We show that in the category of complex algebraic varieties, the Eilenberg–Moore spectral sequence can be endowed with a weight filtration. This implies that it degenerates if all spaces involved have pure cohomology. As application, we compute the rational cohomology of an algebraic -variety ( being a connected algebraic group) in terms of its equivariant cohomology provided that is pure. This is the case, for example, if is smooth and has only finitely many orbits. We work in the category of mixed sheaves; therefore our results apply equally to (equivariant) intersection homology.
Nous montrons que dans la catégorie des variétés algébriques complexes la suite spectrale d’Eilenberg-Moore admet une filtration par le poids, ce qui implique sa dégénérescence si la cohomologie de tous les espaces en jeu est pure. Nous illustrons notre résultat par le calcul de la cohomologie rationnelle d’une -variété algébrique ( étant un group algébrique connexe), à partir de sa cohomologie équivariante, pourvu que soit pur. Cette dernière condition est satisfaite, par exemple, si est lisse et n’a qu’un nombre fini d’orbites. Nous travaillons dans la catégorie des faisceaux mixtes ; nos résultats restent donc également valables pour l’homologie d’intersection.
Keywords: Eilenberg-Moore spectral sequence, weight filtration, equivariant cohomology, intersection cohomology, complex algebraic $G$-varieties
Mot clés : suite spectrale d'Eilenberg-Moore, filtration de poids, cohomologie équivariante, cohomologie d'intersection, variétés algébriques complexes
@article{AIF_2005__55_2_673_0, author = {Franz, Matthias and Weber, Andrzej}, title = {Weights in cohomology and the {Eilenberg-Moore} spectral sequence}, journal = {Annales de l'Institut Fourier}, pages = {673--691}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {55}, number = {2}, year = {2005}, doi = {10.5802/aif.2109}, mrnumber = {2147902}, zbl = {02171520}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.2109/} }
TY - JOUR AU - Franz, Matthias AU - Weber, Andrzej TI - Weights in cohomology and the Eilenberg-Moore spectral sequence JO - Annales de l'Institut Fourier PY - 2005 SP - 673 EP - 691 VL - 55 IS - 2 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.2109/ DO - 10.5802/aif.2109 LA - en ID - AIF_2005__55_2_673_0 ER -
%0 Journal Article %A Franz, Matthias %A Weber, Andrzej %T Weights in cohomology and the Eilenberg-Moore spectral sequence %J Annales de l'Institut Fourier %D 2005 %P 673-691 %V 55 %N 2 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.2109/ %R 10.5802/aif.2109 %G en %F AIF_2005__55_2_673_0
Franz, Matthias; Weber, Andrzej. Weights in cohomology and the Eilenberg-Moore spectral sequence. Annales de l'Institut Fourier, Volume 55 (2005) no. 2, pp. 673-691. doi : 10.5802/aif.2109. http://archive.numdam.org/articles/10.5802/aif.2109/
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