Contramodules

TL;DR

This paper reviews the theory of contramodules, algebraic structures with infinite summation operations, highlighting their definitions, properties, and recent resurgence in various algebraic contexts.

Contribution

It provides a comprehensive overview of contramodules across different algebraic structures, including new perspectives and connections to classical theories.

Findings

Contramodules generalize module-like structures with infinite summation.

The paper discusses the contramodule-contramodule correspondence phenomenon.

Contramodules over various algebraic objects are systematically reviewed.

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.