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Extension DGAs and topological Hochschild homology

Haldun Özgür Bayındır

Algebraic & Geometric Topology 23 (2023) 895–932

We study 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 E for some ring spectrum E. We call these DGAs extension DGAs. We also define and study this notion for E 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.

differential graded algebras, ring spectra, topological Hochschild homology
Mathematical Subject Classification
Primary: 18G35, 55P43, 55U99
Received: 19 May 2021
Revised: 23 September 2021
Accepted: 3 November 2021
Published: 9 May 2023
Haldun Özgür Bayındır
Department of Mathematics
City, University of London
United Kingdom

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