Bivariant algebraic K-Theory
  We show how methods from K-theory of operator algebras can be applied in a completely algebraic setting to define a bivariant, matrix-stable, homotopy-invariant, excisive K-theory of algebras over a fixed unital ground ring H, kk_*(A,B), which is universal in the sense that it maps uniquely to any other such theory.   It turns out kk is related to C. Weibel's homotopy algebraic K-theory, KH. We prove that, if H is commutative and A is central as an H-bimodule, then kk_*(H,A)=KH_*(A).   We show further that some calculations from operator algebra KK-theory, such as the exact sequence of Pimsner-Voiculescu, carry over to algebraic kk. 
Guillermo Corti\~nas, Andreas Thom