### 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.