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.
Inhaltsverzeichnis
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.
