Sur l'homologie des groupes unitaires \`a coefficients polynomiaux

TL;DR

This paper generalizes previous results to compute the stable homology of unitary groups over any ring with polynomial coefficients, expressing it via constant coefficient homology and computable functor homology groups.

Contribution

It extends earlier work to arbitrary rings and polynomial coefficients, providing explicit formulas for stable homology in terms of known homology groups.

Findings

Homology with polynomial coefficients can be derived from constant coefficient homology.

Explicit computation of functor homology groups is possible in certain cases.

Generalization to arbitrary rings broadens the applicability of previous results.

Abstract

We extend the results of the author with C. Vespa (Ann. Sci. ENS 2010) to stable homology of unitary groups over an arbitrary ring twisted by a polynomial functor : we show that it can be computed from the homology with constant coefficients and functor homology groups which can be explicitly computed in some cases.---On g\'en\'eralise les r\'esultats de l'auteur et C. Vespa (Ann. Sci. ENS 2010) \`a l'homologie stable des groupes unitaires sur un anneau quelconque \`a coefficients tordus par un foncteur polynomial, dont on montre qu'elle peut s'exprimer \`a partir de l'homologie \`a coefficients constants et de groupes d'homologie des foncteurs qu'on peut calculer explicitement dans les cas favorables.