Homology of \(\mathrm{GL}_n\): injectivity conjecture for \(\mathrm{GL}_{4}\). (English) Zbl 1129.19005
Summary: The homology of \(\text{GL}_{n}(R)\) and \(\text{SL}_{n}(R)\) is studied, where \(R\) is a commutative ‘ring with many units’. Our main theorem states that the natural map \(H_{4}(\text{GL}_{3}(R), k) \rightarrow H_{4}(\text{GL}_{4}(R), k)\) is injective, where \(k\) is a field with char\((k) \not= 2, 3\). For an algebraically closed field \(F\), we prove a better result, namely, \(H_4(\text{GL}_3(F), \mathbb {Z}) \to H_{4}(\text{GL}_4(F), \mathbb {Z})\) is injective. We prove a similar result replacing GL by SL. This is used to investigate the indecomposable part of the \(K\)-group \(K_{4}(R)\).
MSC:
| 19D55 | \(K\)-theory and homology; cyclic homology and cohomology |
| 18G60 | Other (co)homology theories (MSC2010) |
| 20J05 | Homological methods in group theory |
References:
| [1] | Arlettaz D. (2000). Algebraic K-theory of rings from a topological viewpoint. Publ. Mat. 44(1): 3–84 · Zbl 0956.19003 |
| [2] | Barge J. and Morel F. (1999). Cohomologie des groupes linéaires, K-théorie de Milnor et groupes de Witt. C. R. Acad. Sci. Paris Sér. I Math. 328(3): 191–196 · Zbl 0944.20027 |
| [3] | Bass, H., Tate, J.: The Milnor ring of a global field. Lecture Notes in Mathematics, vol. 342, pp. 349–446 (1973) · Zbl 0299.12013 |
| [4] | Borel A. and Yang J. (1994). The rank conjecture for number fields. Math. Res. Lett. 1(6): 689–699 · Zbl 0841.19002 |
| [5] | Brown K.S. (1994). Cohomology of Groups. Graduate Texts in Mathematics, vol.\(\sim\)87. Springer, New York · Zbl 0823.20049 |
| [6] | Dennis, K.: In search of new ”Homology” functors having a close relationship to K-theory. (preprint, 1976) |
| [7] | De Jeu R. (2002). A remark on the rank conjecture. K-Theory 25(3): 215–231 · Zbl 1016.19003 · doi:10.1023/A:1015656301533 |
| [8] | Dupont, J.-L.: Scissors Congruences, Group Homology and Characteristic Classes. Nankai Tracts in Mathematics, vol.\(\sim\)1. World Scientific Publishing Co., Inc., River Edge (2001) · Zbl 0977.52020 |
| [9] | Elbaz-Vincent P. (1998). The indecomposable K 3 of rings and homology of SL2. J. Pure Appl. Algebra 132(1): 27–71 · Zbl 0926.19001 · doi:10.1016/S0022-4049(97)00080-7 |
| [10] | Guin D. (1989). Homologie du groupe linéaire et K-théorie de Milnor des anneaux. J. Algebra 123(1): 27–59 · Zbl 0669.20037 · doi:10.1016/0021-8693(89)90033-1 |
| [11] | Mirzaii B. (2005). Homology stability for unitary groups II. K-Theory 36(3–4): 305–326 · Zbl 1114.19003 · doi:10.1007/s10977-006-7109-8 |
| [12] | Mirzaii, B.: Third homology of general linear groups. (Preprint) · Zbl 1157.19003 |
| [13] | Mirzaii, B.: Homology of SL n and GL n over an infinite field. (Preprint, available at http://arxiv.org/ abs/math/0605722) · Zbl 1129.19004 |
| [14] | Mirzaii, B.: Homology of GL n over algebraically closed fields. (Preprint, available at http://arxiv.org/ abs/math/0703337) · Zbl 1129.19004 |
| [15] | Nesterenko Yu.P. and Suslin A. (1990). A Homology of the general linear group over a local ring and Milnor’s K-theory. Math. USSR-Izv. 34(1): 121–145 · Zbl 0684.18001 · doi:10.1070/IM1990v034n01ABEH000610 |
| [16] | Sah C. (1989). Homology of classical Lie groups made discrete. III. J. Pure Appl. Algebra 56(3): 269–312 · Zbl 0684.57020 · doi:10.1016/0022-4049(89)90061-3 |
| [17] | Suslin A.A. (1984). On the K-theory of local fields. J. Pure Appl. Algebra 34(2–3): 301–318 · Zbl 0548.12009 · doi:10.1016/0022-4049(84)90043-4 |
| [18] | Suslin A. (1985). A Homology of GL n , characteristic classes and Milnor K-theory. Proc. Steklov Math. 3: 207–225 |
| [19] | Suslin A.: A. K 3 of a field and the Bloch group. Proc. Steklov Inst. Math. 1991 183(4), 217–239 · Zbl 0741.19005 |
| [20] | Soulé C. (1985). Opérations en K-théorie algébrique. Can. J. Math. 37(3): 488–550 · Zbl 0575.14015 · doi:10.4153/CJM-1985-029-x |
| [21] | Van der Kallen W. (1977). The K 2 of rings with many units. Ann. Sci. École Norm. Sup. 4(10): 473–515 · Zbl 0393.18012 |
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.
