Given a kk-algebra AA and an AA-coring CC one can introduce right CC-comodules (M,ρ)(M,\rho) as right AA-modules MM equipped with a coaction which is a right AA-module map ρ:MM AC\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 CC-contramodule is a right AA-module MM equipped with a right AA-module map α:Hom A(C,M)M\alpha: Hom_A(C,M)\to M such that the diagrams

Hom A(C,Hom A(C,M)) Hom A(C,α) Hom A(C,M) α Hom A(C AC,M) Hom A(Δ,M) Hom A(C,M) α M \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\\}
Hom A(A,M) Hom A(ϵ,M) Hom A(C,M) α M \array{ Hom_A(A,M) &&\stackrel{Hom_A(\epsilon,M)}\to && Hom_A(C,M)\\ &\searrow \cong &&\swarrow \alpha &\\ &&M&& \\}


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 (