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# Mathematical Research Letters

## Volume 20 (2013)

### Number 6

### Combinatorial methods for the twisted cohomology of Artin groups

Pages: 1157 – 1175

DOI: http://dx.doi.org/10.4310/MRL.2013.v20.n6.a13

#### Authors

#### Abstract

In this paper we introduce certain “combinatorial sheaves” on posets, which we call *weighted sheaves,* and we relate their cohomology, computed by a *weighted complex,* with the twisted cohomology of Artin groups. It turns out that to each Artin group one can associate a weighted sheaf, where the poset is given by the simplicial complex of all finite parabolic subgroups, and the cohomology of the Artin group with coefficients in a module of Laurent polynomials (interesting for geometrical reasons) is computed by the associated weighted complex. We connect this theory with the so called “Discrete Morse Theory”: a *weighted matching* on the weighted complex gives rise to a Morse complex computing the cohomology. We give a natural filtration of the weighted complex, which is compatible with the weighted matching, so obtaining a converging spectral sequence. We use such machinery to compute the twisted conomology for all exceptional type affine Artin groups.