Comparing assembly maps in algebraic \(K\)-theory. (English) Zbl 1220.19002
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Given a group \(G\) and a ring \(R\), there is an algebraic \(K\)-theory spectrum \(\mathbb{K}^{-\infty}(RG)\) constructed by E. K. Pedersen and C. Weibel [“A nonconnective delooping of algebraic K-theory”. Lect. Notes Math. 1126, 166–181 (1985; Zbl 0591.55002)]. For this spectrum one gets a map \(A_G : BG_+\wedge \mathbb{K}^{-\infty}(R) \to \mathbb{K}^{-\infty}(RG)\) similar to the Loday assembly map \(\alpha_G : BG_+\wedge \mathbf{L}(R) \to \mathbf{L}(RG)\) and knows that these two maps are homotopic. I. Hambleton and E. K. Pedersen sketch a short proof of this fact in \(\S 4\) of [Math. Ann. 328, No. 1-2, 27–57 (2004; Zbl 1051.19002)]. The purpose of this paper is to present a thorough discussion of this sketch and thereby to fill in essential details lacking from the sketch. Here \(\mathbf{L}(R)\) is a spectrum with \(n\)th space \(K_0(S^nR)\times BGL(S^nR)^+\) for \(n\geq 0\) where \(S^n\) indicates the \(n\)-fold suspension of a ring. Using the tensor product pairing \(\mathbb{Z}G\times R \to RG\), the pairing on the Loday spectrum yields a map \(\mathbf{L}(\mathbb{Z}G)\wedge \mathbf{L}(R) \to \mathbf{L}(RG)\). Then by combining this map and the one \(BG \to BGL(\mathbb{Z}G) \subset BGL(\mathbb{Z}G)^+\) induced by the inclusion \(G \to \mathbb{Z}G\), one obtains \(\alpha_G\).
The essential points to be observed are that there are homotopy equivalences \(\theta_R : \mathbb{K}^{-\infty}(R) \to \mathbf{L}(R)\) and that the following diagram \[ \begin{tikzcd} \mathbb{K}^{-\infty}(R)\wedge \mathbb{K}^{-\infty}(R') \rar["\otimes"]\dar["\theta_R\wedge \theta_{R'}" '] & \mathbb{K}^{-\infty}(R\otimes R') \dar["\theta_{R\otimes R'}"] \\ \mathbf{L}(R)\wedge \mathbf{L}(R') \rar["\hat{\gamma}" '] & \mathbf{L}(R\otimes R') \end{tikzcd} \] is homotopy commutative where \(\otimes\) and \(\hat{\gamma}\) denote the Pedersen-Weibel and Loday pairings, respectively. In fact there can be found all the details of the proofs of these, which are attempted to be written in a self-contained manner so that one can read them without going back to the original sources. This will allow readers to more fully understand the original proof above [op. cit.], hence also this paper will be able to serve as a guide for readers who are interested in further studies related to the subject here.
Given a group \(G\) and a ring \(R\), there is an algebraic \(K\)-theory spectrum \(\mathbb{K}^{-\infty}(RG)\) constructed by E. K. Pedersen and C. Weibel [“A nonconnective delooping of algebraic K-theory”. Lect. Notes Math. 1126, 166–181 (1985; Zbl 0591.55002)]. For this spectrum one gets a map \(A_G : BG_+\wedge \mathbb{K}^{-\infty}(R) \to \mathbb{K}^{-\infty}(RG)\) similar to the Loday assembly map \(\alpha_G : BG_+\wedge \mathbf{L}(R) \to \mathbf{L}(RG)\) and knows that these two maps are homotopic. I. Hambleton and E. K. Pedersen sketch a short proof of this fact in \(\S 4\) of [Math. Ann. 328, No. 1-2, 27–57 (2004; Zbl 1051.19002)]. The purpose of this paper is to present a thorough discussion of this sketch and thereby to fill in essential details lacking from the sketch. Here \(\mathbf{L}(R)\) is a spectrum with \(n\)th space \(K_0(S^nR)\times BGL(S^nR)^+\) for \(n\geq 0\) where \(S^n\) indicates the \(n\)-fold suspension of a ring. Using the tensor product pairing \(\mathbb{Z}G\times R \to RG\), the pairing on the Loday spectrum yields a map \(\mathbf{L}(\mathbb{Z}G)\wedge \mathbf{L}(R) \to \mathbf{L}(RG)\). Then by combining this map and the one \(BG \to BGL(\mathbb{Z}G) \subset BGL(\mathbb{Z}G)^+\) induced by the inclusion \(G \to \mathbb{Z}G\), one obtains \(\alpha_G\).
The essential points to be observed are that there are homotopy equivalences \(\theta_R : \mathbb{K}^{-\infty}(R) \to \mathbf{L}(R)\) and that the following diagram \[ \begin{tikzcd} \mathbb{K}^{-\infty}(R)\wedge \mathbb{K}^{-\infty}(R') \rar["\otimes"]\dar["\theta_R\wedge \theta_{R'}" '] & \mathbb{K}^{-\infty}(R\otimes R') \dar["\theta_{R\otimes R'}"] \\ \mathbf{L}(R)\wedge \mathbf{L}(R') \rar["\hat{\gamma}" '] & \mathbf{L}(R\otimes R') \end{tikzcd} \] is homotopy commutative where \(\otimes\) and \(\hat{\gamma}\) denote the Pedersen-Weibel and Loday pairings, respectively. In fact there can be found all the details of the proofs of these, which are attempted to be written in a self-contained manner so that one can read them without going back to the original sources. This will allow readers to more fully understand the original proof above [op. cit.], hence also this paper will be able to serve as a guide for readers who are interested in further studies related to the subject here.
Reviewer: Haruo Minami (Nara)
MSC:
| 19D50 | Computations of higher \(K\)-theory of rings |
| 19D55 | \(K\)-theory and homology; cyclic homology and cohomology |
| 55P42 | Stable homotopy theory, spectra |
References:
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