  We define the Coxeter cochain complex of a Coxeter group (G,S) with
coefficients in a Z[G]-module A. This is closely related to the complex of
simplicial cochains on the abstract simplicial complex I(S) of the commuting
subsets of S. We give some representative computations of Coxeter cohomology
and explain the connection between the Coxeter cohomology for groups of type A,
the (singular) homology of certain configuration spaces, and the (Tor) homology
of certain local Artin rings.
