nLab contramodule

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

$$\array{ Hom_A(C,Hom_A(C,M)) &&\stackrel{Hom_A(C,\alpha)}\to && Hom_A(C,M)\\ \downarrow \cong &&&&\downarrow \alpha\\ Hom_A(C\otimes_A C,M)&\stackrel{Hom_A(\Delta,M)}\to&Hom_A(C,M)&\stackrel{\alpha}\to& M\\}$$

$$\array{ Hom_A(A,M) &&\stackrel{Hom_A(\epsilon,M)}\to && Hom_A(C,M)\\ &\searrow \cong &&\swarrow \alpha &\\ &&M&& \\}$$

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