Given a $k$-algebra $A$ and an $A$-coring $C$ one can introduce right $C$-comodules $(M,\rho)$ as right $A$-modules $M$ equipped with a coaction which is a right $A$-module map $\rho: M\to M\otimes_A C$ satisfying the standard axioms (comodules for coalgebras were introduced by Cartier in the 1950s). There is a rather more recent dualization of the concept. A $C$-contramodule is a right $A$-module $M$ equipped with a right $A$-module map $\alpha: Hom_A(C,M)\to M$ such that the diagrams
commute.
Historically contramodules seem to have first been mentioned a few times in the Eilenberg–Moore treatments of homological algebra in the 1960s and by category theorists (Vazquez, Garcia, Barr). Positselski has used them in his approach to semiinfinite cohomology.
Leonid Positselski, Homological algebra of semimodules and semicontramodules. Semi-infinite homological algebra of associative algebraic structures, arxiv/0708.3398; Contramodules, arxiv/1503.00991; Contraherent cosheaves, arxiv/1209.2995
T. Brzeziński, Contramodules, pdf, slides at International Conference on “Categories in Geometry”, Split, September 24-28, 2007.
T. Brzeziński, Hopf-cyclic homology with contramodule coefficients, To appear in Quantum Groups and Noncommutative Spaces, M Marcolli and D Parashar (eds) Vieweg Verlag (Max-Planck Series), Preprint 2008, arxiv:0806.0389
Last revised on March 4, 2015 at 15:08:42. See the history of this page for a list of all contributions to it.