Report on $\mathbb{E}_\infty$-descendability

TL;DR

This paper introduces the concept of $\\mathbb{E}_\infty$-descendability in ring maps, proves its prevalence among certain classes, and applies it to establish a variant of Tannaka duality.

Contribution

It defines $\\mathbb{E}_\infty$-descendability, shows its occurrence in various ring maps, and connects it to Tannaka duality, advancing the understanding of descent in algebraic geometry.

Findings

Several classes of ring maps are $\\mathbb{E}_\infty$-descendable.

A new variant of Tannaka duality is proved.

The notion of $\\mathbb{E}_\infty$-descendability is introduced and characterized.

Abstract

We introduce the notion of $\mathbb{E}_\infty$-descendability as well as a derived variant. We prove that several classes of descendable maps of commutative rings are $\mathbb{E}_\infty$-descendable. As an application, we prove a variant of Tannaka duality.