Bivariant algebraic K-theory

✍ Scribed by Cortiñas, Guillermo; Thom, Andreas

• Book ID: 118741226
• Publisher: Walter de Gruyter GmbH & Co. KG
• Year: 2007
• Tongue: English
• Weight: 423 KB
• Volume: 2007
• Category: Article
• ISSN: 0075-4102
• DOI: 10.1515/CRELLE.2007.068

✦ Synopsis

We show how methods from K-theory of operator algebras can be applied in a completely algebraic setting to define a bivariant, M y -stable, homotopy-invariant, excisive K-theory of algebras over a fixed unital ground ring H, ðA; BÞ 7 ! kk à ðA; BÞ, 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 à ðH; AÞ ¼ KH à ðAÞ:

We show further that some calculations from operator algebra KK-theory, such as the exact sequence of Pimsner-Voiculescu, carry over to algebraic kk.

Cortin ˜as and Thom, Bivariant algebraic K-theory