Successive Spectral Sequences

TL;DR

This paper introduces a generalized framework for spectral sequences based on filtered chain complexes over arbitrary posets, revealing finer invariants and unifying multiple spectral sequence types with new theoretical insights.

Contribution

It develops a structure theory for generalized spectral sequences, including new construction methods, invariants, and unification of various spectral sequences, extending classical results.

Findings

Finer invariants than ordinary spectral sequences

A criterion for product structures in Grothendieck's spectral sequences

Unified framework for successive spectral sequences

Abstract

In this paper, we develop a structure theory for generalized spectral sequences, which are derived from chain complexes that are filtered over arbitrary partially ordered sets. Also, a more general construction method reminiscent of exact couples is studied, together with examples where they arise naturally. As for ordinary spectral sequences we will see differentials and group extensions, however the real power comes from the appearance of natural isomorphims between pages of differing indices. The constructions reveal finer invariants than ordinary spectral sequences, and they connect to other fields such as Fary functors and perverse sheaves. They are based on a natural index scheme, which allows us to obtain new results even in the standard case of Z-filtered chain complexes, e.g. a useful criterion for a product structure for Grothendieck's spectral sequences, and news paths to connect the first or second page to the limit. This turns out to yield the right framework for unifying several spectral sequences that one would usually apply one after another. Examples that we work out are successive Leray--Serre spectral sequences, the Adams--Novikov spectral sequence following the chromatic spectral sequence, successive Grothendieck spectral sequences, and successive Eilenberg--Moore spectral sequences.