nLab étale morphism of E-∞ rings

Contents

Contents

Definition

Definition

A morphism f:ABf \colon \mathbf{A} \to \mathbf{B} of E-∞ rings is an étale morphism if the underlying homomorphism of commutative rings π 0f:π 0Aπ 0B\pi_0 f\colon \pi_0 \mathbf{A}\to \pi_0 \mathbf{B} is an étale morphism, and the map π n(A) π 0(A)π 0(B)π n(B)\pi_n(A) \otimes_{\pi_0(A)} \pi_0(B) \to \pi_n(B) is an isomorphism of abelian groups for every integer nn.

(Lurie, def.7.5.1.4)

Properties

étale morphisms of underlying rings lift essentially uniquely to étale morphisms of E-∞ rings:

Proposition

For A\mathbf{A} an E-∞ ring and π 0AB\pi_0 \mathbf{A} \to B a homomorphism to an ordinary ring BB, then there is an essentially unique E E_\infty-ring B\mathbf{B} with π 0BB\pi_0 \mathbf{B} \simeq B and étale morphism AB\mathbf{A}\to \mathbf{B}.

(Lurie, theorem 7.5.0.6)

Remark

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.

Remark

(localization of E E_\infty-rings)

Proposition serves to lift localization of rings from rings to E E_\infty-rings: for A\mathbf{A} an E-∞ ring and aπ 0Aa\in \pi_0 A an element, then the map π 0A(π 0A)[a 1]\pi_0 \mathbf{A} \to (\pi_0 \mathbf{A})[a^{-1}] of localization of a ring away from aa lifts to yield an E-∞ ring A[a 1]\mathbf{A}[a^{-1}] with étale morphism AA[a 1]\mathbf{A} \to \mathbf{A}[a^{-1}]. See also at localization of a module for more on this.

References

Last revised on May 7, 2021 at 09:48:45. See the history of this page for a list of all contributions to it.