Third homology of \(\mathrm{SL}_2\) over number fields: the norm-Euclidean quadratic imaginary case. (English) Zbl 07784725
For a field \(F\), let \(H_3(\mathrm{SL}_2(F),\mathbb{Z})_0\) denote the kernel of the natural surjective homomorphism \(H_3(\mathrm{SL}_2(F),\mathbb{Z})\to K_3^{\mbox{ \small ind}}(F)\). If \(F\) is a global field, then there is a natural surjective map
\[
S: H_3(\mathrm{SL}_2(F),\mathbb{Z}[1/2])_0\to \oplus_v P(k(v))\otimes \mathbb{Z}[1/2]
\]
where the sum is over all discrete valuations \(v\) of \(F\), \(k(v)\) denotes the residue field and \(P(k)\) is the scissors congruence group (or pre-Bloch group) of the field \(k\) [the reviewer, J. Pure Appl. Algebra 217, No. 11, 2003–2035 (2013; Zbl 1281.19003)]. The reviewer proved that this map is an isomorphism in the case \(F=\mathbb{Q}\) [J. Algebra 570, 366–396 (2021; Zbl 1462.20020)]. In this article, the author proves that it is an isomorphism in the case that \(F=\mathbb{Q}(\sqrt{-m})\), \(m=2,3,7,11\). Furthermore, he proves that it becomes an isomorphism in the case \(F=\mathbb{Q}(\sqrt{-1})\) after first taking coinvariants for the action of \(\sqrt{-1}\) on the left-hand side.
Reviewer: Kevin Hutchinson (Dublin)
MSC:
| 20G10 | Cohomology theory for linear algebraic groups |
| 19D55 | \(K\)-theory and homology; cyclic homology and cohomology |
| 19F99 | \(K\)-theory in number theory |
| 20J05 | Homological methods in group theory |
References:
| [1] | Brown, K. S., Cohomology of Groups. Graduate Text in Mathematics (1982), Springer Verlag: Springer Verlag New York · Zbl 0584.20036 |
| [2] | Dupont, Johan L.; Han, Sah Chih, Scissors congruences. II. J. Pure Appl. Algebra, 2, 159-195 (1982) · Zbl 0496.52004 |
| [3] | Hutchinson, Kevin; Liqun, Tao, The third homology of the special linear group of a field. J. Pure Appl. Algebra, 1665-1680 (2009) · Zbl 1187.19002 |
| [4] | Hutchinson, Kevin, A Bloch Wigner complex for \(S L_2\). J. K-Theory, 15-68 (2013) · Zbl 1284.19003 |
| [5] | Hutchinson, Kevin, A refined Bloch group and the third homology of \(\operatorname{SL}_2\) of a field. J. Pure Appl. Algebra, 2003-2035 (2013) · Zbl 1281.19003 |
| [6] | Hutchinson, Kevin, The third homology of \(S L_2\) of fields with discrete valuation. J. Pure Appl. Algebra, 5, 1076-1111 (2017) · Zbl 1357.19003 |
| [7] | Hutchinson, Kevin, The third homology of \(S L_2(\mathbb{Q})\). J. Algebra, 366-396 (2021), ISSN 0021-8693 · Zbl 1462.20020 |
| [8] | Behrooz, Mirzaii, Third homology of general linear groups. J. Algebra, 5, 1851-1877 (2008) · Zbl 1157.19003 |
| [9] | Behrooz, Mirzaii; Mokari Fatemeh, Y., A Bloch-Wigner theorem over rings with many units II. J. Pure Appl. Algebra, 11, 5078-5096 (2015) · Zbl 1329.19002 |
| [10] | Chin-Han, Sah, Homology of classical Lie groups made discrete, III. J. Pure Appl. Algebra, 3, 269-312 (1989) · Zbl 0684.57020 |
| [11] | Suslin, A. A., \( K_3\) of a field and the Bloch group. Proc. Steklov Inst. Math., 4, 217-239 (1991) · Zbl 0741.19005 |
| [12] | Weibel, Charles, The \(K\)-Book: an Introduction to Algebraic \(K\)-Theory. Graduate Studies in Math. (2013), AMS · Zbl 1273.19001 |
| [13] | Zickert Christian, K., The extended Bloch group and algebraic \(K\)-theory. J. Reine Angew. Math., 21-54 (2015) · Zbl 1334.19004 |
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.
