Algebraic shifting and exterior and symmetric algebra methods

Abstract

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 Gröbner basis theory over the exterior algebra, Cartan homology, degree functions, and Alexander duality.

26 pages. Revised version. Proofs of theorems 2.10 (now 2.11) and 2.11 (now 2.12) corrected. Exposition improved with some reordering. Minor changes