Descent spectral sequences through synthetic spectra

TL;DR

This paper investigates the conditions under which the synthetic analogue functor preserves limits and descent spectral sequences, introducing a new synthetic spectrum and enhancing the theoretical framework of synthetic spectra in algebraic topology.

Contribution

It provides a necessary and sufficient criterion for limit preservation by the synthetic functor and demonstrates how to implement descent spectral sequences within synthetic spectra.

Findings

Criteria for the preservation of limits by the synthetic functor

Construction of a new MU-synthetic spectrum called Smf

Framework for descent spectral sequences in synthetic spectra

Abstract

The synthetic analogue functor $\nu$ from spectra to synthetic spectra does not preserve all limits. In this paper, we give a necessary and sufficient criterion for $\nu$ to preserve the global sections of a derived stack. Even when these conditions are not satisfied, our framework still yields synthetic spectra that implement the descent spectral sequence for the structure sheaf, thus placing descent spectral sequences on good footing in the $\infty$-category of synthetic spectra. As an example, we introduce a new $\mathrm{MU}$-synthetic spectrum $\mathrm{Smf}$.