Extension DGAs and topological Hochschild homology

TL;DR

This paper investigates a special class of differential graded algebras (DGAs) called extension DGAs, their topological Hochschild homology, and conditions under which various equivalences coincide, providing new insights into their structure and properties.

Contribution

It introduces the concept of extension DGAs arising from ring spectra, analyzes their THH splitting, and compares different notions of equivalence for these algebras.

Findings

Extension DGAs have a THH spectrum that splits conveniently.

Formal DGAs with nice homology are often extension DGAs.

In certain cases, topological equivalences and quasi-isomorphisms coincide for extension DGAs.

Abstract

In this work, we study those differential graded algebras (DGAs) that arise from ring spectra through the extension of scalars functor. Namely, we study DGAs whose corresponding Eilenberg-Mac Lane ring spectrum is equivalent to $H\mathbb{Z} \wedge E$ for some ring spectrum $E$. We call these DGAs extension DGAs. We also define and study this notion for $E_\infty$ DGAs. The topological Hochschild homology (THH) spectrum of an extension DGA splits in a convenient way. We show that formal DGAs with nice homology rings are extension and therefore their THH groups can be obtained from their Hochschild homology groups in many cases of interest. We also provide interesting examples of DGAs that are not extension. In the second part, we study properties of extension DGAs. We show that in various cases, topological equivalences and quasi-isomorphisms agree for extension DGAs. From this, we obtain that dg Morita equivalences and Morita equivalences also agree in these cases.