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.