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
,
we consider which classes of ring homomorphisms
can be realised by a
map
in the category
of global
–modules,
and what multiplicative structures can be placed on
. If
witnesses as a
projective
–module,
then such an
exists as a map between homotopy commutative global
–algebras. If
is in addition
étale or
is a
–algebra, then
can be upgraded
to a map of
–global
–algebras or a map of
––algebras,
respectively. Various global spectra and
–global
ring spectra are then obtained from classical homotopy-theoretic and algebraic
constructions, with a controllable global homotopy type.
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