Etale morphisms #
An R
-algebra A
is formally étale if for every R
-algebra,
every square-zero ideal I : Ideal B
and f : A →ₐ[R] B ⧸ I
, there exists
exactly one lift A →ₐ[R] B
.
It is étale if it is formally étale and of finite presentation.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under R
-algebra homomorphisms and compositions.
We show that étale is stable under algebra isomorphisms, composition and localization at an element.
TODO: #
- Show that étale is stable under base change.
An R
algebra A
is formally étale if for every R
-algebra, every square-zero ideal
I : Ideal B
and f : A →ₐ[R] B ⧸ I
, there exists exactly one lift A →ₐ[R] B
.
- comp_bijective : ∀ ⦃B : Type u⦄ [inst : CommRing B] [inst_1 : Algebra R B] (I : Ideal B), I ^ 2 = ⊥ → Function.Bijective (AlgHom.comp (Ideal.Quotient.mkₐ R I))
Instances
Equations
- ⋯ = ⋯
Equations
- ⋯ = ⋯
Equations
- ⋯ = ⋯
An R
-algebra A
is étale if it is formally étale and of finite presentation.
- formallyEtale : Algebra.FormallyEtale R A
- finitePresentation : Algebra.FinitePresentation R A
Instances
Being étale is transported via algebra isomorphisms.
Étale is stable under composition.
Localization at an element is étale.