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Realising $\pi_\ast^eR$–algebras by global ring spectra

Jack Morgan Davies

Algebraic & Geometric Topology 21 (2021) 1745–1790

We approach a problem of realising algebraic objects in a certain universal equivariant stable homotopy theory, the global homotopy theory of Schwede (2018). Specifically, for a global ring spectrum R, we consider which classes of ring homomorphisms η: πeR S can be realised by a map η: R S in the category of global R–modules, and what multiplicative structures can be placed on S. If η witnesses S as a projective πeR–module, then such an η exists as a map between homotopy commutative global R–algebras. If η is in addition étale or S0 is a –algebra, then η can be upgraded to a map of 𝔼–global R–algebras or a map of 𝔾R–algebras, respectively. Various global spectra and 𝔼–global ring spectra are then obtained from classical homotopy-theoretic and algebraic constructions, with a controllable global homotopy type.

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global homotopy theory, stable homotopy theory, higher algebra, equivariant homotopy theory, étale morphisms, realising algebra
Mathematical Subject Classification
Primary: 55P43, 55P91, 55P92, 55Q91
Received: 12 September 2019
Revised: 20 June 2020
Accepted: 27 July 2020
Published: 18 August 2021
Jack Morgan Davies
Mathematisch Instituut
Universiteit Utrecht