A morphism of E-∞ rings is an étale morphism if the underlying homomorphism of commutative rings is an étale morphism, and the map is an isomorphism of abelian groups for every integer .
étale morphisms of underlying rings lift essentially uniquely to étale morphisms of E-∞ rings:
For an E-∞ ring and a homomorphism to an ordinary ring , then there is an essentially unique -ring with and étale morphism .
Proposition is a central ingredient in the characterization of the moduli stack of derived elliptic curves as having underlying it the ordinaty moduli stack of elliptic curves.
(localization of -rings)
Proposition serves to lift localization of rings from rings to -rings: for an E-∞ ring and an element, then the map of localization of a ring away from lifts to yield an E-∞ ring with étale morphism . See also at localization of a module for more on this.
Last revised on May 7, 2021 at 09:48:45. See the history of this page for a list of all contributions to it.