Depth and homology decompositions

TL;DR

This paper explores how depth conditions influence the vanishing of higher derived limits in homology decompositions, with applications to Stanley-Reisner algebras, group cohomology, and polynomial invariants.

Contribution

It introduces a unified approach to analyze depth conditions affecting homology decompositions and applies it to various algebraic and topological contexts.

Findings

Depth of Stanley-Reisner algebras characterized by simplicial complexes

Depth of group cohomology linked to centralizer subgroup properties

Depth of polynomial invariants related to stabilizer subgroup invariants

Abstract

Homology decomposition techniques are a powerful tool used in the analysis of the homotopy theory of (classifying) spaces. The associated Bousfield-Kan spectral sequences involve higher derived limits of the inverse limit functor. We study the impact of depth conditions on the vanishing of these higher limits and apply our theory in several cases. We will show that the depth of Stanley-Reisner algebras can be characterized in combinatorial terms of the underlying simplicial complexes, the depth of group cohomology in terms of depth of group cohomology of centralizers of elementary abelian subgroups, and the depth of polynomial invariants in terms of depth of polynomial invariants of point-wise stabilizer subgroups. The latter two applications follow from the analysis of an algebraic version of centralizer decompositions in terms of Lannes' $T$-functor.