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Comparaison des homologies du groupe linéaire et de son algèbre de Lie. (Comparison of homologies of a linear group and its Lie algebra). (French) Zbl 0619.20025

The homology of the discrete group \(GL_ n(R)\) for a local ring R behaves like the homology of the Lie algebra \(gl_ n(A)\) for A an associative algebra over a characteristic zero field. The aim of this article is to survey the known results (without giving any proof) about these homology groups and to connect them with algebraic K-theory, cyclic homology and motivic cohomology. Some questions are raised and a definition for an ”additive motivic cohomology theory” is suggested.

MSC:

20G10 Cohomology theory for linear algebraic groups
17B45 Lie algebras of linear algebraic groups
17B56 Cohomology of Lie (super)algebras
18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
20G35 Linear algebraic groups over adèles and other rings and schemes
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