### Abstract

Let G be a simple Chevalley group defined over F_{q}. We show that if r does not divide q and k is an algebraically closed field of characteristic r, then very few irreducible kG-modules have nonzero H^{1}(G, V). We also give an explicit upper bound for dim H^{1}(G, V) for V an irreducible kG-module that does not depend on q, but only on the rank of the group. Cline, Parshall and Scott showed that such a bound exists when r|q. We obtain extremely strong bounds in the case that a Borel subgroup has no fixed points on V.

Original language | English (US) |
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Pages (from-to) | 543-559 |

Number of pages | 17 |

Journal | Annals of Mathematics |

Volume | 174 |

Issue number | 1 |

DOIs | |

State | Published - Jul 1 2011 |

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

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## Cite this

Guralnick, R. M., & Tiep, P. H. (2011). First cohomology groups of Chevalley groups in cross characteristic.

*Annals of Mathematics*,*174*(1), 543-559. https://doi.org/10.4007/annals.2011.174.1.16