The generalized Burnside ring and the \(K\)-theory of a ring with roots of unity. (English) Zbl 0780.19001
Let \(\ell\) be an odd prime and let \(p\neq\ell\) be a prime which generates the \(\ell\)-adic units. Let \(\zeta_ a\) be a primitive \(\ell^ a\)-th root of unity and let \(\mu\) be the group of \(\ell\)-primary roots of unity in \(\mathbb{Z}[\zeta_ a]\). Then there is a natural map \(h: Q_ 0(B\mu_ +)\to\text{BGL}(\mathbb{Z}[\zeta_ a])^ +\), where \(B\mu_ +\) is the classifying space of \(\mu\) with an added base point, \(Q_ 0(\;)\) denotes the infinite loop space associated to the suspension spectrum and the superscript \(+\) denotes Quillen’s plus construction. In \(\mathbb{Z}[\zeta_ a]\) there is a unique prime above \(p\) with residue class field \(\mathbb{F}_ q:=\mathbb{F}_ p[\zeta_ a]\).
Generalizing an earlier result of Quillen, it was shown by B. Harris and G. Segal that the composition \(Q_ 0(B\mu_ +)\to\text{BGL}(\mathbb{Z}[\zeta_ a])^ +\to\text{BGL}(\mathbb{F}_ q)^ +\) induces a surjection on homotopy groups with coefficients in \(\mathbb{Z}/\ell\). In fact, after localization at \(\ell\) the above map can be identified up to homotopy with the first projection \(r\) in a product decomposition.
The aim of the present paper is to show that the map \(h: Q_ 0(B\mu_ +)\to\text{BGL}(\mathbb{Z}[\zeta_ a])^ +\) does not detect more in homotopy \(\text{mod }\ell\) than this surjection, by showing that after localization at \(\ell\) it is homotopic to \(h\circ s\circ r\), where \(s\) is a suitable right inverse to \(r\). This is proven by showing that for any finite \(\ell\)-group G and any map \(\text{BG}\to Q_ 0(B\mu_ +)\) the compositions with \(h\) and \(h\circ s\circ r\) are homotopic, using the generalized Burnside ring and representation rings.
Generalizing an earlier result of Quillen, it was shown by B. Harris and G. Segal that the composition \(Q_ 0(B\mu_ +)\to\text{BGL}(\mathbb{Z}[\zeta_ a])^ +\to\text{BGL}(\mathbb{F}_ q)^ +\) induces a surjection on homotopy groups with coefficients in \(\mathbb{Z}/\ell\). In fact, after localization at \(\ell\) the above map can be identified up to homotopy with the first projection \(r\) in a product decomposition.
The aim of the present paper is to show that the map \(h: Q_ 0(B\mu_ +)\to\text{BGL}(\mathbb{Z}[\zeta_ a])^ +\) does not detect more in homotopy \(\text{mod }\ell\) than this surjection, by showing that after localization at \(\ell\) it is homotopic to \(h\circ s\circ r\), where \(s\) is a suitable right inverse to \(r\). This is proven by showing that for any finite \(\ell\)-group G and any map \(\text{BG}\to Q_ 0(B\mu_ +)\) the compositions with \(h\) and \(h\circ s\circ r\) are homotopic, using the generalized Burnside ring and representation rings.
Reviewer: A.Cap (Wien)
MSC:
| 19A22 | Frobenius induction, Burnside and representation rings |
| 19D06 | \(Q\)- and plus-constructions |
References:
| [1] | Bousfield, A. K., and Kan, D. M.:Homotopy Limits, Completions and Localizations, Lecture Notes in Math. 304, Springer, Berlin, 1972. · Zbl 0259.55004 |
| [2] | Friedlander, E. and Mislin, G.: Galois descent and cohomology for algebraic groups,Math. Z. 205 (1990), 177-190. · Zbl 0763.14023 · doi:10.1007/BF02571234 |
| [3] | Harris, B. and Segal, G.:K i groups of rings of algebraic integers,Ann. of Math. 101 (1975), 20-33. · Zbl 0331.18015 · doi:10.2307/1970984 |
| [4] | Lewis, G., May, P. and McClure, J.: Classifying G-spaces and the Segal conjecture,Current Trends in Algebraic Topology, CMS Conference Proc. No. 2, 1982, pp. 165-179. · Zbl 0572.55006 |
| [5] | May, J. P.: Stable maps between classifying spaces,Contemp. Math. vol. 37, American Math. Soc., Providence, 1985, pp. 121-129. · Zbl 0565.55021 |
| [6] | Mitchell, S. A.: The MoravaK-theory of algebraicK-theory spectra,K-Theory 3 (1990), 607-626. · Zbl 0709.55011 · doi:10.1007/BF01054453 |
| [7] | Mitchell, S. A.: On the Lichtenbaum-Quillen conjectures from a stable homotopy-theoretic viewpoint, preprint (1990). |
| [8] | Rector, D.: Modular characters andK-theory with coefficients in a finite field,J. Pure Appl. Algebra 4 (1974), 135-158. · Zbl 0323.18009 · doi:10.1016/0022-4049(74)90019-X |
| [9] | Reiner, I.:Maximal Orders, Academic Press, London, 1975. · Zbl 0305.16001 |
| [10] | Quillen, D. G.: Letter to J. Milnor,Lecture Notes in Math. 551, Springer, New York, 1976, pp. 182-188. |
| [11] | Roquette, P.: Realiserung von Darstellungen endlicher nilpotente Gruppen,Arch. Math. (Basel) 9 (1958), 224-250. · Zbl 0083.25002 · doi:10.1007/BF01900587 |
| [12] | Tornehave, J.: Relations among permutation representations of finite p-groups, preprint (1984). |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
