Quillen's Fundamental Spectral Sequences Revisited

TL;DR

This paper provides a detailed and rigorous proof of Quillen's fundamental spectral sequences, which connect Andre9-Quillen homology and cohomology to Tor and Ext, crucial for understanding regular and complete intersection local rings.

Contribution

It offers the first comprehensive and detailed proof of Quillen's spectral sequences, filling gaps in the existing literature and clarifying their structures and applications.

Findings

Rigorous proof of Quillen's spectral sequences

Characterization of regular and complete intersection rings

Clarification of the structures involved in the spectral sequences

Abstract

Quillen's fundamental spectral sequences relate Andr\'{e}-Quillen homology and cohomology to Tor and Ext functors. The five-term exact sequences arising from these spectral sequences are leveraged to characterize regular and complete intersection local rings. Despite their immense importance in the theory of Andr\'{e}-Quillen homology and cohomology, these spectral sequences are not treated in the literature as they merit. A sketchy argument with gaps for a special case of the first spectral sequence appears in an unpublished manuscript of Quillen, while the second spectral sequence is only stated without proof in another paper of Quillen. A rigorous proof of the spectral sequences requires a delicate investigation of the subtle structures involved. The goal of this expository article is to present a comprehensive and detailed proof of Quillen's fundamental spectral sequences in a readable and well-structured manner.