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Journal of the London Mathematical Society Advance Access originally published online on November 12, 2007
Journal of the London Mathematical Society 2007 76(3):605-621; doi:10.1112/jlms/jdm081
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© 2007 London Mathematical Society

Homology of GLn over algebraically closed fields

B. Mirzaii

Department of Pure Mathematics
Queen's University Belfast
Belfast BT7 1NN
United Kingdom

In this paper we define higher pre-Bloch groups pn(F) of a field F. When the base field is algebraically closed, we study its connection to the homology of the general linear groups with coefficients in Z/l Z, where l is a positive integer. As a result of our investigation we give a necessary and sufficient condition for the natural map Hn(GLn–1(F), Z/l Z) -> Hn(GLn(F), Z/l Z) to be bijective. We prove that this map is bijective for n≤4. We also demonstrate that a certain property of pn(C) is equivalent to the validity of the Friedlander–Milnor isomorphism conjecture for (n+1)th homology of GLn(C).

2000 Mathematics Subject Classification 19D55 (primary), 18G60, 20J05 (secondary).

Received November 3, 2006; revised May 3, 2007; published online November 12, 2007.


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