This entry is about étale morphisms between schemes. The term étale map is preferred in the context of topology and differential geometry, see étalé space for the topological version.
higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
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The notion of étale morphism of schemes is the realization of the general notion of étale morphism for maps between schemes, hence it captures roughly the idea of a map of schemes which is a local homeomorphism/local diffeomorphism.
A central use of étale morphisms of schemes is that they appear as coverings in the Grothendieck topology of the étale site. The abelian sheaf cohomology with respect to these étale covers of schemes is accordingly called étale cohomology.
A morphism of schemes is an étale morphism if the following equivalent conditions hold:
it is
it is
of relative dimension $0$.
it is
it is
(A number of other equivalent definitions are listed at wikipedia.)
For morphisms $f \colon X \longrightarrow Y$ between algebraic varieties over an algebraically closed field this means that for all points $p \in X$ the induced morphism on tangent cones
is an isomorphism. This is analogous to the corresponding characterization of local diffeomorphisms of smooth manifolds.
Relaxing the finiteness condition in item 4 of 1 yields the notion of weakly étale morphism.
étale morphism $\Rightarrow$ pro-étale morphism $\Rightarrow$ weakly étale morphism $\Rightarrow$ formally étale morphism
A jointly surjective collection of étale morphisms $\{U_i \to X\}$ is called an étale cover.
Most of the pairs of conditions in def. 1 can be read as constraining the fiber of the morphism to be first suitably surjective/bundle-like (smooth, flat) and second suitably locally injective (unramified).
Specifically the first condition has an infinitesimal anlog: a formally étale morphism is a formally smooth and formally unramified morphism. These notions also have an interpretation in synthetic differential geometry and there they correspond to the statement that a local diffeomorphism is a submersion which is also an immersion of smooth manifolds.
A morphism is formally étale morphism if it is
formally smooth (satisfying an infinitesimal lifting property)
and formally unramified.
These are sheaf-like properties, which can be formalized in the language of Q-categories (monopresheaf and epipresheaf properties on the $Q$-category of nilpotent thickenings).
See at differential cohesion and at infinitesimal shape modality.
(e.g. Milne, prop. 2.11)
Use that an étale morphism is a formally étale morphism with finite fibers, and that $f \colon X \to Y$ is formally étale precisely if the infinitesimal shape modality unit naturality square
is a pullback square. Then the three properties to be shown are equivalently the pasting law for pullback diagrams.
A smooth morphism of schemes is étale iff there is an étale cover of the base scheme by open subschemes such that the pullback of the projection to each of them is an open local isomorphism of locally ringed spaces (and in particular, the pullback of the projection morphism is an étale map of the corresponding underlying topological spaces).
This disjointness picture of étale covers make them suitable for having nontrivial cohomology in situations where Zariski covers give vanishing cohomology.
Let $k$ be a field. A morphism of schemes $Y \to Spec k$ is étale precisely if $Y$ is a coproduct $Y \simeq \coprod_i Spec k_i$ for each $k_i$ a finite and separable field extension of $k$.
This appears for instance as de Jong, prop. 3.1 i).
Such étale morphisms are classified by the classical Galois theory of field extensions.
A ring homomorphism of affine varieties $Spec(A) \to Spec(B)$ for $Spec(B)$ non-singular and for $A \simeq B[x_1, \cdots, x_n]/(f_1, \cdots, f_n)$ with polynomials $f_i$ is étale precisel if the Jacobian $det(\frac{\partial f_i}{\partial x_j})$ is invertible.
This appears for instance as (Milne, prop. 2.1).
A sheaf $F$ on a scheme $X$ corresponds to an étale morphism $Y \to X$ precisely if there is an étale cover $\{U_i \to X\}$ such that each restriction
is isomorphic to a constant sheaf on a set $K_i$.
A proof is in (Deligne).
A finite separable field extension $K \hookrightarrow L$ corresponds dually to an étale morphism $Spec L \to Spec K$. These are the morphisms classified by classical Galois theory.
Every open immersion of schemes is an étale morphism of schemes. In particular a standard open inclusion (a cover in the Zariski topology) induced by the localization of a commutative ring
is étale.
(e.g. Stacks Project, lemma 28.37.9)
By one of the equivalent characterizations of étale morphism it is sufficient to check that the map $Spec(R[S^{-1}]) \longrightarrow Spec(R)$ is a formally étale morphism and locally of finite presentation.
The latter is clear, since the very definition of
exhibits a finitely presented algebra over $R$.
To see that it is formally étale we need to check that for every commutative ring $T$ with nilpotent ideal $J$ we have a pullback diagram
Now by the universal property of the localization, a homomorphism $R[S^{-1}] \longrightarrow T$ is a homomorphism $R \longrightarrow T$ which sends all elements in $S \hookrightarrow R$ to invertible elements in $T$. But no element in a nilpotent ideal can be invertible, Therefore the fiber product of the bottom and right map is the set of maps from $R$ to $T$ such that $S$ is taken to invertibles, which is indeed the top left set.
étale morphism of schemes
The classical references are
Lecture notes include
The local structure theorems are discussed in
Discussion of etale morphisms between E-infinity rings/spectral schemes is in
and generally in E-∞ geometry in