Contramodules

Contramodules are module-like algebraic structures endowed with infinite summation (or, occasionally, integration) operations satisfying natural axioms. Introduced originally by Eilenberg and Moore in 1965 in the case of coalgebras over commutative rings, contramodules experience a small renaissance now after being all but forgotten for three decades between 1970-2000. Here we present a review of various definitions and results related to contramodules (drawing mostly from our monographs and preprints arXiv:0708.3398, arXiv:0905.2621, arXiv:1202.2697, arXiv:1209.2995, arXiv:1512.08119, arXiv:1710.02230, arXiv:1705.04960, arXiv:1808.00937) - including contramodules over corings, topological associative rings, topological Lie algebras and topological groups, semicontramodules over semialgebras, and a "contra version" of the Bernstein-Gelfand-Gelfand category O. Several underived manifestations of the comodule-contramodule correspondence phenomenon are discussed.Comment: LaTeX 2e with pb-diagram and xy-pic; 93 pages, 3 commutative diagrams; v.4: updated to account for the development of the theory over the four years since Spring 2015: introduction updated, references added, Remark 2.2 inserted, Section 3.3 rewritten, Sections 3.7-3.8 adde

Homological algebra of semimodules and semicontramodules: Semi-infinite homological algebra of associative algebraic structures

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. An application to the correspondence between complexes of representations of an infinite-dimensional Lie algebra on the complementary central charge levels ($c$ and $26-c$ for the Virasoro) is worked out.Comment: Dedicated to the memory of my father. LaTeX 2e, 310 pages. With appendices coauthored by S.Arkhipov and D.Rumynin. v.12: changes in the Introduction, additions to Section 0 and Appendix D, small improvements in Appendix C and elsewhere, subtitle added -- this is intended as the final arXiv version; v.13: abstract updated, LaTeX file unchanged (the publisher's version is more complete

Kleene Algebras and Semimodules for Energy Problems

With the purpose of unifying a number of approaches to energy problems found in the literature, we introduce generalized energy automata. These are finite automata whose edges are labeled with energy functions that define how energy levels evolve during transitions. Uncovering a close connection between energy problems and reachability and B\"uchi acceptance for semiring-weighted automata, we show that these generalized energy problems are decidable. We also provide complexity results for important special cases