nLab étale morphism of E-∞ rings

Definition

Definition

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

(Lurie, def.7.5.1.4)

Properties

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

Proposition

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

(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 $E_\infty$-rings)

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

References

• Jacob Lurie, Higher Algebra