2006/03/31 by Guillermo Cortiñas, Andreas Thom
Mathematics · #math.KT #math.RA #msc:19K35 #msc:19D25 #msc:18E30
published as J. reine angew. Math. 510 (2007) 71-124 · 40 pages, no figures. Comparison with Kassel's K-group added (see 6.7). Final version to appear in Crelle's Journal, including galley proof corrections
arxiv created 2007/04/23 · arxiv updated 2011/08/03
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.