Contramodules

Positselski, Leonid

Confluentes Mathematici, Tome 13 (2021) no. 2, pp. 93-182.

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, papers, and preprints [69, 70, 81, 71, 66, 92, 78, 82])—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 $\mathsf{O}$. Several underived manifestations of the comodule-contramodule correspondence phenomenon are discussed.

Classification : 16T15, 16W60, 16W80, 18E10, 17B65, 22D12, 22E65
Mots clés : contramodules, comodules, discrete modules, smooth modules, torsion modules, coalgebras, corings, semialgebras, topological rings, adic completions, topological groups, pro-algebraic groups, topological Lie algebras, Tate Harish-Chandra pairs

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