Abstract
The homology of GL n (R) and SL n (R) is studied, where R is a commutative ‘ring with many units’. Our main theorem states that the natural map H 4(GL3(R), k) → H 4(GL4(R), k) is injective, where k is a field with char(k) ≠ 2, 3. For an algebraically closed field F, we prove a better result, namely, \(H_4({\rm GL}_3(F), \mathbb {Z}) \to H_{4}({\rm GL}_4(F), \mathbb {Z})\) is injective. We will prove a similar result replacing GL by SL. This is used to investigate the indecomposable part of the K-group K 4(R).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arlettaz D. (2000). Algebraic K-theory of rings from a topological viewpoint. Publ. Mat. 44(1): 3–84
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
Bass, H., Tate, J.: The Milnor ring of a global field. Lecture Notes in Mathematics, vol. 342, pp. 349–446 (1973)
Borel A. and Yang J. (1994). The rank conjecture for number fields. Math. Res. Lett. 1(6): 689–699
Brown K.S. (1994). Cohomology of Groups. Graduate Texts in Mathematics, vol.~87. Springer, New York
Dennis, K.: In search of new “Homology” functors having a close relationship to K-theory. (preprint, 1976)
De Jeu R. (2002). A remark on the rank conjecture. K-Theory 25(3): 215–231
Dupont, J.-L.: Scissors Congruences, Group Homology and Characteristic Classes. Nankai Tracts in Mathematics, vol.~1. World Scientific Publishing Co., Inc., River Edge (2001)
Elbaz-Vincent P. (1998). The indecomposable K 3 of rings and homology of SL2. J. Pure Appl. Algebra 132(1): 27–71
Guin D. (1989). Homologie du groupe linéaire et K-théorie de Milnor des anneaux. J. Algebra 123(1): 27–59
Mirzaii B. (2005). Homology stability for unitary groups II. K-Theory 36(3–4): 305–326
Mirzaii, B.: Third homology of general linear groups. (Preprint)
Mirzaii, B.: Homology of SL n and GL n over an infinite field. (Preprint, available at http://arxiv.org/ abs/math/0605722)
Mirzaii, B.: Homology of GL n over algebraically closed fields. (Preprint, available at http://arxiv.org/ abs/math/0703337)
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
Sah C. (1989). Homology of classical Lie groups made discrete. III. J. Pure Appl. Algebra 56(3): 269–312
Suslin A.A. (1984). On the K-theory of local fields. J. Pure Appl. Algebra 34(2–3): 301–318
Suslin A. (1985). A Homology of GL n , characteristic classes and Milnor K-theory. Proc. Steklov Math. 3: 207–225
Suslin A.: A. K 3 of a field and the Bloch group. Proc. Steklov Inst. Math. 1991 183(4), 217–239
Soulé C. (1985). Opérations en K-théorie algébrique. Can. J. Math. 37(3): 488–550
Van der Kallen W. (1977). The K 2 of rings with many units. Ann. Sci. École Norm. Sup. 4(10): 473–515