Extending locally truncated chamber systems by sheaves

Special issue in honour of A. Barlotti. We generalize the theory of sheaves to chamber systems. We prove that, given a chamber system W and a family R of proper residues of C containing all residues of rank less than or equal to 1, every sheaf defined over R admits a completion which extends W. We also prove that, under suitable hypotheses, a sheaf defined over a truncation of W can be extended to a sheaf for W. In the last section of this paper, we apply these results to a number of special cases.