### Abstract

We give several resolutions of the Steinberg representation St_{n} for the general linear group over a principal ideal domain, in particular over Z{double-struck}. We compare them, and use these results to prove that the computations in Avner Ash et al. (2011) [AGM11] are definitive. In particular, in Avner Ash et al. (2011) [AGM11] we use two complexes to compute certain cohomology groups of congruence subgroups of SL(4,Z{double-struck}). One complex is based on Voronoi's polyhedral decomposition of the symmetric space for SL(n,R{double-struck}), whereas the other is a larger complex that has an action of the Hecke operators. We prove that both complexes allow us to compute the relevant cohomology groups, and that the use of the Voronoi complex does not introduce any spurious Hecke eigenclasses.

Original language | English (US) |
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Pages (from-to) | 380-390 |

Number of pages | 11 |

Journal | Journal of Algebra |

Volume | 349 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2012 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

### Keywords

- Cohomology of arithmetic groups
- Modular symbols
- Steinberg module
- Voronoi complex

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## Cite this

*Journal of Algebra*,

*349*(1), 380-390. https://doi.org/10.1016/j.jalgebra.2011.09.018