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 $h⩽2$.

中文翻译：

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幂运算的代数理论

我们开发并阐述了一些一般代数，可用于处理稳定同伦理论中出现的某些代数结构，例如那些编码表现良好的幂运算理论的代数结构${\mathbb{乙}}_{\mathrm{无穷大}}$环光谱。特别是，我们在代数理论、过剩理论和代数加法理论的 Koszul 解析的背景下考虑 Quillen 上同调。通过将这种一般代数与障碍理论机制相结合，我们获得了计算工具${\mathbb{乙}}_{\mathrm{无穷大}}$代数结束${\mathbb{F}}_p$以及鲁宾-泰特光谱。作为一个应用程序，我们证明了存在${\mathbb{乙}}_{\mathrm{无穷大}}$周期性复数高度方向$H⩽2$。