  We establish results about algebraic shifting of simplicial complexes and use
them to compare different shifting operations. In particular, we show that each
shifting operation does not decrease the number of facets, and that the
exterior shift is the best among the exterior shifting operations in the sense
that it increases the number of facets the least. Methods of proof include
Gr\"obner basis theory over the exterior algebra, Cartan homology, degree
functions, and Alexander duality.
