Andre-Quillen (co)homology and Equivariant Stable Homotopy Theory

TL;DR

This paper extends André-Quillen (co)homology to equivariant stable homotopy theory by developing a cotangent complex for incomplete Tambara functors and establishing a fundamental spectral sequence, advancing the algebraic tools in equivariant contexts.

Contribution

It introduces a cotangent complex for incomplete Tambara functors and constructs an analogue of the fundamental spectral sequence in equivariant stable homotopy theory.

Findings

Established a cotangent complex for incomplete Tambara functors.

Developed an equivariant analogue of the fundamental spectral sequence.

Clarified conditions for generating the spectral sequence with cofibrations.

Abstract

Andr\'e and Quillen introduced a (co)homology theory for augmented commutative rings. Strickland initially proposed some issues with the analogue of the abelianization functor in the equivariant setting. These were resolved by Hill who further gave the notion of a genuine derivation and a module of K\"ahler differentials. We build on this endeavor by expanding to incomplete Tambara functors, introducing the cotangent complex and its various properties, and producing an analogue of the fundamental spectral sequence. Note: This thesis is reposted with a correction that appears at the end of chapter 3. Namely, we make an additional assumption that a map be a cofibration in order to generate the transitivity long exact sequence. This change influences chapter 4 and when we can generate the fundamental spectral sequence.