Algebraic Shifting and Spectral Sequences

Abstract

There is a canonical spectral sequence associated to any filtration of simplicial complexes. Algebraically shifting a finite filtration of simplicial complexes produces a new filtration of shifted complexes.

We prove that certain sums of the dimensions of the limit terms of the spectral sequence of a filtration weakly decrease by algebraically shifting the filtration. A key step is the combinatorial interpretation of the dimensions of the limit terms of the spectral sequence of a filtration consisting of near-cones.