic structures