Bi-relative algebraic K-theory and topological cyclic homology

By Thomas H. Geisser, L. Hesselholt. 2004

49 citations recorded by the source corpus.

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 one defines bi-relative algebraic K-theory so as to measure the deviation. It was proved recently by Cortinas [6] that, rationally, bi-relative algebraic K-theory and bi-relative negative 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 k 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. Let F be a functor that to a unital associative ring A associates a symmetric spectrum F (A). If I ⊂ A is a two-sided ideal, the relative term F (A, I) is defined to be the homotopy fiber of the map F (A) → F (A/I) induced from the canonical projection. If further f : A → B is a ring homomorphism such that f : I → f(I) is an isomorphism onto a two-sided ideal of B, there is an induced map F (A, I)→ F (B, f(I)), and the bi-relative term F (A,B, I) is defined to be the homotopy fiber of this map. Then there is a distinguished triangle in the stable homotopy category

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/fad4d4310a5e2ef016cb93e88da71fc30772246e

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Citations in the corpus

K-theory and topological cyclic homology of henselian pairs (2018, methodology): “A key ingredient in the proof will be to use results of Geisser–Hesselholt [29] about excision in K-theory and topological cyclic homology to relate the setup of a general henselian pair to the case (Z ⋉ I, I), where we can appeal to the finiteness results of the previous subsection.”
Toric varieties, monoid schemes and $cdh$ descent (2011, result): “B], [12] shows that this homotopy fiber (regarded as a pro-presheaf of spectra) satisfies the hypotheses of Theorem 0.”
The arc-topology (2018, background): “The question of excision has played a crucial role in algebraic K-theory, see for instance [46, 10, 19].”
The Local Structure of Algebraic K-Theory (2012, result): “These notes have existed for quite a while on the net, and we are grateful for the helpful comments we have received from a number of people, in particular from Morten Brun, Lars Hesselholt, Harald Kittang, Birgit Richter, John Rognes, Stefan Schwede and Paul Arne 8 Østvær.”