Bivariant algebraic K-Theory

Details Bivariant algebraic K-Theory Bivariant algebraic K-Theory

2006-03-22

arxiv.org/abs/math/0603531v6

J. reine angew. Math. 510 (2007) 71-124

Mathematics

K-Theory and Homology

40 pages, no figures. Comparison with Kassel's K-group added (see 6.7). Final version to appear in Crelle's Journal, including

Scientific paper

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.

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