Bi-relative algebraic K-theory and topological cyclic homology
Abstract
It is well-known that algebraic K-theory preserves products of rings. However, in general, algebraic K-theory does not preserve fiber-products of rings, and bi-relative algebraic K-theory measures the deviation. It was proved by Cortinas that,rationally, bi-relative algebraic K-theory and bi-relative cyclic homology agree. In this paper, we show that, with finite coefficients, bi-relative algebraic K-theory and bi-relative topological cyclic homology agree. As an application, we show that for a, possibly singular, curve over a perfect field of positive characteristic p, the cyclotomic trace map induces an isomorphism of the p-adic algebraic K-groups and the p-adic topological cyclic homology groups in non-negative degrees. As a further application, we show that the difference between the p-adic K-groups of the integral group ring of a finite group and the p-adic K-groups of a maximal Z-order in the rational group algebra can be expressed entirely in terms of topological cyclic homology.
- Publication:
-
Inventiones Mathematicae
- Pub Date:
- May 2006
- DOI:
- 10.1007/s00222-006-0515-y
- arXiv:
- arXiv:math/0409122
- Bibcode:
- 2006InMat.166..359G
- Keywords:
-
- Mathematics - Number Theory;
- Mathematics - K-Theory and Homology
- E-Print:
- Invent. Math. 166 (2006), 359-395.