Algebraic K-theory of group rings and the cyclotomic trace map

作者：Lueck Wolfgang; Reich Holger; Rognes John; Varisco Marco^*

来源：Advances in Mathematics, 2017, 304: 930-1020.

DOI：10.1016/j.aim.2016.09.004

• Algebraic K-theory
• Farrell-Jones conjecture
• Assembly maps
• Cyclotomic trace
• Topological cyclic homology
• Topological Hochschild homology

We prove that the Farrell-Jones assembly map for connective algebraic K-theory is rationally injective, under mild homological finiteness conditions on the group and assuming that a,weak version of the Leopoldt Schneider conjecture holds for cyclotomic fields. ThiS generalizes a result of Bokstedt, Hsiang, and Madsen, and leads to a concrete description of a huge direct summand of K-n (L[C]) D in terms of group homology. In many cases the number theoretic conjectures are true, so we obtain rational injectivity results about assembly maps, in particular for Whitehead groups, under homological finiteness assumptions on the group only. The proof uses the cyclotomic trace map to topological cyclic homology, Bokstedt Hsiang Madsen's functor C, and new general isomorphism and injectivity results about the assembly maps for topological Hochschild homology and C.

• 出版日期2017-1-2