Comparaison des homologies du groupe linéaire et de son algèbre de Lie

Loday, Jean-Louis

Annales de l'Institut Fourier, Tome 37 (1987) no. 4, pp. 167-190.

Pour un anneau local $R$ l’homologie du groupe discret $G{L}_n\left(R\right)$ a un comportement tout à fait analogue à l’homologie de l’algèbre de Lie $\mathbf{g}{\mathbf{l}}_n\left(A\right)$ lorsque $A$ est une algèbre associative sur un corps de caractéristique zéro. L’objet de cet article est de faire une synthèse (sans démonstration) des résultats connus sur ces groupes d’homologie en exhibant leurs liens avec la $K$-théorie algébrique, l’homologie cyclique et la cohomologie motivique. On y pose un certain nombre de questions et on propose une définition pour l’analogue additif de la cohomologie motivique.

The homology of the discrete groupe $G{L}_n\left(R\right)$ for a local ring $R$ behaves like the homology of the Lie algebra $\mathbf{g}{\mathbf{l}}_n\left(A\right)$ 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 “addivitive motivic cohomology theory” is suggested.

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