Homological Algebra of Semimodules and Semicontramodules
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 . (For further instances of semi-in?nite mathematics see, e. g.,  and , 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 , as the semialgebrasare the natural objects dual to corings.
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 applicationsExotic 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