## Abstract

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 ηe W πe eR ! Se can be realised by a map ηW R ! S in the category of global R– modules, and what multiplicative structures can be placed on S. If ηe witnesses Se as a projective πe eR–module, then such an η exists as a map between homotopy commutative global R–algebras. If ηe is in addition étale or S0 is a Q–algebra, then η can be upgraded to a map of E1–global R–algebras or a map of G1–R–algebras, respectively. Various global spectra and E1–global ring spectra are then obtained from classical homotopy-theoretic and algebraic constructions, with a controllable global homotopy type.

Original language | English |
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Pages (from-to) | 1745–1790 |

Number of pages | 46 |

Journal | Algebraic and Geometric Topology |

Volume | 21 |

Issue number | 4 |

DOIs | |

Publication status | Published - 18 Aug 2021 |

## Keywords

- equivariant homotopy theory
- étale morphisms
- global homotopy theory
- higher algebra
- realising algebra
- stable homotopy theory

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