Algebraic theories of power operations

TL;DR

This paper develops algebraic tools for understanding power operations in stable homotopy theory, enabling computations and applications such as periodic complex orientations at low heights.

Contribution

It introduces a unified algebraic framework combining Quillen cohomology, plethories, and Koszul resolutions for $ ext{E}_ ext{infty}$ algebras, facilitating new computations.

Findings

Tools for computing with $ ext{E}_ ext{infty}$ algebras over $ ext{F}_p$ and Lubin-Tate spectra

Demonstration of $ ext{E}_ ext{infty}$ periodic complex orientations at heights $h \,\leq\, 2$

Enhanced understanding of power operations in stable homotopy theory

Abstract

We develop and exposit some general algebra useful for working with certain algebraic structures that arise in stable homotopy theory, such as those encoding well-behaved theories of power operations for $\mathbb{E}_\infty$ ring spectra. In particular, we consider Quillen cohomology in the context of algebras over algebraic theories, plethories, and Koszul resolutions for algebras over additive theories. By combining this general algebra with obstruction-theoretic machinery, we obtain tools for computing with $\mathbb{E}_\infty$ algebras over $\mathbb{F}_p$ and over Lubin-Tate spectra. As an application, we demonstrate the existence of $\mathbb{E}_\infty$ periodic complex orientations at heights $h\leq 2$.