Homological Algebra of Semimodules and Semicontramodules: Semi-infinite Homological Algebra of Associative Algebraic Structures by Leonid PositselskiHomological Algebra of Semimodules and Semicontramodules: Semi-infinite Homological Algebra of Associative Algebraic Structures by Leonid Positselski

Homological Algebra of Semimodules and Semicontramodules: Semi-infinite Homological Algebra of…

byLeonid Positselski

Paperback | November 5, 2012

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ThesubjectofthisbookisSemi-In?niteAlgebra,ormorespeci?cally,Semi-In?nite Homological Algebra. The term "semi-in?nite" is loosely associated with objects that can be viewed as extending in both a "positive" and a "negative" direction, withsomenaturalpositioninbetween,perhapsde?nedupto a"?nite"movement. Geometrically, this would mean an in?nite-dimensional variety with a natural class of "semi-in?nite" cycles or subvarieties, having always a ?nite codimension in each other, but in?nite dimension and codimension in the whole variety [37]. (For further instances of semi-in?nite mathematics see, e. g. , [38] and [57], and references below. ) Examples of algebraic objects of the semi-in?nite type range from certain in?nite-dimensional Lie algebras to locally compact totally disconnected topolo- cal groups to ind-schemes of ind-in?nite type to discrete valuation ?elds. From an abstract point of view, these are ind-pro-objects in various categories, often - dowed with additional structures. One contribution we make in this monograph is the demonstration of another class of algebraic objects that should be thought of as "semi-in?nite", even though they do not at ?rst glance look quite similar to the ones in the above list. These are semialgebras over coalgebras, or more generally over corings - the associative algebraic structures of semi-in?nite nature. The subject lies on the border of Homological Algebra with Representation Theory, and the introduction of semialgebras into it provides an additional link with the theory of corings [23], as the semialgebrasare the natural objects dual to corings.
Title:Homological Algebra of Semimodules and Semicontramodules: Semi-infinite Homological Algebra of…Format:PaperbackDimensions:352 pages, 23.5 × 15.5 × 0.17 inPublished:November 5, 2012Publisher:Springer-Verlag/Sci-Tech/TradeLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:3034803133

ISBN - 13:9783034803137


Table of Contents

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.

Editorial Reviews

From the reviews:"The book is written for experts in several areas such as homological algebra, representation theory of Lie algebras and Hopf algebras. The theoretical results are proven with enough detail and with an eye towards specific applications in mind. . give an excellent overview of the whole book and its connections with the relevant results in the literature." (Atabey Kaygun, Mathematical Reviews, Issue 2012 c)