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.

References in corpus (2)

• On the K-theory of local fields
• The obstruction to excision in K-theory and in cyclic homology

Cited by in corpus (9)

• Algebraic K-theory and descent for blow-ups
• Excision in algebraic K-theory revisited
• On the K-theory of the coordinate axes in the plane
• On the relative and bi-relative K-theory of rings of finite characteristic
• Some recent advances in topological Hochschild homology
• Categorical Milnor squares and K-theory of algebraic stacks
• K-theory of one-dimensional rings via pro-excision
• On the K-theory of coordinate axes in affine space
• Assembly maps for topological cyclic homology of group algebras