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

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• 作者：Wolfgang Lü ; ck^a ; ^{wolfgang.lueck@him.uni-bonn.de" class="auth_mail" title="E-mail the corresponding author} ; ; Holger Reich^b ; ^{holger.reich@fu-berlin.de" class="auth_mail" title="E-mail the corresponding author} ; ; John Rognes^c ; ^{rognes@math.uio.no" class="auth_mail" title="E-mail the corresponding author} ; ; Marco Varisco^d ; ^{mvarisco@albany.edu" class="auth_mail" title="E-mail the corresponding author} ;
• 关键词：19D50 ; 19D55 ; 19B28 ; 55P91 ; 55P42
• 刊名：Advances in Mathematics
• 出版年：2017
• 出版时间：2 January 2017
• 年：2017
• 卷：304
• 期：Complete
• 页码：930-1020
• 全文大小：1369 K

文摘

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 Bökstedt, Hsiang, and Madsen, and leads to a concrete description of a large direct summand of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816301177&_mathId=si1.gif&_user=111111111&_pii=S0001870816301177&_rdoc=1&_issn=00018708&md5=6df297f013e7a9ba08cd5d4bedb66b37" title="Click to view the MathML source">K_n(Z[G])⊗_ZQclass="mathContainer hidden">class="mathCode">$K_n(\mathbb{Z}[G]){\otimes }_{\mathbb{Z}}\mathbb{Q}$ 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, Bökstedt–Hsiang–Madsen's functor C, and new general isomorphism and injectivity results about the assembly maps for topological Hochschild homology and C.