Weights in cohomology and the Eilenberg-Moore spectral sequence

Matthias Franz[1]; Andrzej Weber

  • [1] Université de Genève, section de Mathématiques, CP 240, 1211 Genève 24 (Switzerland), Uniwersytet Warszawski, Instytut Matematyki, ul. Banacha 2, 02-097 Warszawa (POLAND)

Annales de l’institut Fourier (2005)

  • Volume: 55, Issue: 2, page 673-691
  • ISSN: 0373-0956

Abstract

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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 * ( X ) 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.

How to cite

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Franz, Matthias, and Weber, Andrzej. "Weights in cohomology and the Eilenberg-Moore spectral sequence." Annales de l’institut Fourier 55.2 (2005): 673-691. <http://eudml.org/doc/116202>.

@article{Franz2005,
abstract = {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 $H_G^*(X)$ 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.},
affiliation = {Université de Genève, section de Mathématiques, CP 240, 1211 Genève 24 (Switzerland), Uniwersytet Warszawski, Instytut Matematyki, ul. Banacha 2, 02-097 Warszawa (POLAND)},
author = {Franz, Matthias, Weber, Andrzej},
journal = {Annales de l’institut Fourier},
keywords = {Eilenberg-Moore spectral sequence; weight filtration; equivariant cohomology; intersection cohomology; complex algebraic $G$-varieties},
language = {eng},
number = {2},
pages = {673-691},
publisher = {Association des Annales de l'Institut Fourier},
title = {Weights in cohomology and the Eilenberg-Moore spectral sequence},
url = {http://eudml.org/doc/116202},
volume = {55},
year = {2005},
}

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
PB - Association des Annales de l'Institut Fourier
VL - 55
IS - 2
SP - 673
EP - 691
AB - 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 $H_G^*(X)$ 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.
LA - eng
KW - Eilenberg-Moore spectral sequence; weight filtration; equivariant cohomology; intersection cohomology; complex algebraic $G$-varieties
UR - http://eudml.org/doc/116202
ER -

References

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