Homological Algebra of Semimodules and Semicontramodules

ThesubjectofthisbookisSemi-In?niteAlgebraormorespeci?callySemi-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 withsomenaturalpositioninbetweenperhapsde?nedupto a?nitemovement. 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.