Homology of linear groups with coefficients in the adjoint action and \(K\)-theory
A result of C.-H. Sah for infinite fields \(F\), \(K_3(F)^{\text{ind}}\cong H_0(F^\times ,H_3(SL(2,F)))\bmod{\mathcal C}_6\) has been generalized to commutative rings `with many units' [\textit{Ph. Elbaz-Vincent}, ``The indecomposable \(K_3\) of rings and homology of \(SL_2\)'', J. Pure Appl. Algebra 132, 27-71 (1998)]. The main purpose of the present article is to apply this result to get explicit formulas for the relative \(K\)-group \(K_3(R[\varepsilon ],(\varepsilon))\), where \(R\) is commutative `with many units'. In particular: \[ K_3^{(2)}(R[\varepsilon ],(\varepsilon))_{\mathbb{Q}}\cong \bigoplus_{i=1}^3H_0(R^\times ,H_{3-i}(SL_2(R);\Lambda_{\mathbb{Q}}^i (sl_2(R))), \] where \(R[\varepsilon ]\) is the ring of dual numbers. The proof follows from the existence of a spectral decomposition of the rational homology groups of \(SL_k(R[\varepsilon ])\) [see, e.g., Lemma 2.1 in \textit{J. L. Dupont} and \textit{C.-H. Sah}, ``Homology of Euclidean groups of motions made discrete and Euclidean scissors congruences'', Acta Math. 164, No. 1/2, 1-27 (1990; Zbl 0724.57027)]. The results in this article have been announced [\textit{Ph. Elbaz-Vincent}, ``Homology of the special linear group and \(K\)-theory'', C. R. Acad. Sci., Paris, Sér. I 322, No. 9, 813-817 (1996; Zbl 0849.19004)].
scientific article; zbMATH DE number 1271717