Higher order derived functors and the Adams spectral sequence

TL;DR

This paper introduces higher order track algebras and chain complexes to generalize classical homological algebra, revealing that the E_m-term of the Adams spectral sequence corresponds to higher order Ext-groups derived from these structures.

Contribution

It extends classical homological algebra by defining higher order structures and shows their role in determining the E_m-term of the Adams spectral sequence.

Findings

Higher order resolutions exist in higher track categories.

Higher order Ext-groups determine the E_m-term of the Adams spectral sequence.

Track algebras encode higher cohomology operations.

Abstract

Classical homological algebra considers chain complexes, resolutions, and derived functors in additive categories. We describe "track algebras in dimension n", which generalize additive categories, and we define higher order chain complexes, resolutions, and derived functors. We show that higher order resolutions exist in higher track categories, and that they determine higher order Ext-groups. In particular, the E_m-term of the Adams spectral sequence (m<n+3) is a higher order Ext-group, which is determined by the track algebra of higher cohomology operations.