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étale morphism of E-∞ rings

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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 or rings.

(Lurie, def.8.5.0.4)

Properties

étale morphisms of underlying rings lift essentially uniquely to étale morphosms 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 8.5.0.6)

Remark

Proposition 1 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 1 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

Revised on August 18, 2014 21:05:11 by Urs Schreiber (89.204.153.66)