Cellularity of Chromatic Synthetic Spectra

TL;DR

The paper addresses when the $ abla$∞-category of $R$-synthetic spectra, $\text{Syn}_R$, is cellular and, in particular, proves cellularity for Morava $E$-theory, yielding a filtered-module description $ ext{Syn}_E \simeq \text{Mod}_{\mathrm{map}(\nu \mathbf{S}, \mathrm{Wh}(\tau^{-1}\nu \mathbf{S}))}(\mathrm{FilSp})$ with $\nu X \mapsto \mathrm{Tot}(\mathrm{Wh}(E^{\otimes[\bullet]} \otimes X))$ for $E$-nilpotent complete $X$. The main technique uses a general construction of filtered deformations from monoidal t-structures: defining $\mathrm{FilSp}$, employing the Whitehead filtration $\mathrm{Wh}$, and establishing a t-strictness criterion that yields a symmetric monoidal equivalence $\mathcal{C} \simeq \mathrm{Mod}_{\mathrm{map}(\mathbf{1},\mathrm{Wh}A)}(\mathrm{FilSp})$ under suitable hypotheses. This framework provides a filtered-model description of $\text{Syn}_E$ via a signature functor and extends to general Adams-type rings $R$ through filtered Adams resolutions, relating synthetic spectra to cosimplicial $R$-resolutions and their associated filtrations. Overall, the work couples cellularity of Morava-synthetic spectra with a robust deformation-theoretic approach to filtered spectral algebra, enabling computable filtrations and module-structure descriptions.

Abstract

We show that the $\infty$-category of synthetic spectra based on Morava E-theory is generated by the bigraded spheres and identify it with the $\infty$-category of modules over a filtered ring spectrum. The latter we show using a general method for constructing filtered deformations from t-structures on symmetric monoidal stable $\infty$-categories.

Theorems & Definitions (12)

• Theorem 1
• Definition 1.1
• Definition 1.2
• Proposition 1.3
• Theorem 1.4
• Lemma 1.5
• Theorem 2.1: Filtered Schwede--Shipley; Pstrągowski pstragowski_perfect_even_filtration, Proposition 3.16
• Definition 2.2
• Lemma 2.3
• Theorem 2.4