# nLab contramodule

Given a $k$-algebra $A$ and an $A$-coring $C$ one can introduce right $C$-comodules $\left(M,\rho \right)$ 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 $\alpha :{\mathrm{Hom}}_{A}\left(C,M\right)\to M$ such that the diagrams

$\begin{array}{ccccc}{\mathrm{Hom}}_{A}\left(C,{\mathrm{Hom}}_{A}\left(C,M\right)\right)& & \stackrel{{\mathrm{Hom}}_{A}\left(C,\alpha \right)}{\to }& & {\mathrm{Hom}}_{A}\left(C,M\right)\\ ↓\cong & & & & ↓\alpha \\ {\mathrm{Hom}}_{A}\left(C{\otimes }_{A}C,M\right)& \stackrel{{\mathrm{Hom}}_{A}\left(\Delta ,M\right)}{\to }& {\mathrm{Hom}}_{A}\left(C,M\right)& \stackrel{\alpha }{\to }& M\\ \end{array}$\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\\}
$\begin{array}{ccccc}{\mathrm{Hom}}_{A}\left(A,M\right)& & \stackrel{{\mathrm{Hom}}_{A}\left(ϵ,M\right)}{\to }& & {\mathrm{Hom}}_{A}\left(C,M\right)\\ & ↘\cong & & ↙\alpha & \\ & & M& & \\ \end{array}$\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

• 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

Revised on July 22, 2010 01:51:58 by Toby Bartels (173.190.159.38)