Ramification and descent in homotopy theory and derived algebraic geometry

TL;DR

This paper generalizes ramification concepts in homotopy theory and derived algebraic geometry, providing new tools for calculating topological Hochschild homology and enabling ramified descent analysis.

Contribution

It introduces a unified notion of ramification applicable to various algebraic structures and links these to THH calculations and descent techniques.

Findings

Defined unramified and totally ramified maps in broad contexts.

Established equivalence with classical definitions for number field rings.

Derived a formula for THH of rings of integers using ramification concepts.

Abstract

We introduce notions of unramified and totally ramified maps in great generality - for commutative rings, schemes, ring spectra, or derived schemes. We prove that the definition is equivalent to the classical definition in the case of rings of integers in number fields. The new definition leads directly (without computational techniques) to a calculation of topological Hochschild homology for rings of integers. We show that THH(R) is the homotopy cofiber of a map $R[\Omega S^3]\otimes_R\Omega^1_{R/\mathbb{Z}}\to R[\Omega S^3\langle 3\rangle]$, so there is a long exact sequence $\cdots\to H_\ast(\Omega S^3;\Omega^1_{R/\mathbb{Z}})\to H_\ast(\Omega S^3\langle 3\rangle;R)\to THH_\ast(R)\to\cdots$. Any time an extension Y/X is a composite of unramified and totally ramified extensions, our results allow for the study of THH(X) in terms of THH(Y) by a kind of weak etale descent (ramified descent).