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 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.

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

Autorentext

Leonid Positselski received his Ph.D. in Mathematics from Harvard University in 1998. He did his postdocs at the Institute for Advanced Study (Princeton), Institut des Hautes Etudes Scientifiques (Bures-sur-Yvette), Max-Planck-Institut fuer Mathematik (Bonn), the University of Stockholm, and the Independent University of Moscow in 1998-2003. He taught as an Associate Professor at the Mathematics Faculty of the National Research University Higher School of Economics in Moscow in 2011-2014. In Spring 2014 he moved from Russia to Israel, and since 2018 he work as a Researcher at the Institute of Mathematics of the Czech Academy of Sciences in Prague. He is an algebraist specializing in homological algebra. His research papers span a wide area including algebraic geometry, representation theory, commutative algebra, algebraic K-theory, and algebraic number theory. He is the author of four books and memoirs, including "Quadratic Algebras" (joint with A.Polishchuk, AMS University Lecture Series, 2005), "Homological algebra of semimodules and semicontramodules: Semi-infinite homological algebra of associative algebraic structures" (Monografie Matematyczne IMPAN, Birkhauser Basel, 2010), "Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence" (AMS Memoir, 2011), and "Weakly curved A-infinity algebras over a topological local ring" (Memoir of the French Math. Society, 2018-19).

Inhalt

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.