Algebraic K-theory of group rings and the cyclotomic trace map. (English) Zbl 1357.19002
This article establishes a number of important results related to the Farrell-Jones conjectures. This review will only address the less technical ones and not mention the precise strong statements about the assembly map for topological Hochschild homology, nor the precise form of what the authors call the rationalized Farrell-Jones assembly map. Let us only say that this map goes from a direct sum taken over the set of conjugacy classes of finite cyclic subgroups of a group \(G\) to \(K_n(\mathbb Z [G]) \otimes_{\mathbb Z} \mathbb Q\). When restricted to the factor corresponding to the trivial subgroup one gets back the rationalized classical assembly map
\[ \bigoplus_{s+t=n} H_s(BG; \mathbb Q) \otimes_{\mathbb Q} \left( K_t(\mathbb Z) \otimes_{\mathbb Z} \mathbb Q \right) \rightarrow K_n(\mathbb Z [G]) \otimes_{\mathbb Z} \mathbb Q \]
The main theorem about the (generalized) assembly map is its injectivity for any group satisfying two conditions. The first condition is a purely group theoretical one, requiring that all integral homology groups of the centralizer of any finite cyclic subgroup of \(G\) is finitely generated. The second condition is a conjecturally true statement about the \(\mathbb Q\)-injectivity of a map in \(K\)-theory depending only on the order of finite subgroups of \(G\), and is referred to as a weak version of the Leopoldt-Schneider conjecture.
This conjecture is known to be true for the trivial subgroup (the case of order \(1\)), a direct corollary is thus the M. Bökstedt et al. Theorem [Invent. Math. 111, No. 3, 465–539 (1993; Zbl 0804.55004)], establishing the injectivity of the rationalized classical assembly map for any group \(G\) with finitely generated integral homology groups. The weak Leopoldt-Schneider conjecture is also true in small degrees (on \(K_0\) and \(K_1\)), from which the authors deduce cool consequences about the Whitehead group of many groups (satisfying very mild finiteness homological conditions), namely that rationally \(Wh(G)\) is assembled from all Whitehead groups \(Wh(H)\) where \(H\) is a finite subgroup of \(G\).
The introduction is very well written and gives an almost ten page long account on the results of this paper and related work. It continues with a thorough discussion of the two conditions we have mentioned above, in particular to which groups they are known to apply. An extra section presents the strategy of the proof and contains in particular a “roadmap” summarizing the main steps in the proof.
\[ \bigoplus_{s+t=n} H_s(BG; \mathbb Q) \otimes_{\mathbb Q} \left( K_t(\mathbb Z) \otimes_{\mathbb Z} \mathbb Q \right) \rightarrow K_n(\mathbb Z [G]) \otimes_{\mathbb Z} \mathbb Q \]
The main theorem about the (generalized) assembly map is its injectivity for any group satisfying two conditions. The first condition is a purely group theoretical one, requiring that all integral homology groups of the centralizer of any finite cyclic subgroup of \(G\) is finitely generated. The second condition is a conjecturally true statement about the \(\mathbb Q\)-injectivity of a map in \(K\)-theory depending only on the order of finite subgroups of \(G\), and is referred to as a weak version of the Leopoldt-Schneider conjecture.
This conjecture is known to be true for the trivial subgroup (the case of order \(1\)), a direct corollary is thus the M. Bökstedt et al. Theorem [Invent. Math. 111, No. 3, 465–539 (1993; Zbl 0804.55004)], establishing the injectivity of the rationalized classical assembly map for any group \(G\) with finitely generated integral homology groups. The weak Leopoldt-Schneider conjecture is also true in small degrees (on \(K_0\) and \(K_1\)), from which the authors deduce cool consequences about the Whitehead group of many groups (satisfying very mild finiteness homological conditions), namely that rationally \(Wh(G)\) is assembled from all Whitehead groups \(Wh(H)\) where \(H\) is a finite subgroup of \(G\).
The introduction is very well written and gives an almost ten page long account on the results of this paper and related work. It continues with a thorough discussion of the two conditions we have mentioned above, in particular to which groups they are known to apply. An extra section presents the strategy of the proof and contains in particular a “roadmap” summarizing the main steps in the proof.
Reviewer: Jérôme Scherer (Lausanne)
MSC:
| 19D55 | \(K\)-theory and homology; cyclic homology and cohomology |
| 19D50 | Computations of higher \(K\)-theory of rings |
| 19B28 | \(K_1\) of group rings and orders |
| 55P91 | Equivariant homotopy theory in algebraic topology |
| 55P42 | Stable homotopy theory, spectra |
Keywords:
algebraic \(K\)-theory; Farrell-Jones conjecture; assembly maps; cyclotomic trace; topological cyclic homology; topological Hochschild homologyCitations:
Zbl 0804.55004References:
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