Homological Algebra of Semimodules and Semicontramodules

This book provides comprehensive coverage on semi-infinite homology and cohomology of associative algebraic structures. It features rich representation-theoretic and algebro-geometric examples and applications.

This is a monograph in semi-infinite homological algebra, concentrated mostly on the semi-infinite theory of associative algebraic structures, but including also some material on the semi-infinite homology and cohomology of Lie algebras and topological groups. The main objects of study are the double-sided derived functors SemiExt and SemiTor, and the phenomenon of comodule-contramodule correspondence, connecting them with the more conventional, one-sided Ext and CtrTor. Contramodules, introduced originally by Eilenberg and Moore in 1960's but almost forgotten for four decades, play a very prominent role in this book, with many versions of them introduced and discussed. Among other topics considered in the monograph there are the semi-infinite model category structures and the relative nohomogeneous Koszul duality.

Intended as a definitive treatment of the subject of semi-infinite homology and cohomology of associative algebraic structures, this book contains also rich representation-theoretic and algebro-geometric examples and applications Exotic derived categories, contramodules, semialgebras, infinite-dimensional Lie algebras, algebraic Harish-Chandra pairs, and locally compact totally disconnected topological groups all interplay in the theories developed in this monograph Includes supplementary material: sn.pub/extras

Texte du rabat
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.

Contenu

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 Arkhipov's Ext^{\infty/2+} and Sevostyanov's 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.