Classification of module spectra and Franke's algebraicity conjecture

Abstract: This is all joint work with Piotr Pstrągowski. Given an $E_1$-ring $R$ such that the graded homotopy ring $\pi_* R$ is $q$-sparse and the global projective dimension $d$ of $\pi_* R$ is less than $q$, we show that the homotopy $(q – d)$-category of Mod($R$) is equivalent to the homotopy $(q – d)$-category of differential graded modules over $\pi_*R$. Thus for such $E_1$-rings the homotopy theory of their modules is algebraic up to the level $(q – d)$. Examples include appropriate Morava $K$-theories, Johnson–Wilson theories, truncated Brown–Peterson theories and some variations of topological $K$-theory spectra. We also show that the result is optimal in the sense that $(q – d)$ is the best possible level in general where algebraicity happens. At the end of the talk we will outline how the results for modules can be generalised to the settings where we do not have compact projective generators. This proves Franke's algebraicity conjecture, which provides a general result when certain nice homology theories provide algebraic models for homotopy theories.

Mathematics

Audience: researchers in the topic

Opening Workshop (IRP Higher Homotopy Structures 2021, CRM-Bellaterra)