The third homology of \(\mathrm{SL}_2\) of fields with discrete valuation. (English) Zbl 1357.19003
Let \(F\) be a field with discrete valuation \(v\) and residue field \(k\). Assume there is \(n\geq1\) such that every \(x\in F^\times\) with \(v(x)=0\) and \(v(1-x)\geq n\) is a square. The author relates the third homology of \(\mathrm{SL}_2(F)\) with \(\mathbb Z[1/2]\) coefficients to the third homology of \(\mathrm{SL}_2(k)\) and a certain ‘refined scissors congruence group’ of \(k\). Actions of the group ring of \(F^\times/(F^{\times})^2\) are used extensively. Higher-dimensional local fields are also treated.
Reviewer: Wilberd van der Kallen (Utrecht)
MSC:
| 19D99 | Higher algebraic \(K\)-theory |
| 20J05 | Homological methods in group theory |
| 20G10 | Cohomology theory for linear algebraic groups |
References:
| [1] | Dupont, J. L.; Sah, C-H., Scissors congruences. II, J. Pure Appl. Algebra, 25, 2, 159-195 (1982) · Zbl 0496.52004 |
| [2] | Fesenko, I. B.; Vostokov, S. V., Local Fields and Their Extensions, Translations of Mathematical Monographs, vol. 121 (1993), American Mathematical Society: American Mathematical Society Providence, RI, A constructive approach, With a foreword by I.R. Shafarevich · Zbl 0929.14028 |
| [3] | Gille, S.; Scully, S.; Zhong, C., Milnor-Witt \(K\)-groups of local rings, Adv. Math., 286, 729-753 (2016) · Zbl 1335.11030 |
| [4] | Hutchinson, K., Scissors congruence groups and the third homology of \(SL_2\) of local rings and fields |
| [5] | Hutchinson, K., The third homology of \(SL_2\) of fields with discrete valuation |
| [6] | Hutchinson, K., A refined Bloch group and the third homology of \(SL_2\) of a field, J. Pure Appl. Algebra, 217, 2003-2035 (2013) · Zbl 1281.19003 |
| [7] | Hutchinson, K., A Bloch-Wigner complex for \(SL_2\), J. K-Theory, 12, 1, 15-68 (2013) · Zbl 1284.19003 |
| [8] | Hutchinson, K., On the low-dimensional homology of \(SL_2(k [t, t^{- 1}])\), J. Algebra, 425, 324-366 (2015) · Zbl 1310.19004 |
| [9] | Hutchinson, K.; Wendt, M., On the third homology of \(S L_2\) and weak homotopy invariance, Trans. Am. Math. Soc., 367, 10, 7481-7513 (2015) · Zbl 1330.20069 |
| [10] | Kato, K., Class field theory and algebraic \(K\)-theory, (Algebraic Geometry. Algebraic Geometry, Tokyo/Kyoto, 1982. Algebraic Geometry. Algebraic Geometry, Tokyo/Kyoto, 1982, Lecture Notes in Mathematics, vol. 1016 (1983), Springer: Springer Berlin), 109-126 · Zbl 0544.12010 |
| [11] | Levine, M., The indecomposable \(K_3\) of fields, Ann. Sci. Éc. Norm. Supér. (4), 22, 2, 255-344 (1989) · Zbl 0705.19001 |
| [12] | Mirzaii, B., Third homology of general linear groups, J. Algebra, 320, 5, 1851-1877 (2008) · Zbl 1157.19003 |
| [13] | Neukirch, J., Algebraic Number Theory, Grundlehren der Mathematischen Wissenschaften, vol. 322 (1999), Springer-Verlag: Springer-Verlag Berlin, Translated from the 1992 German original and with a note by Norbert Schappacher, with a foreword by G. Harder · Zbl 0956.11021 |
| [14] | Parry, W.; Sah, C-H., Third homology of \(SL(2, R)\) made discrete, J. Pure Appl. Algebra, 30, 2, 181-209 (1983) · Zbl 0527.18006 |
| [15] | Parshin, A. N., Local class field theory, Algebraic Geometry and Its Applications. Algebraic Geometry and Its Applications, Tr. Mat. Inst. Steklova, 165, 143-170 (1984) · Zbl 0535.12013 |
| [16] | Schlichting, M., Euler class groups, and the homology of elementary and special linear groups · Zbl 1387.19002 |
| [17] | Suslin, A. A., Torsion in \(K_2\) of fields, \(K\)-Theory, 1, 1, 5-29 (1987) · Zbl 0635.12015 |
| [18] | Suslin, A. A., \(K_3\) of a field, and the Bloch group, Galois Theory, Rings, Algebraic Groups and Their Applications. Galois Theory, Rings, Algebraic Groups and Their Applications, Tr. Mat. Inst. Steklova. Galois Theory, Rings, Algebraic Groups and Their Applications. Galois Theory, Rings, Algebraic Groups and Their Applications, Tr. Mat. Inst. Steklova, Proc. Steklov Inst. Math., 4, 217-239 (1991), (in Russian). Translated in · Zbl 0741.19005 |
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