Contramodules

Abstract

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.

LaTeX 2e with pb-diagram and xy-pic; 95 pages, 3 commutative diagrams; v.4: updated to account for the development of the theory over the four years since Spring 2015: Remark 2.2 inserted, Section 3.3 rewritten, Sections 3.7-3.8 added; v.5: Remark 1.7 and Example 2.2 inserted; v.6: references added; v.7: several misprints corrected, references updated -- this is intended as the final version

References in corpus (27)

• New Approach to Arakelov Geometry
• On purity and applications to coderived and singularity categories
• Covers, envelopes, and cotorsion theories in locally presentable abelian categories and contramodule categories
• Dedualizing complexes and MGM duality
• Silting objects
• $S$-almost perfect commutative rings
• Triangulated Matlis equivalence
• The tilting-cotilting correspondence
• Coherent rings, fp-injective modules, dualizing complexes, and covariant Serre-Grothendieck duality
• Contramodules over pro-perfect topological rings
• Infinity-tilting theory
• Koszulity of cohomology = $K(\pi ,1)$-ness + quasi-formality
• Positive energy representations of affine vertex algebras
• Flat morphisms of finite presentation are very flat
• Topologically semisimple and topologically perfect topological rings
• On strongly flat and weakly cotorsion modules
• Matlis category equivalences for a ring epimorphism
• Abelian right perpendicular subcategories in module categories
• Covers and direct limits: a contramodule-based approach
• Flat ring epimorphisms of countable type
• Remarks on derived complete modules and complexes
• Smooth duality and co-contra correspondence
• Relative nonhomogeneous Koszul duality
• Flat connections and (co)modules
• Nearly locally presentable categories
• Exact categories of topological vector spaces with linear topology
• Remarks on topological algebras

Cited by in corpus (12)

• The tilting-cotilting correspondence
• Differential graded Koszul duality: an introductory survey
• Remarks on derived complete modules and complexes
• Flat comodules and contramodules as directed colimits, and cotorsion periodicity
• Contramodules for algebraic groups: the existence of mock projectives
• Contramodules for algebraic groups: induction and projective covers
• On Vector Spaces with Formal Infinite Sums
• The categories of corings and coalgebras over a ring are locally countably presentable
• A bounded below, noncontractible, acyclic complex of projective modules
• A contramodule generalization of Neeman's flat and projective module theorem
• Homological full-and-faithfulness of comodule inclusion and contramodule forgetful functors
• Topologically semiperfect topological rings