Some combinatorial constructions and relations with Artin groups.

We give a general construction on poset diagrams over a ring, introducing weighted sheaves over posets and the associated weighted complexes. In particular, such complexes compute the local cohomology of Artin groups. We find an unexp[ected relation between the homology of such complexes, in case $A_n,$ and the homology of very particular graph complexes, called "independence graphs", which have been used in completely different situations. The main tool is the use of a variation of Discrete Morse theory, useful in our situation.