Mathematics – Category Theory
Scientific paper
2007-08-27
Monografie Matematyczne, vol.70, Birkhauser Basel, 2010, xxiv+349 pp
Mathematics
Category Theory
Dedicated to the memory of my father. LaTeX 2e, 310 pages. With appendices coauthored by S.Arkhipov and D.Rumynin. v.11: impro
Scientific paper
10.1007/978-3-0346-0436-9
We develop the basic constructions of homological algebra in the (appropriately defined) unbounded derived categories of modules over algebras over coalgebras over noncommutative rings (which we call semialgebras over corings). We define double-sided derived functors SemiTor and SemiExt of the functors of semitensor product and semihomomorphisms, and construct an equivalence between the exotic derived categories of semimodules and semicontramodules. Certain (co)flatness and/or (co)projectivity conditions have to be imposed on the coring and semialgebra to make the module categories abelian (and the cotensor product associative). Besides, for a number of technical reasons we mostly have to assume that the basic ring has a finite homological dimension (no such assumptions about the coring and semialgebra are made). In the final sections we construct model category structures on the categories of complexes of semi(contra)modules, and develop relative nonhomogeneous Koszul duality theory for filtered semialgebras and quasi-differential corings. Our motivating examples come from the semi-infinite cohomology theory. Comparison with the semi-infinite (co)homology of Tate Lie algebras and graded associative algebras is established in appendices; and the semi-infinite homology of a locally compact topological group relative to an open profinite subgroup is defined.
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