Homological Algebra of Semimodules and Semicontramodules : Semi infinite Homological Algebra of Associative Algebraic Structures This monograph deals with semi infinite homological algebra. Intended as the definitive treatment of the subject of semi infinite homology and cohomology of associative algebraic structures it also contains material on the semi infinite (co)homology of Lie algebras and topological groups the derived comodule contramodule correspondence its application to the duality between representations of infinite dimensional Lie algebras with complementary central charges and relative non homogeneous Koszul duality. The book explains with great clarity what the associative version of semi infinite cohomology is why it exists and for what kind of objects it is defined. Semialgebras contramodules exotic derived categories Tate Lie algebras algebraic Harish Chandra pairs and locally compact totally disconnected topological groups all interplay in the theories developed in this monograph. Contramodules introduced originally by Eilenberg and Moore in the 1960s but almost forgotten for four decades are featured prominently in this book with many versions of them introduced and discussed. Rich in new ideas on homological algebra and the theory of corings and their analogues this book also makes a contribution to the foundational aspects of representation theory. In particular it will be a valuable addition to the algebraic literature available to mathematical physicists. Preface Introduction 0 Preliminaries and Summary 1 Semialgebras and Semitensor Product 2 Derived Functor SemiTor 3 Semicontramodules and Semihomomorphisms 4 Derived Functor SemiExt 5 Comodule Contramodule Correspondence 6 Semimodule Semicontramodule Correspondence 7 Functoriality in the Coring 8 Functoriality in the Semialgebra 9 Closed Model Category Structures 10 A Construction of Semialgebras 11 Relative Nonhomogeneous Koszul Duality Appendix A Contramodules over Coalgebras over Fields Appendix B Comparison with Arkhipovs Ext^{\\infty 2+*} and Sevostyanovs Tor_{\\infty 2+*} Appendix C Semialgebras Associated to Harish Chandra Pairs Appendix D Tate Harish Chandra Pairs and Tate Lie Algebras Appendix E Groups with Open Profinite Subgroups Appendix F Algebraic Groupoids with Closed Subgroupoids Bibliography Index.