\(\mathrm{K}_1\) of a \(p\)-adic group ring. II: The determinantal kernel \(\mathrm{SK}_1\). (English) Zbl 1308.19002
Summary: We describe the group \(\mathrm{SK}_1(R [G])\) for group rings \(R[G]\) where \(G\) is an arbitrary finite group and where the coefficient ring \(R\) is a \(p\)-adically complete Noetherian integral domain of characteristic zero which admits a lift of Frobenius and which also satisfies a number of further mild conditions. Our results extend previous work of R. Oliver [Invent. Math. 57, 183–204 (1980; Zbl 0428.18011); Math. Scand. 47, 195–231 (1980; Zbl 0456.16027); Lect. Notes Math. 854, 299–337 (1981; Zbl 0454.16010); Proc. Lond. Math. Soc. (3) 46, 1–37 (1983; Zbl 0499.16017)] who obtained such results for the valuation rings of finite extensions of the \(p\)-adic field.
For part I, cf. [the authors, J. Algebra 326, No. 1, 74–112 (2011; Zbl 1219.19005)].
For part I, cf. [the authors, J. Algebra 326, No. 1, 74–112 (2011; Zbl 1219.19005)].
MSC:
| 19B28 | \(K_1\) of group rings and orders |
| 19C99 | Steinberg groups and \(K_2\) |
| 16E20 | Grothendieck groups, \(K\)-theory, etc. |
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