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We introduce a refinement of the Bloch-Wigner complex of a field F. This is acomplex of modules over the multiplicative group of the field. Instead ofcomputing K_2 and indecomposable K_3 - as the classical Bloch-Wigner complexdoes - it calculates the second and third integral homology of SL_2 of thefield. On passing to coinvariants for the action of the multiplicative group werecover the classical Bloch-Wigner complex. The case of finite fields isincluded throughout the article.

A Bloch-Wigner complex for SL_2