Algebraic shifting and exterior and symmetric algebra methods

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.