  We try to understand the behavior of exterior algebraic shifting with respect
to basic constructions on simplicial complexes, like union and join. In
particular we give a complete combinatorial description of the shifting of a
disjoint union, and more generally of a union along a simplex, in terms of the
shifting of its components. As a corollary, we prove the following, conjectured
by Kalai: $\Delta(K \cup L) = \Delta (\Delta(K) \cup \Delta(L))$, where $K,L$
are complexes, $\cup$ means disjoint union, and $\Delta$ is the exterior
shifting operator. We give an example showing that replacing the operation
'union' with the operation 'join' in the above equation is wrong, disproving a
conjecture made by Kalai.
  We adopt a homological point of view on the algebraic shifting operator,
which is used throughout this work.
