Abstract:
For rings with a large number of units the authors prove a strengthened theorem on homological stabilization: the homomorphism $H_k(\operatorname{GL}_n(A))\to H_k(\operatorname{GL}(A))$ is surjective for $n\geqslant k+\operatorname{sr}A-1$ and bijective for $n\geqslant k+\operatorname{sr}A$.
If $A$ is a local ring with an infinite residue field, then this result admits further refinement: the homomorphism $H_n(\operatorname{GL}_n(A))\to H_n(\operatorname{GL}(A))$ is bijective and the factor group $H_n(\operatorname{GL}(A))/H_n(\operatorname{GL}_{n-1}(A))$ is canonically isomorphic to Milnor's $n$ th $K$-group of the ring $A$. The results are applied to compute the Chow groups of algebraic varieties.
Bibliography: 16 titles.
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