Introduction

The Direct Sum Theorem for geometries states that a geometry belonging to a disconnected diagram is the direct sum of subgeometries corresponding to the connected components of that diagram. No analogous statement holds for chamber systems in general.

This situation has some uncomfortable consequences. For instance, we cannot reduce a classification problem for a class of chamber systems to cases with connected diagram, except when we have previously proved that the Direct Sum Theorem holds for the chamber systems of that class. Or, if we apply the celebrated criterion by Tits on rank 3 residues of spherical type to see if a given chamber system C belonging to a Coxeter diagram is covered by a building, we should check if the residues of C corresponding to disconnected rank 3 subdiagrams split as direct sums of subsystems of rank 1 or 2.

Unfortunately, some of the authors who have written on chamber systems seem to have been not awared of these problems. It would be stupid making a list of those who occasionaly said something wrong because of this oversight. I am not going to do that. Rather, I want to show that this situation is not really so bad as it might look. To support my optimistic opinion, I will show that in some important cases the counterexamples to the statement of the Direct Sum Theorem are quite sporadic, so that things can be kept under control in those cases.