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This paper describes the generalization to arbitrary associative algebras of the complex of "typical curves on algebraic      K K      -theory" and shows, in particular, that for certain      Q {\mathbf {Q}}      -algebras,      A A      , the complex is isomorphic to the "generalized de Rham complex,"      ( H H ∗  ( A ) , B ) (H{H_*}(A),B)      , in which      B B       is Connes’ operator acting on the Hochschild homology groups of      A A      .

Curves on $K$-theory and the de Rham homology of associative algebras