Weights in cohomology and the Eilenberg-Moore spectral sequence

Franz, Matthias; Weber, Andrzej

Annales de l'Institut Fourier, Tome 55 (2005) no. 2, pp. 673-691.

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 $G$-variété algébrique $X$ ($G$ étant un group algébrique connexe), à partir de sa cohomologie équivariante, pourvu que $H_G^{*}\left(X\right)$ soit pur. Cette dernière condition est satisfaite, par exemple, si $X$ 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.

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 $G$-variety $X$ ($G$ being a connected algebraic group) in terms of its equivariant cohomology provided that $H_G^{*}\left(X\right)$ is pure. This is the case, for example, if $X$ 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.

Classification : 32S35, 14L30, 14F43, 55N33
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

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